2.1 (c+d x)^m (a+b (F^(g (e+f x)))^n)^p.nb

(* Content-type: application/mathematica *)

(*** Wolfram Notebook File ***)
(* http://www.wolfram.com/nb *)

(* CreatedBy='Mathematica 7.0' *)

(*CacheID: 234*)
(* Internal cache information:
NotebookFileLineBreakTest
NotebookFileLineBreakTest
NotebookDataPosition[       145,          7]
NotebookDataLength[    305979,       6201]
NotebookOptionsPosition[    297974,       5992]
NotebookOutlinePosition[    299732,       6046]
CellTagsIndexPosition[    299689,       6043]
WindowFrame->Normal*)

(* Beginning of Notebook Content *)
Notebook[{
Cell[BoxData[
 StyleBox[
  RowBox[{Cell[TextData[StyleBox["Rules for integrands of the form",
    FontFamily->"Arial",
    FontSize->16]], "None"], 
   SuperscriptBox[
    RowBox[{"(", 
     RowBox[{"c", "+", 
      RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
   SuperscriptBox[
    RowBox[{"(", 
     RowBox[{"a", "+", 
      RowBox[{"b", " ", 
       SuperscriptBox[
        RowBox[{"(", 
         SuperscriptBox["F", 
          RowBox[{"g", 
           RowBox[{"(", 
            RowBox[{"e", "+", 
             RowBox[{"f", " ", "x"}]}], ")"}]}]], ")"}], "n"]}]}], ")"}], 
    "p"]}],
  FontSize->16]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193415004828687`*^9, 3.5193415113404875`*^9}, {
   3.5193513965602303`*^9, 3.519351397420232*^9}, {3.5194037839416766`*^9, 
   3.5194037847117205`*^9}, {3.5233155055329347`*^9, 3.523315506402936*^9}, {
   3.523318726949605*^9, 3.5233187276796064`*^9}, {3.523920400879534*^9, 
   3.5239204094439487`*^9}, 3.52392096260852*^9, 3.523921038171853*^9, 
   3.5340477459008017`*^9, 3.5340515540554085`*^9, {3.536542511100503*^9, 
   3.53654251626051*^9}, {3.536775961244521*^9, 3.5367759616145215`*^9}, {
   3.544373142375139*^9, 3.5443731599851637`*^9}, {3.5721095751430073`*^9, 
   3.5721095751430073`*^9}, {3.57578648209792*^9, 3.575786503859958*^9}, 
   3.575786540270422*^9, {3.5757866379421935`*^9, 3.5757866609522343`*^9}, 
   3.5758478960540276`*^9, 3.6735458028230343`*^9, {3.6735666862897754`*^9, 
   3.673566701369638*^9}, {3.6738103578923755`*^9, 3.673810361661591*^9}, 
   3.674579202779709*^9, 3.6745803236138167`*^9, 3.6745819151528473`*^9, 
   3.674582629346697*^9},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{Cell[TextData[StyleBox["1.",
   FontFamily->"Arial"]], "None"], "  ", 
  RowBox[{"\[Integral]", 
   RowBox[{
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"c", "+", 
       RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"b", " ", 
       SuperscriptBox["F", 
        RowBox[{"g", " ", 
         RowBox[{"(", 
          RowBox[{"e", "+", 
           RowBox[{"f", " ", "x"}]}], ")"}]}]]}], ")"}], "n"], 
    RowBox[{"\[DifferentialD]", "x"}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193415004828687`*^9, 3.5193415113404875`*^9}, {
   3.5193513965602303`*^9, 3.519351397420232*^9}, {3.5194037839416766`*^9, 
   3.5194037847117205`*^9}, 3.52105258275843*^9, 3.5210531741726685`*^9, {
   3.521057461672199*^9, 3.5210574760242243`*^9}, {3.521059881564049*^9, 
   3.521059886088057*^9}, {3.5213902158895197`*^9, 3.521390217543123*^9}, {
   3.521390827878595*^9, 3.521390829376197*^9}, {3.5214688792561245`*^9, 
   3.521468879443325*^9}, {3.523320201951167*^9, 3.523320202247567*^9}, {
   3.5288596764894824`*^9, 3.5288596768094826`*^9}, {3.529196266260353*^9, 
   3.5291962665099535`*^9}, {3.5291997050223927`*^9, 
   3.5291997052563934`*^9}, {3.529201346457276*^9, 3.529201347252877*^9}, {
   3.529285041073242*^9, 3.5292850412632427`*^9}, {3.529285215443486*^9, 
   3.529285215443486*^9}, {3.5295115356999426`*^9, 3.5295115433443794`*^9}, {
   3.529521412127842*^9, 3.529521414647986*^9}, {3.529529429975619*^9, 
   3.529529432955224*^9}, 3.5340074894011116`*^9, 3.5415537711400824`*^9, {
   3.5415538454742126`*^9, 3.54155384951462*^9}, {3.5417297197063265`*^9, 
   3.5417297252443366`*^9}, {3.546297494083555*^9, 3.5462974982435613`*^9}, {
   3.5462989134855423`*^9, 3.5462989153655453`*^9}, 3.548544146508506*^9, 
   3.5488738855784535`*^9, 3.548877085095433*^9, {3.548885157919084*^9, 
   3.5488851581390843`*^9}, {3.5501050720442505`*^9, 3.550105091724278*^9}, 
   3.552842303236889*^9, {3.5528426826774197`*^9, 3.552842683117421*^9}, 
   3.552842964957815*^9, 3.5528435586686463`*^9, {3.5528438442590466`*^9, 
   3.5528438442590466`*^9}, {3.5528470750635695`*^9, 
   3.5528470992336035`*^9}, {3.552925992292306*^9, 3.5529260013247223`*^9}, 
   3.553222405588056*^9, 3.5795486748848715`*^9, {3.579548905405194*^9, 
   3.579548907775197*^9}, 3.5795492772757144`*^9, 3.579549350275817*^9, 
   3.5795585715262985`*^9, 3.5795588167566414`*^9, {3.6738105630461097`*^9, 
   3.67381058140716*^9}, {3.67458301822494*^9, 3.674583037320032*^9}, {
   3.6745840615256133`*^9, 3.6745840707781425`*^9}, 3.6745849272331285`*^9, {
   3.67458651161775*^9, 3.674586511830763*^9}},
 FontSize->12,
 FontWeight->"Bold"],

Cell[TextData[{
 "If the control variable ",
 Cell[BoxData["$UseGamma"], "Subsubtitle",
  CellDingbat->None,
  CellChangeTimes->{{3.4796579723816*^9, 3.4796580261689425`*^9}, 
    3.479661191290163*^9, {3.479686720164157*^9, 3.479686720204214*^9}, {
    3.479686831554328*^9, 3.479686834819022*^9}, {3.4940970712485504`*^9, 
    3.4940971130286083`*^9}, 3.4940971455886545`*^9, 3.4953221152495327`*^9, 
    3.495322146199576*^9, {3.495322261579737*^9, 3.4953222704097495`*^9}, 
    3.4953325041434045`*^9, 3.4953325401034546`*^9, {3.4953327138936977`*^9, 
    3.4953327232437115`*^9}, {3.4953341614457245`*^9, 3.4953341867757597`*^9},
     3.5501063615760555`*^9, {3.5501064648262005`*^9, 3.550106490466236*^9}, {
    3.550106525066285*^9, 3.550106540276306*^9}, {3.5501065790863605`*^9, 
    3.5501066156964116`*^9}, {3.5528423117869005`*^9, 3.552842314026904*^9}, {
    3.5528426917374325`*^9, 3.5528426937974358`*^9}, {3.5528429682378197`*^9, 
    3.5528429694078217`*^9}, {3.5528433415383425`*^9, 
    3.5528433442283463`*^9}, {3.5528435601586485`*^9, 
    3.5528435612786503`*^9}, {3.552844695100238*^9, 3.552844701190246*^9}, {
    3.5528471263836412`*^9, 3.552847134093652*^9}, {3.5795487900050325`*^9, 
    3.579548792195035*^9}, {3.5795492795057173`*^9, 3.579549280715719*^9}, {
    3.5795493517058187`*^9, 3.5795493524758196`*^9}, {3.579549671386266*^9, 
    3.579549675026271*^9}, 3.5795497284863462`*^9, {3.5795585728663*^9, 
    3.5795585738863015`*^9}, {3.579558822086649*^9, 3.579558823556651*^9}, {
    3.579558905576766*^9, 3.5795589085567703`*^9}, {3.579558982036873*^9, 
    3.5795590045369043`*^9}, {3.579559110317053*^9, 3.579559113487057*^9}},
  TextAlignment->Center,
  FontSize->12,
  FontWeight->"Bold"],
 " is ",
 Cell[BoxData["True"], "Subsubtitle",
  CellDingbat->None,
  CellChangeTimes->{{3.4796579723816*^9, 3.4796580261689425`*^9}, 
    3.479661191290163*^9, {3.479686720164157*^9, 3.479686720204214*^9}, {
    3.479686831554328*^9, 3.479686834819022*^9}, {3.4940970712485504`*^9, 
    3.4940971130286083`*^9}, 3.4940971455886545`*^9, 3.4953221152495327`*^9, 
    3.495322146199576*^9, {3.495322261579737*^9, 3.4953222704097495`*^9}, 
    3.4953325041434045`*^9, 3.4953325401034546`*^9, {3.4953327138936977`*^9, 
    3.4953327232437115`*^9}, {3.4953341614457245`*^9, 3.4953341867757597`*^9},
     3.5501063615760555`*^9, {3.5501064648262005`*^9, 3.550106490466236*^9}, {
    3.550106525066285*^9, 3.550106540276306*^9}, {3.5501065790863605`*^9, 
    3.5501066156964116`*^9}, {3.5528423117869005`*^9, 3.552842314026904*^9}, {
    3.5528426917374325`*^9, 3.5528426937974358`*^9}, {3.5528429682378197`*^9, 
    3.5528429694078217`*^9}, {3.5528433415383425`*^9, 
    3.5528433442283463`*^9}, {3.5528435601586485`*^9, 
    3.5528435612786503`*^9}, {3.552844695100238*^9, 3.552844701190246*^9}, {
    3.5528471263836412`*^9, 3.552847134093652*^9}, {3.5795487900050325`*^9, 
    3.579548792195035*^9}, {3.5795492795057173`*^9, 3.579549280715719*^9}, {
    3.5795493517058187`*^9, 3.5795493524758196`*^9}, {3.579549671386266*^9, 
    3.579549675026271*^9}, 3.5795497284863462`*^9, {3.5795585728663*^9, 
    3.5795585738863015`*^9}, {3.579558822086649*^9, 3.579558823556651*^9}, {
    3.579558905576766*^9, 3.5795589085567703`*^9}, {3.579558982036873*^9, 
    3.5795590045369043`*^9}, {3.579559110317053*^9, 3.579559113487057*^9}},
  TextAlignment->Center,
  FontSize->12,
  FontWeight->"Bold"],
 ", antiderivatives of expressions of the form ",
 Cell[BoxData[
  RowBox[{
   SuperscriptBox[
    RowBox[{"(", 
     RowBox[{"d", "+", 
      RowBox[{"e", " ", "x"}]}], ")"}], "m"], 
   SuperscriptBox[
    RowBox[{"(", 
     SuperscriptBox["F", 
      RowBox[{"c", 
       RowBox[{"(", 
        RowBox[{"a", "+", 
         RowBox[{"b", " ", "x"}]}], ")"}]}]], ")"}], "n"]}]], "Subsubtitle",
  CellDingbat->None,
  CellChangeTimes->{{3.4796579723816*^9, 3.4796580261689425`*^9}, 
    3.479661191290163*^9, {3.479686720164157*^9, 3.479686720204214*^9}, {
    3.479686831554328*^9, 3.479686834819022*^9}, {3.4940970712485504`*^9, 
    3.4940971130286083`*^9}, 3.4940971455886545`*^9, 3.4953221152495327`*^9, 
    3.495322146199576*^9, {3.495322261579737*^9, 3.4953222704097495`*^9}, 
    3.4953325041434045`*^9, 3.4953325401034546`*^9, {3.4953327138936977`*^9, 
    3.4953327232437115`*^9}, {3.4953341614457245`*^9, 3.4953341867757597`*^9},
     3.5501063615760555`*^9, {3.5501064648262005`*^9, 3.550106490466236*^9}, {
    3.550106525066285*^9, 3.550106540276306*^9}, {3.5501065790863605`*^9, 
    3.5501066156964116`*^9}, {3.5528423117869005`*^9, 3.552842314026904*^9}, {
    3.5528426917374325`*^9, 3.5528426937974358`*^9}, {3.5528429682378197`*^9, 
    3.5528429694078217`*^9}, {3.5528433415383425`*^9, 
    3.5528433442283463`*^9}, {3.5528435601586485`*^9, 
    3.5528435612786503`*^9}, {3.552844695100238*^9, 3.552844701190246*^9}, {
    3.5528471263836412`*^9, 3.552847134093652*^9}, {3.5795487900050325`*^9, 
    3.579548792195035*^9}, {3.5795492795057173`*^9, 3.579549280715719*^9}, {
    3.5795493517058187`*^9, 3.5795493524758196`*^9}, {3.579549671386266*^9, 
    3.579549675026271*^9}, 3.5795497284863462`*^9, {3.5795585728663*^9, 
    3.5795585738863015`*^9}, {3.579558822086649*^9, 3.579558823556651*^9}, {
    3.579558905576766*^9, 3.5795589085567703`*^9}, {3.579558982036873*^9, 
    3.5795590045369043`*^9}, {3.579559110317053*^9, 3.579559113487057*^9}},
  TextAlignment->Center,
  FontSize->12,
  FontWeight->"Bold"],
 " will be much more compactly expressed in terms of the ",
 Cell[BoxData["Gamma"], "Subsubtitle",
  CellDingbat->None,
  CellChangeTimes->{{3.4796579723816*^9, 3.4796580261689425`*^9}, 
    3.479661191290163*^9, {3.479686720164157*^9, 3.479686720204214*^9}, {
    3.479686831554328*^9, 3.479686834819022*^9}, {3.4940970712485504`*^9, 
    3.4940971130286083`*^9}, 3.4940971455886545`*^9, 3.4953221152495327`*^9, 
    3.495322146199576*^9, {3.495322261579737*^9, 3.4953222704097495`*^9}, 
    3.4953325041434045`*^9, 3.4953325401034546`*^9, {3.4953327138936977`*^9, 
    3.4953327232437115`*^9}, {3.4953341614457245`*^9, 3.4953341867757597`*^9},
     3.5501063615760555`*^9, {3.5501064648262005`*^9, 3.550106490466236*^9}, {
    3.550106525066285*^9, 3.550106540276306*^9}, {3.5501065790863605`*^9, 
    3.5501066156964116`*^9}, {3.5528423117869005`*^9, 3.552842314026904*^9}, {
    3.5528426917374325`*^9, 3.5528426937974358`*^9}, {3.5528429682378197`*^9, 
    3.5528429694078217`*^9}, {3.5528433415383425`*^9, 
    3.5528433442283463`*^9}, {3.5528435601586485`*^9, 
    3.5528435612786503`*^9}, {3.552844695100238*^9, 3.552844701190246*^9}, {
    3.5528471263836412`*^9, 3.552847134093652*^9}, {3.5795487900050325`*^9, 
    3.579548792195035*^9}, {3.5795492795057173`*^9, 3.579549280715719*^9}, {
    3.5795493517058187`*^9, 3.5795493524758196`*^9}, {3.579549671386266*^9, 
    3.579549675026271*^9}, 3.5795497284863462`*^9, {3.5795585728663*^9, 
    3.5795585738863015`*^9}, {3.579558822086649*^9, 3.579558823556651*^9}, {
    3.579558905576766*^9, 3.5795589085567703`*^9}, {3.579558982036873*^9, 
    3.5795590045369043`*^9}, {3.579559110317053*^9, 3.579559113487057*^9}},
  TextAlignment->Center,
  FontSize->12,
  FontWeight->"Bold"],
 " function instead of elementary functions."
}], "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.495322033659418*^9, 3.495322068589467*^9}, {
   3.495332560963484*^9, 3.4953325798335104`*^9}, {3.4953342240158124`*^9, 
   3.4953342584258604`*^9}, {3.5488732775776024`*^9, 
   3.5488732775776024`*^9}, {3.548873326337671*^9, 3.548873326337671*^9}, {
   3.5488733961777687`*^9, 3.548873400247774*^9}, {3.548882497145359*^9, 
   3.5488825231053953`*^9}, {3.548882954315999*^9, 3.548882954315999*^9}, {
   3.548883883077299*^9, 3.5488839072773333`*^9}, {3.5488840277975016`*^9, 
   3.5488840277975016`*^9}, {3.550092011002294*^9, 3.550092011712295*^9}, {
   3.5500922678526535`*^9, 3.5500922678526535`*^9}, {3.550093724424693*^9, 
   3.550093724424693*^9}, {3.5500971557107277`*^9, 3.5500971557107277`*^9}, {
   3.5501063662860622`*^9, 3.550106398116107*^9}, {3.550106454446186*^9, 
   3.550106454446186*^9}, 3.5501069180668344`*^9, {3.5501072155072513`*^9, 
   3.5501072155072513`*^9}, {3.5501082711687293`*^9, 
   3.5501082711687293`*^9}, {3.5528426907374315`*^9, 
   3.5528426907374315`*^9}, {3.552843846249049*^9, 3.552843846249049*^9}, {
   3.552926049123206*^9, 3.5529260500280075`*^9}, {3.614435460468696*^9, 
   3.6144354830209866`*^9}, {3.6144355645106473`*^9, 
   3.6144356670275106`*^9}, {3.6144384893899403`*^9, 3.614438642770714*^9}, {
   3.614438809104227*^9, 3.614438852651718*^9}, {3.614439207616021*^9, 
   3.6144392175165873`*^9}, {3.673810385263941*^9, 3.673810385263941*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"$UseGamma", "=", "False"}], ";"}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, 
   3.4953222454097147`*^9, 3.4953324633733473`*^9, 3.4953326632436275`*^9, 
   3.4953341350856876`*^9, {3.548816716117523*^9, 3.54881672825754*^9}, {
   3.5488168441577024`*^9, 3.548816845257704*^9}, {3.5488174267385178`*^9, 
   3.548817476678588*^9}, {3.5488175358286705`*^9, 3.548817537178673*^9}, 
   3.548818879032552*^9, 3.5488189948567142`*^9, {3.548873179327465*^9, 
   3.5488731994574933`*^9}, {3.5488733093376465`*^9, 3.5488733155476556`*^9}, 
   3.5488825439754243`*^9, {3.5488826089855156`*^9, 3.548882630935546*^9}, 
   3.5488838853873024`*^9, 3.5488840304775057`*^9, {3.550091804762005*^9, 
   3.5500918642520885`*^9}, {3.5500920434623394`*^9, 
   3.5500921913025465`*^9}, {3.5500935055643864`*^9, 
   3.5500935449644413`*^9}, {3.5500935868345003`*^9, 3.550093588744503*^9}, {
   3.550093644764581*^9, 3.550093654344595*^9}, 3.550097159490733*^9, 
   3.550106818926696*^9, {3.5501068633667583`*^9, 3.5501068987568083`*^9}, {
   3.5501071702771883`*^9, 3.5501071712671895`*^9}, {3.550107218837256*^9, 
   3.5501072235372624`*^9}, 3.5501082039586353`*^9, 3.550109236880081*^9, {
   3.552842314946905*^9, 3.5528423179569097`*^9}, 3.552842477407133*^9, 
   3.5528425238471975`*^9, {3.5528426947374372`*^9, 3.552842703087449*^9}, {
   3.5528430422779236`*^9, 3.552843043307925*^9}, 3.552843088757989*^9, 
   3.5528431567680836`*^9, {3.552843347728351*^9, 3.5528433497783537`*^9}, {
   3.552843562598652*^9, 3.5528435632086525`*^9}, 3.5528436835188212`*^9, 
   3.552844216459567*^9, 3.5528442843696623`*^9, {3.552844706140253*^9, 
   3.552844714870265*^9}, 3.552845986792046*^9, {3.5528471936237354`*^9, 
   3.5528472087337565`*^9}, {3.552926054099615*^9, 3.552926054910816*^9}, {
   3.5529275303886056`*^9, 3.5529275747238836`*^9}, 3.5795494000658865`*^9, 
   3.579549439275941*^9, 3.579549476805994*^9, {3.5795497467963715`*^9, 
   3.5795497486763744`*^9}, {3.5795497909664335`*^9, 3.579549801246448*^9}, {
   3.579553111741083*^9, 3.579553118591092*^9}, {3.579553619284293*^9, 
   3.5795536265743036`*^9}, {3.5795586243963723`*^9, 
   3.5795586248963733`*^9}, {3.579558671406438*^9, 3.5795587013164797`*^9}, 
   3.579558739216533*^9, {3.5795589205967875`*^9, 3.579558927636797*^9}, 
   3.579558988336882*^9, {3.5795590471569643`*^9, 3.579559048416966*^9}, 
   3.6144353068819113`*^9, {3.614435550434842*^9, 3.6144355564221845`*^9}, {
   3.6144384265533466`*^9, 3.614438431660639*^9}, {3.614438786225919*^9, 
   3.6144387908331823`*^9}, {3.6144391978404617`*^9, 3.614439198755514*^9}},
 Background->GrayLevel[0.85]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t", 
  RowBox[{Cell[TextData[StyleBox["1:",
    FontFamily->"Arial",
    FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"b", " ", 
        SuperscriptBox["F", 
         RowBox[{"g", " ", 
          RowBox[{"(", 
           RowBox[{"e", "+", 
            RowBox[{"f", " ", "x"}]}], ")"}]}]]}], ")"}], "n"], 
     RowBox[{"\[DifferentialD]", "x"}], "  ", 
     StyleBox["when",
      FontFamily->"Arial",
      FontWeight->"Plain"], 
     StyleBox["  ",
      FontFamily->"Arial",
      FontWeight->"Plain"], Cell[TextData[Cell[BoxData[
      RowBox[{
       RowBox[{"m", ">", "0"}], " ", "\[And]", " ", 
       RowBox[{
        RowBox[{"2", "m"}], "\[Element]", "\[DoubleStruckCapitalZ]"}]}]]]], 
      "None"]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193422443984776`*^9, 3.5193422447260776`*^9}, 
   3.5193424488212357`*^9, {3.5193434708254333`*^9, 3.519343473103037*^9}, {
   3.5193435097631016`*^9, 3.51934353169674*^9}, {3.519345739319023*^9, 
   3.519345739599823*^9}, 3.5193462601259375`*^9, {3.5193475061060295`*^9, 
   3.5193475233908596`*^9}, {3.519351391880224*^9, 3.5193513931402254`*^9}, {
   3.519403773021052*^9, 3.5194037798564425`*^9}, 3.5194106793250647`*^9, {
   3.519537578341393*^9, 3.5195375792305946`*^9}, {3.5195376277778797`*^9, 
   3.51953767927357*^9}, {3.5197586260636144`*^9, 3.5197586260636144`*^9}, {
   3.51978053295651*^9, 3.519780542862528*^9}, {3.5197806526087203`*^9, 
   3.5197806657595434`*^9}, {3.5197807490168896`*^9, 3.5197807642581167`*^9}, 
   3.5197809004619555`*^9, {3.519782653514635*^9, 3.5197826623286505`*^9}, {
   3.5200207810164795`*^9, 3.520020785046485*^9}, {3.5213126833374643`*^9, 
   3.5213126833374643`*^9}, {3.521328235411002*^9, 3.521328235411002*^9}, {
   3.523315567893022*^9, 3.523315567893022*^9}, {3.5233157569332867`*^9, 
   3.5233157613032923`*^9}, {3.5323023198133526`*^9, 3.532302355723403*^9}, {
   3.5323026417638035`*^9, 3.5323026417638035`*^9}, {3.5323026980138817`*^9, 
   3.5323026980138817`*^9}, {3.532636287104328*^9, 3.532636287104328*^9}, {
   3.5330803686035347`*^9, 3.533080368863535*^9}, {3.533248909289961*^9, 
   3.5332489110299635`*^9}, {3.534962469565694*^9, 3.5349624910757236`*^9}, {
   3.5349625224057674`*^9, 3.534962522655768*^9}, {3.534965265519608*^9, 
   3.534965265519608*^9}, {3.5349668288168063`*^9, 3.5349668292668066`*^9}, {
   3.5349747358481646`*^9, 3.5349747358481646`*^9}, {3.534975457949176*^9, 
   3.534975457949176*^9}, {3.5368633963532934`*^9, 3.536863396571694*^9}, {
   3.541644322511304*^9, 3.5416443577513533`*^9}, {3.5416458269234104`*^9, 
   3.541645839633428*^9}, {3.5416460312236967`*^9, 3.541646031483697*^9}, 
   3.541646227783972*^9, {3.541647080335165*^9, 3.5416471077052035`*^9}, {
   3.541648997607849*^9, 3.5416489978278494`*^9}, 3.541649088407976*^9, {
   3.5417848711596775`*^9, 3.541784902429721*^9}, {3.541786145331461*^9, 
   3.5417861455914617`*^9}, {3.5417870037166634`*^9, 3.5417870039366636`*^9}, 
   3.5462979620242105`*^9, {3.546298264094633*^9, 3.546298265344635*^9}, {
   3.546299000235664*^9, 3.546299001505666*^9}, {3.546315119909564*^9, 
   3.5463151267659564`*^9}, {3.547227356319705*^9, 3.547227356319705*^9}, 
   3.547918559331044*^9, {3.547918610170952*^9, 3.5479186304411116`*^9}, 
   3.548537755517557*^9, 3.5485441629685287`*^9, 3.5485517520154133`*^9, {
   3.5488740727887154`*^9, 3.5488740727887154`*^9}, {3.548874206618903*^9, 
   3.548874206808903*^9}, {3.5488771226154857`*^9, 3.5488771332055006`*^9}, {
   3.548878122226885*^9, 3.548878122366885*^9}, 3.548878188846978*^9, {
   3.548882350145153*^9, 3.548882373605186*^9}, 3.548882435765273*^9, {
   3.5488824861553435`*^9, 3.5488824861553435`*^9}, {3.5488825297954044`*^9, 
   3.5488825297954044`*^9}, {3.54888256919546*^9, 3.5488825868054843`*^9}, {
   3.548883870637282*^9, 3.548883870637282*^9}, {3.5488839030373273`*^9, 
   3.5488839030373273`*^9}, {3.5488839818074374`*^9, 3.548884015947485*^9}, {
   3.5500922442826204`*^9, 3.5500922592326417`*^9}, {3.5500925229930105`*^9, 
   3.550092523183011*^9}, {3.550092560253063*^9, 3.5500925604230633`*^9}, {
   3.550093689654644*^9, 3.5500937159446807`*^9}, {3.550097151700722*^9, 
   3.550097151700722*^9}, {3.5501061958958235`*^9, 3.5501061960958242`*^9}, {
   3.5501062579559107`*^9, 3.5501062691259265`*^9}, {3.5501064214861393`*^9, 
   3.5501064483561773`*^9}, {3.5501072046872363`*^9, 
   3.5501072092172422`*^9}, {3.5501082663287225`*^9, 3.5501082663287225`*^9}, 
   3.5501084178989344`*^9, 3.5528423044768906`*^9, {3.5528426858774247`*^9, 
   3.552842686577426*^9}, 3.5528429659778166`*^9, 3.552843559598648*^9, {
   3.552843845529048*^9, 3.552843845529048*^9}, 3.552847117793629*^9, {
   3.5529260063791313`*^9, 3.5529260293891716`*^9}, 3.5532224074321613`*^9, 
   3.5795487431649666`*^9, 3.579548951725259*^9, 3.5795492786357164`*^9, 
   3.579549351275818*^9, 3.5795585724362993`*^9, 3.5795588185066442`*^9, {
   3.614435512487672*^9, 3.614435523088278*^9}, {3.6144620601640797`*^9, 
   3.614462083817433*^9}, 3.614463316997967*^9, {3.6144633904911704`*^9, 
   3.6144633981086063`*^9}, {3.6738105688774433`*^9, 
   3.6738106146200595`*^9}, {3.6738145986429324`*^9, 
   3.6738145997229943`*^9}, {3.6738152103199186`*^9, 3.6738152170003004`*^9}, 
   3.6745830200420437`*^9, 3.674583052075876*^9, 3.674584118078848*^9, 
   3.674584928017174*^9},
 FontSize->12,
 FontWeight->"Bold"],

Cell["Derivation: Integration by parts", "Subsubsection",
 CellChangeTimes->{{3.495321995009364*^9, 3.4953219984993687`*^9}, {
  3.4953340975556355`*^9, 3.495334101755641*^9}}],

Cell[TextData[{
 "Basis: ",
 Cell[BoxData[
  RowBox[{
   SuperscriptBox[
    RowBox[{"(", 
     RowBox[{"b", " ", 
      SuperscriptBox["F", 
       RowBox[{"g", " ", 
        RowBox[{"(", 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], ")"}]}]]}], ")"}], "n"], "\[Equal]", 
   RowBox[{
    SubscriptBox["\[PartialD]", "x"], 
    FractionBox[
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"b", " ", 
        SuperscriptBox["F", 
         RowBox[{"g", " ", 
          RowBox[{"(", 
           RowBox[{"e", "+", 
            RowBox[{"f", " ", "x"}]}], ")"}]}]]}], ")"}], "n"], 
     RowBox[{"f", " ", "g", " ", "n", " ", 
      RowBox[{"Log", "[", "F", "]"}]}]]}]}]]]
}], "Subsubsection",
 CellChangeTimes->{{3.495321995009364*^9, 3.4953220146593914`*^9}, {
   3.495322060049455*^9, 3.4953220695394683`*^9}, 3.4953324529433327`*^9, 
   3.4953341264356756`*^9, {3.5488752985704317`*^9, 3.5488752985704317`*^9}, {
   3.548882905755931*^9, 3.548882921645953*^9}, {3.550106698996528*^9, 
   3.5501067522966027`*^9}, {3.552842305866893*^9, 3.552842307596895*^9}, {
   3.552842966747818*^9, 3.552842967457819*^9}, {3.552843335158334*^9, 
   3.552843335158334*^9}, {3.5528448228604164`*^9, 3.552844824730419*^9}, {
   3.552847150523675*^9, 3.552847155623682*^9}, {3.579549628426206*^9, 
   3.5795496593962493`*^9}, {3.5795588195966454`*^9, 
   3.5795588203866467`*^9}, {3.579558891916747*^9, 3.5795588928967485`*^9}, {
   3.67381065799654*^9, 3.67381065799654*^9}, {3.674583127046164*^9, 
   3.674583127046164*^9}, {3.6745841802004013`*^9, 3.6745841802004013`*^9}, {
   3.6745849287692165`*^9, 3.674584929330249*^9}}],

Cell[TextData[{
 "Rule: If  ",
 Cell[BoxData[
  RowBox[{
   RowBox[{"m", ">", "0"}], " ", "\[And]", " ", 
   RowBox[{
    RowBox[{"2", "m"}], "\[Element]", "\[DoubleStruckCapitalZ]"}]}]]],
 ", then"
}], "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.495322033659418*^9, 3.495322068589467*^9}, {
   3.495332560963484*^9, 3.4953325798335104`*^9}, {3.4953342240158124`*^9, 
   3.4953342584258604`*^9}, {3.5488732775776024`*^9, 
   3.5488732775776024`*^9}, {3.548873326337671*^9, 3.548873326337671*^9}, {
   3.5488733961777687`*^9, 3.548873400247774*^9}, {3.548882497145359*^9, 
   3.5488825231053953`*^9}, {3.548882954315999*^9, 3.548882954315999*^9}, {
   3.548883883077299*^9, 3.5488839072773333`*^9}, {3.5488840277975016`*^9, 
   3.5488840277975016`*^9}, {3.550092011002294*^9, 3.550092011712295*^9}, {
   3.5500922678526535`*^9, 3.5500922678526535`*^9}, {3.550093724424693*^9, 
   3.550093724424693*^9}, {3.5500971557107277`*^9, 3.5500971557107277`*^9}, {
   3.5501063662860622`*^9, 3.550106398116107*^9}, {3.550106454446186*^9, 
   3.550106454446186*^9}, 3.5501069180668344`*^9, {3.5501072155072513`*^9, 
   3.5501072155072513`*^9}, {3.5501082711687293`*^9, 
   3.5501082711687293`*^9}, {3.5528426907374315`*^9, 
   3.5528426907374315`*^9}, {3.552843846249049*^9, 3.552843846249049*^9}, {
   3.552926049123206*^9, 3.5529260500280075`*^9}, {3.614435535412983*^9, 
   3.614435535412983*^9}, {3.614462092009901*^9, 3.614462092009901*^9}, {
   3.614463598140047*^9, 3.614463598140047*^9}, {3.673815230143052*^9, 
   3.6738152301440525`*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"\[Integral]", 
   RowBox[{
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"c", "+", 
       RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"b", " ", 
       SuperscriptBox["F", 
        RowBox[{"g", " ", 
         RowBox[{"(", 
          RowBox[{"e", "+", 
           RowBox[{"f", " ", "x"}]}], ")"}]}]]}], ")"}], "n"], 
    RowBox[{"\[DifferentialD]", 
     RowBox[{"x", "  ", "\[LongRightArrow]", "  ", 
      FractionBox[
       RowBox[{
        SuperscriptBox[
         RowBox[{"(", 
          RowBox[{"c", "+", 
           RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
        SuperscriptBox[
         RowBox[{"(", 
          RowBox[{"b", " ", 
           SuperscriptBox["F", 
            RowBox[{"g", " ", 
             RowBox[{"(", 
              RowBox[{"e", "+", 
               RowBox[{"f", " ", "x"}]}], ")"}]}]]}], ")"}], "n"]}], 
       RowBox[{"f", " ", "g", " ", "n", " ", 
        RowBox[{"Log", "[", "F", "]"}]}]]}]}]}]}], "-", 
  RowBox[{
   FractionBox[
    RowBox[{"d", " ", "m"}], 
    RowBox[{"f", " ", "g", " ", "n", " ", 
     RowBox[{"Log", "[", "F", "]"}]}]], 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", " ", "x"}]}], ")"}], 
      RowBox[{"m", "-", "1"}]], " ", 
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"b", " ", 
        SuperscriptBox["F", 
         RowBox[{"g", " ", 
          RowBox[{"(", 
           RowBox[{"e", "+", 
            RowBox[{"f", " ", "x"}]}], ")"}]}]]}], ")"}], "n"], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579723816*^9, 3.4796580261689425`*^9}, 
   3.479661191290163*^9, {3.479686720164157*^9, 3.479686720204214*^9}, {
   3.479686831554328*^9, 3.479686834819022*^9}, {3.4940970712485504`*^9, 
   3.4940971130286083`*^9}, 3.4940971455886545`*^9, 3.4953221152495327`*^9, 
   3.495322146199576*^9, {3.495322261579737*^9, 3.4953222704097495`*^9}, 
   3.4953325041434045`*^9, 3.4953325401034546`*^9, {3.4953327138936977`*^9, 
   3.4953327232437115`*^9}, {3.4953341614457245`*^9, 3.4953341867757597`*^9}, 
   3.5501063615760555`*^9, {3.5501064648262005`*^9, 3.550106490466236*^9}, {
   3.550106525066285*^9, 3.550106540276306*^9}, {3.5501065790863605`*^9, 
   3.5501066156964116`*^9}, {3.5528423117869005`*^9, 3.552842314026904*^9}, {
   3.5528426917374325`*^9, 3.5528426937974358`*^9}, {3.5528429682378197`*^9, 
   3.5528429694078217`*^9}, {3.5528433415383425`*^9, 
   3.5528433442283463`*^9}, {3.5528435601586485`*^9, 
   3.5528435612786503`*^9}, {3.552844695100238*^9, 3.552844701190246*^9}, {
   3.5528471263836412`*^9, 3.552847134093652*^9}, {3.5795487900050325`*^9, 
   3.579548792195035*^9}, {3.5795492795057173`*^9, 3.579549280715719*^9}, {
   3.5795493517058187`*^9, 3.5795493524758196`*^9}, {3.579549671386266*^9, 
   3.579549675026271*^9}, 3.5795497284863462`*^9, {3.5795585728663*^9, 
   3.5795585738863015`*^9}, {3.579558822086649*^9, 3.579558823556651*^9}, {
   3.579558905576766*^9, 3.5795589085567703`*^9}, {3.579558982036873*^9, 
   3.5795590045369043`*^9}, {3.579559110317053*^9, 3.579559113487057*^9}, {
   3.6738105903226695`*^9, 3.673810632865103*^9}, {3.6738107463915963`*^9, 
   3.6738107498977966`*^9}, {3.674583020060045*^9, 3.674583020068045*^9}, {
   3.674583142642056*^9, 3.6745832043315845`*^9}, 3.674584133059705*^9, {
   3.6745841890679083`*^9, 3.6745841986884584`*^9}, {3.674584930315305*^9, 
   3.6745849313543644`*^9}},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}, {3.550106725296565*^9, 3.550106727366568*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{"c_.", "+", 
        RowBox[{"d_.", "*", "x_"}]}], ")"}], "^", "m_."}], "*", 
     RowBox[{
      RowBox[{"(", 
       RowBox[{"b_.", "*", 
        RowBox[{"F_", "^", 
         RowBox[{"(", 
          RowBox[{"g_.", "*", 
           RowBox[{"(", 
            RowBox[{"e_.", "+", 
             RowBox[{"f_.", "*", "x_"}]}], ")"}]}], ")"}]}]}], ")"}], "^", 
      "n_."}]}], ",", "x_Symbol"}], "]"}], " ", ":=", "\n", "  ", 
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", "*", "x"}]}], ")"}], "^", "m"}], "*", 
     RowBox[{
      RowBox[{
       RowBox[{"(", 
        RowBox[{"b", "*", 
         RowBox[{"F", "^", 
          RowBox[{"(", 
           RowBox[{"g", "*", 
            RowBox[{"(", 
             RowBox[{"e", "+", 
              RowBox[{"f", "*", "x"}]}], ")"}]}], ")"}]}]}], ")"}], "^", 
       "n"}], "/", 
      RowBox[{"(", 
       RowBox[{"f", "*", "g", "*", "n", "*", 
        RowBox[{"Log", "[", "F", "]"}]}], ")"}]}]}], " ", "-", " ", 
    "\[IndentingNewLine]", "  ", 
    RowBox[{"d", "*", 
     RowBox[{"m", "/", 
      RowBox[{"(", 
       RowBox[{"f", "*", "g", "*", "n", "*", 
        RowBox[{"Log", "[", "F", "]"}]}], ")"}]}], "*", 
     RowBox[{"Int", "[", 
      RowBox[{
       RowBox[{
        RowBox[{
         RowBox[{"(", 
          RowBox[{"c", "+", 
           RowBox[{"d", "*", "x"}]}], ")"}], "^", 
         RowBox[{"(", 
          RowBox[{"m", "-", "1"}], ")"}]}], "*", 
        RowBox[{
         RowBox[{"(", 
          RowBox[{"b", "*", 
           RowBox[{"F", "^", 
            RowBox[{"(", 
             RowBox[{"g", "*", 
              RowBox[{"(", 
               RowBox[{"e", "+", 
                RowBox[{"f", "*", "x"}]}], ")"}]}], ")"}]}]}], ")"}], "^", 
         "n"}]}], ",", "x"}], "]"}]}]}], " ", "/;", "\n", 
   RowBox[{
    RowBox[{"FreeQ", "[", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{
       "F", ",", "b", ",", "c", ",", "d", ",", "e", ",", "f", ",", "g", ",", 
        "n"}], "}"}], ",", "x"}], "]"}], " ", "&&", " ", 
    RowBox[{"GtQ", "[", 
     RowBox[{"m", ",", "0"}], "]"}], " ", "&&", " ", 
    RowBox[{"IntegerQ", "[", 
     RowBox[{"2", "*", "m"}], "]"}], " ", "&&", " ", 
    RowBox[{"Not", "[", 
     RowBox[{"TrueQ", "[", "$UseGamma", "]"}], "]"}]}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, 
   3.4953222454097147`*^9, 3.4953324633733473`*^9, 3.4953326632436275`*^9, 
   3.4953341350856876`*^9, {3.548816716117523*^9, 3.54881672825754*^9}, {
   3.5488168441577024`*^9, 3.548816845257704*^9}, {3.5488174267385178`*^9, 
   3.548817476678588*^9}, {3.5488175358286705`*^9, 3.548817537178673*^9}, 
   3.548818879032552*^9, 3.5488189948567142`*^9, {3.548873179327465*^9, 
   3.5488731994574933`*^9}, {3.5488733093376465`*^9, 3.5488733155476556`*^9}, 
   3.5488825439754243`*^9, {3.5488826089855156`*^9, 3.548882630935546*^9}, 
   3.5488838853873024`*^9, 3.5488840304775057`*^9, {3.550091804762005*^9, 
   3.5500918642520885`*^9}, {3.5500920434623394`*^9, 
   3.5500921913025465`*^9}, {3.5500935055643864`*^9, 
   3.5500935449644413`*^9}, {3.5500935868345003`*^9, 3.550093588744503*^9}, {
   3.550093644764581*^9, 3.550093654344595*^9}, 3.550097159490733*^9, 
   3.550106818926696*^9, {3.5501068633667583`*^9, 3.5501068987568083`*^9}, {
   3.5501071702771883`*^9, 3.5501071712671895`*^9}, {3.550107218837256*^9, 
   3.5501072235372624`*^9}, 3.5501082039586353`*^9, 3.550109236880081*^9, {
   3.552842314946905*^9, 3.5528423179569097`*^9}, 3.552842477407133*^9, 
   3.5528425238471975`*^9, {3.5528426947374372`*^9, 3.552842703087449*^9}, {
   3.5528430422779236`*^9, 3.552843043307925*^9}, 3.552843088757989*^9, 
   3.5528431567680836`*^9, {3.552843347728351*^9, 3.5528433497783537`*^9}, {
   3.552843562598652*^9, 3.5528435632086525`*^9}, 3.5528436835188212`*^9, 
   3.552844216459567*^9, 3.5528442843696623`*^9, {3.552844706140253*^9, 
   3.552844714870265*^9}, 3.552845986792046*^9, {3.5528471936237354`*^9, 
   3.5528472087337565`*^9}, {3.552926054099615*^9, 3.552926054910816*^9}, {
   3.5529275303886056`*^9, 3.5529275747238836`*^9}, 3.5795494000658865`*^9, 
   3.579549439275941*^9, 3.579549476805994*^9, {3.5795497467963715`*^9, 
   3.5795497486763744`*^9}, {3.5795497909664335`*^9, 3.579549801246448*^9}, {
   3.579553111741083*^9, 3.579553118591092*^9}, {3.579553619284293*^9, 
   3.5795536265743036`*^9}, {3.5795586243963723`*^9, 
   3.5795586248963733`*^9}, {3.579558671406438*^9, 3.5795587013164797`*^9}, 
   3.579558739216533*^9, {3.5795589205967875`*^9, 3.579558927636797*^9}, 
   3.579558988336882*^9, {3.5795590471569643`*^9, 3.579559048416966*^9}, 
   3.6144353068819113`*^9, {3.614435550434842*^9, 3.6144355564221845`*^9}, {
   3.6144384265533466`*^9, 3.614438431660639*^9}, {3.6144621285929937`*^9, 
   3.614462146147998*^9}, {3.6144636037563686`*^9, 3.6144636183762045`*^9}, {
   3.614471679656844*^9, 3.6144716799288597`*^9}, {3.6145568877653627`*^9, 
   3.6145569221873317`*^9}, 3.6145577730379972`*^9, 3.614558329958852*^9, 
   3.614558469893855*^9, {3.614651041094572*^9, 3.614651046535883*^9}, {
   3.673810731537747*^9, 3.6738107347639313`*^9}, {3.673810789096039*^9, 
   3.673810814723505*^9}, 3.6738120383694935`*^9, {3.673815236132395*^9, 
   3.673815257081593*^9}, {3.6745832453569307`*^9, 3.6745832779947977`*^9}, 
   3.6745833363151336`*^9, {3.674584247227235*^9, 3.6745842499783916`*^9}, 
   3.6745843006032877`*^9, {3.6745843336981807`*^9, 3.6745843403535614`*^9}, 
   3.715109317369136*^9, 3.747859849051644*^9},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9}]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t", 
  RowBox[{Cell[TextData[StyleBox["2:",
    FontFamily->"Arial",
    FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"b", " ", 
        SuperscriptBox["F", 
         RowBox[{"g", " ", 
          RowBox[{"(", 
           RowBox[{"e", "+", 
            RowBox[{"f", " ", "x"}]}], ")"}]}]]}], ")"}], "n"], 
     RowBox[{"\[DifferentialD]", "x"}], "  ", 
     StyleBox["when",
      FontFamily->"Arial",
      FontWeight->"Plain"], 
     StyleBox["  ",
      FontFamily->"Arial",
      FontWeight->"Plain"], Cell[TextData[Cell[BoxData[
      RowBox[{
       RowBox[{"m", "<", 
        RowBox[{"-", "1"}]}], " ", "\[And]", " ", 
       RowBox[{
        RowBox[{"2", "m"}], "\[Element]", "\[DoubleStruckCapitalZ]"}]}]]]], 
      "None"]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193422443984776`*^9, 3.5193422447260776`*^9}, 
   3.5193424488212357`*^9, {3.5193434708254333`*^9, 3.519343473103037*^9}, {
   3.5193435097631016`*^9, 3.51934353169674*^9}, {3.519345739319023*^9, 
   3.519345739599823*^9}, 3.5193462601259375`*^9, {3.5193475061060295`*^9, 
   3.5193475233908596`*^9}, {3.519351391880224*^9, 3.5193513931402254`*^9}, {
   3.519403773021052*^9, 3.5194037798564425`*^9}, 3.5194106793250647`*^9, {
   3.519537578341393*^9, 3.5195375792305946`*^9}, {3.5195376277778797`*^9, 
   3.51953767927357*^9}, {3.5197586260636144`*^9, 3.5197586260636144`*^9}, {
   3.51978053295651*^9, 3.519780542862528*^9}, {3.5197806526087203`*^9, 
   3.5197806657595434`*^9}, {3.5197807490168896`*^9, 3.5197807642581167`*^9}, 
   3.5197809004619555`*^9, {3.519782653514635*^9, 3.5197826623286505`*^9}, {
   3.5200207810164795`*^9, 3.520020785046485*^9}, {3.5213126833374643`*^9, 
   3.5213126833374643`*^9}, {3.521328235411002*^9, 3.521328235411002*^9}, {
   3.523315567893022*^9, 3.523315567893022*^9}, {3.5233157569332867`*^9, 
   3.5233157613032923`*^9}, {3.5323023198133526`*^9, 3.532302355723403*^9}, {
   3.5323026417638035`*^9, 3.5323026417638035`*^9}, {3.5323026980138817`*^9, 
   3.5323026980138817`*^9}, {3.532636287104328*^9, 3.532636287104328*^9}, {
   3.5330803686035347`*^9, 3.533080368863535*^9}, {3.533248909289961*^9, 
   3.5332489110299635`*^9}, {3.534962469565694*^9, 3.5349624910757236`*^9}, {
   3.5349625224057674`*^9, 3.534962522655768*^9}, {3.534965265519608*^9, 
   3.534965265519608*^9}, {3.5349668288168063`*^9, 3.5349668292668066`*^9}, {
   3.5349747358481646`*^9, 3.5349747358481646`*^9}, {3.534975457949176*^9, 
   3.534975457949176*^9}, {3.5368633963532934`*^9, 3.536863396571694*^9}, {
   3.541644322511304*^9, 3.5416443577513533`*^9}, {3.5416458269234104`*^9, 
   3.541645839633428*^9}, {3.5416460312236967`*^9, 3.541646031483697*^9}, 
   3.541646227783972*^9, {3.541647080335165*^9, 3.5416471077052035`*^9}, {
   3.541648997607849*^9, 3.5416489978278494`*^9}, 3.541649088407976*^9, {
   3.5417848711596775`*^9, 3.541784902429721*^9}, {3.541786145331461*^9, 
   3.5417861455914617`*^9}, {3.5417870037166634`*^9, 3.5417870039366636`*^9}, 
   3.5462979620242105`*^9, {3.546298264094633*^9, 3.546298265344635*^9}, {
   3.546299000235664*^9, 3.546299001505666*^9}, {3.546315119909564*^9, 
   3.5463151267659564`*^9}, {3.547227356319705*^9, 3.547227356319705*^9}, 
   3.547918559331044*^9, {3.547918610170952*^9, 3.5479186304411116`*^9}, 
   3.548537755517557*^9, 3.5485441629685287`*^9, 3.5485517520154133`*^9, {
   3.5488740727887154`*^9, 3.5488740727887154`*^9}, {3.548874206618903*^9, 
   3.548874206808903*^9}, {3.5488771226154857`*^9, 3.5488771332055006`*^9}, {
   3.548878122226885*^9, 3.548878122366885*^9}, 3.548878187466976*^9, {
   3.5488823466851482`*^9, 3.548882375975189*^9}, 3.5488824384852767`*^9, {
   3.5488824881153464`*^9, 3.5488824881153464`*^9}, {3.5488825728354645`*^9, 
   3.548882573075465*^9}, {3.5488839747174273`*^9, 3.5488839747174273`*^9}, {
   3.5488841172076273`*^9, 3.548884119867631*^9}, {3.5488842417478013`*^9, 
   3.5488842733678455`*^9}, {3.5500922898926845`*^9, 
   3.5500923738128023`*^9}, {3.550092524353013*^9, 3.550092524543013*^9}, {
   3.550092561623065*^9, 3.5500925617830653`*^9}, {3.550106200715831*^9, 
   3.550106200905831*^9}, {3.550106983486926*^9, 3.5501069912769375`*^9}, {
   3.550107070567048*^9, 3.5501070863370705`*^9}, 3.5501071950172224`*^9, {
   3.550108275278735*^9, 3.550108275278735*^9}, {3.5501084552089868`*^9, 
   3.550108455358987*^9}, 3.552842337716937*^9, {3.5528427514575167`*^9, 
   3.5528427523975177`*^9}, 3.5528429751278296`*^9, 3.552843572758666*^9, {
   3.552843866239077*^9, 3.552843866239077*^9}, {3.5528477607345295`*^9, 
   3.5528477617845306`*^9}, 3.5529260766260543`*^9, {3.5529262622819805`*^9, 
   3.5529262628747816`*^9}, 3.5532224151376023`*^9, 3.5795487528749804`*^9, 
   3.579548954225262*^9, 3.5795492927257357`*^9, 3.579549356335825*^9, 
   3.579558578856309*^9, 3.5795588424566774`*^9, {3.6144380880559855`*^9, 
   3.6144380880559855`*^9}, {3.614462452603526*^9, 3.614462469070468*^9}, 
   3.614463323088315*^9, {3.614463449380539*^9, 3.614463485057579*^9}, {
   3.6144667861233892`*^9, 3.6144668037143955`*^9}, 3.673811487902008*^9, {
   3.673811839304107*^9, 3.673811848530635*^9}, {3.6738146008820605`*^9, 
   3.6738146010380697`*^9}, {3.67381527941187*^9, 3.673815283505104*^9}, 
   3.6745830200790462`*^9, 3.6745833713001347`*^9, 3.6745841212050266`*^9, 
   3.674584932369423*^9},
 FontSize->12,
 FontWeight->"Bold"],

Cell["Derivation: Integration by parts", "Subsubsection",
 CellChangeTimes->{{3.495321995009364*^9, 3.4953219984993687`*^9}, {
  3.4953340975556355`*^9, 3.495334101755641*^9}}],

Cell[TextData[{
 "Basis: ",
 Cell[BoxData[
  RowBox[{
   SuperscriptBox[
    RowBox[{"(", 
     RowBox[{"c", "+", 
      RowBox[{"d", " ", "x"}]}], ")"}], "m"], "\[Equal]", 
   RowBox[{
    SubscriptBox["\[PartialD]", "x"], 
    FractionBox[
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", " ", "x"}]}], ")"}], 
      RowBox[{"m", "+", "1"}]], 
     RowBox[{"d", 
      RowBox[{"(", 
       RowBox[{"m", "+", "1"}], ")"}]}]]}]}]]]
}], "Subsubsection",
 CellChangeTimes->{{3.495321995009364*^9, 3.4953220146593914`*^9}, {
   3.495322060049455*^9, 3.4953220695394683`*^9}, 3.4953324529433327`*^9, 
   3.4953341264356756`*^9, {3.5488752985704317`*^9, 3.5488752985704317`*^9}, {
   3.548882905755931*^9, 3.548882921645953*^9}, {3.550106698996528*^9, 
   3.5501067522966027`*^9}, {3.550107289067354*^9, 3.550107289067354*^9}, {
   3.5501073942375016`*^9, 3.5501073969375052`*^9}, {3.552842755937523*^9, 
   3.5528427573375244`*^9}, {3.552843573298667*^9, 3.5528435738286676`*^9}, {
   3.5795492937857375`*^9, 3.579549294295738*^9}, {3.579549356585826*^9, 
   3.579549356845826*^9}, {3.5795585794263096`*^9, 3.5795585798063097`*^9}, {
   3.5795598725706387`*^9, 3.5795598725706387`*^9}, 3.6745830200990467`*^9, {
   3.6745834134395447`*^9, 3.6745834134395447`*^9}}],

Cell[TextData[{
 "Rule: If  ",
 Cell[BoxData[
  RowBox[{
   RowBox[{"m", "<", 
    RowBox[{"-", "1"}]}], " ", "\[And]", " ", 
   RowBox[{
    RowBox[{"2", "m"}], "\[Element]", "\[DoubleStruckCapitalZ]"}]}]]],
 ", then"
}], "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.495322033659418*^9, 3.495322068589467*^9}, {
   3.495332560963484*^9, 3.4953325798335104`*^9}, {3.4953342240158124`*^9, 
   3.4953342584258604`*^9}, {3.4953343353559685`*^9, 
   3.4953343804860315`*^9}, {3.5488732348875427`*^9, 3.548873262597582*^9}, {
   3.548873422497805*^9, 3.548873422497805*^9}, {3.5488839614474087`*^9, 
   3.5488839614474087`*^9}, {3.5501071130071077`*^9, 
   3.5501071130071077`*^9}, {3.550108278348739*^9, 3.550108278348739*^9}, {
   3.552842758197526*^9, 3.552842758197526*^9}, {3.5528438669090776`*^9, 
   3.5528438669090776`*^9}, {3.552926249271558*^9, 3.552926249895559*^9}, {
   3.6144380953064003`*^9, 3.6144380953064003`*^9}, {3.6144624944659204`*^9, 
   3.6144624944659204`*^9}, {3.614463537205562*^9, 3.614463537205562*^9}, {
   3.6738152878953557`*^9, 3.6738152878953557`*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"\[Integral]", 
   RowBox[{
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"c", "+", 
       RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"b", " ", 
       SuperscriptBox["F", 
        RowBox[{"g", " ", 
         RowBox[{"(", 
          RowBox[{"e", "+", 
           RowBox[{"f", " ", "x"}]}], ")"}]}]]}], ")"}], "n"], 
    RowBox[{"\[DifferentialD]", 
     RowBox[{"x", "  ", "\[LongRightArrow]", "  ", 
      FractionBox[
       RowBox[{
        SuperscriptBox[
         RowBox[{"(", 
          RowBox[{"c", "+", 
           RowBox[{"d", " ", "x"}]}], ")"}], 
         RowBox[{"m", "+", "1"}]], " ", 
        SuperscriptBox[
         RowBox[{"(", 
          RowBox[{"b", " ", 
           SuperscriptBox["F", 
            RowBox[{"g", " ", 
             RowBox[{"(", 
              RowBox[{"e", "+", 
               RowBox[{"f", " ", "x"}]}], ")"}]}]]}], ")"}], "n"]}], 
       RowBox[{"d", 
        RowBox[{"(", 
         RowBox[{"m", "+", "1"}], ")"}]}]]}]}]}]}], "-", 
  RowBox[{
   FractionBox[
    RowBox[{"f", " ", "g", " ", "n", " ", 
     RowBox[{"Log", "[", "F", "]"}]}], 
    RowBox[{"d", 
     RowBox[{"(", 
      RowBox[{"m", "+", "1"}], ")"}]}]], 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", " ", "x"}]}], ")"}], 
      RowBox[{"m", "+", "1"}]], " ", 
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"b", " ", 
        SuperscriptBox["F", 
         RowBox[{"g", " ", 
          RowBox[{"(", 
           RowBox[{"e", "+", 
            RowBox[{"f", " ", "x"}]}], ")"}]}]]}], ")"}], "n"], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579723816*^9, 3.4796580261689425`*^9}, 
   3.479661191290163*^9, {3.479686720164157*^9, 3.479686720204214*^9}, {
   3.479686831554328*^9, 3.479686834819022*^9}, {3.4940970712485504`*^9, 
   3.4940971130286083`*^9}, 3.4940971455886545`*^9, 3.4953221152495327`*^9, 
   3.495322146199576*^9, {3.495322261579737*^9, 3.4953222704097495`*^9}, 
   3.4953325041434045`*^9, 3.4953325401034546`*^9, {3.4953327138936977`*^9, 
   3.4953327232437115`*^9}, {3.4953341614457245`*^9, 3.4953341867757597`*^9}, 
   3.4953344255060945`*^9, {3.550107259017312*^9, 3.550107263687319*^9}, {
   3.5501073156473913`*^9, 3.550107376467477*^9}, {3.552842341676943*^9, 
   3.552842343686946*^9}, {3.552842758827527*^9, 3.5528427609175296`*^9}, {
   3.552842976237831*^9, 3.552842977737833*^9}, 3.552843419398452*^9, {
   3.552843574568669*^9, 3.5528435753786697`*^9}, {3.5528448701004825`*^9, 
   3.552844871900485*^9}, {3.5528477643645344`*^9, 3.552847772404546*^9}, {
   3.5795487966050415`*^9, 3.5795487979350433`*^9}, {3.579549294865739*^9, 
   3.579549296925742*^9}, {3.579549357115826*^9, 3.579549357665827*^9}, 
   3.579549687396289*^9, {3.5795502376170588`*^9, 3.579550241107064*^9}, {
   3.5795585802863107`*^9, 3.579558581126312*^9}, {3.5795588433266788`*^9, 
   3.579558844736681*^9}, {3.5795598818306513`*^9, 3.5795598912806644`*^9}, {
   3.673811855908057*^9, 3.6738118613153667`*^9}, {3.673811896046353*^9, 
   3.6738118965283804`*^9}, 3.6745830201070476`*^9, {3.6745833762114153`*^9, 
   3.6745833873870544`*^9}, {3.674583418307823*^9, 3.674583489005867*^9}, {
   3.6745835586198487`*^9, 3.67458355899487*^9}, {3.6745836452518034`*^9, 
   3.674583645447815*^9}, {3.6745843702902737`*^9, 3.674584378464741*^9}, {
   3.674584933315477*^9, 3.674584936315648*^9}},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{"c_.", "+", 
        RowBox[{"d_.", "*", "x_"}]}], ")"}], "^", "m_"}], "*", 
     RowBox[{
      RowBox[{"(", 
       RowBox[{"b_.", "*", 
        RowBox[{"F_", "^", 
         RowBox[{"(", 
          RowBox[{"g_.", "*", 
           RowBox[{"(", 
            RowBox[{"e_.", "+", 
             RowBox[{"f_.", "*", "x_"}]}], ")"}]}], ")"}]}]}], ")"}], "^", 
      "n_."}]}], ",", "x_Symbol"}], "]"}], " ", ":=", "\n", "  ", 
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", "*", "x"}]}], ")"}], "^", 
      RowBox[{"(", 
       RowBox[{"m", "+", "1"}], ")"}]}], "*", 
     RowBox[{
      RowBox[{
       RowBox[{"(", 
        RowBox[{"b", "*", 
         RowBox[{"F", "^", 
          RowBox[{"(", 
           RowBox[{"g", "*", 
            RowBox[{"(", 
             RowBox[{"e", "+", 
              RowBox[{"f", "*", "x"}]}], ")"}]}], ")"}]}]}], ")"}], "^", 
       "n"}], "/", 
      RowBox[{"(", 
       RowBox[{"d", "*", 
        RowBox[{"(", 
         RowBox[{"m", "+", "1"}], ")"}]}], ")"}]}]}], " ", "-", " ", 
    "\[IndentingNewLine]", "  ", 
    RowBox[{"f", "*", "g", "*", "n", "*", 
     RowBox[{
      RowBox[{"Log", "[", "F", "]"}], "/", 
      RowBox[{"(", 
       RowBox[{"d", "*", 
        RowBox[{"(", 
         RowBox[{"m", "+", "1"}], ")"}]}], ")"}]}], "*", 
     RowBox[{"Int", "[", 
      RowBox[{
       RowBox[{
        RowBox[{
         RowBox[{"(", 
          RowBox[{"c", "+", 
           RowBox[{"d", "*", "x"}]}], ")"}], "^", 
         RowBox[{"(", 
          RowBox[{"m", "+", "1"}], ")"}]}], "*", 
        RowBox[{
         RowBox[{"(", 
          RowBox[{"b", "*", 
           RowBox[{"F", "^", 
            RowBox[{"(", 
             RowBox[{"g", "*", 
              RowBox[{"(", 
               RowBox[{"e", "+", 
                RowBox[{"f", "*", "x"}]}], ")"}]}], ")"}]}]}], ")"}], "^", 
         "n"}]}], ",", "x"}], "]"}]}]}], " ", "/;", "\n", 
   RowBox[{
    RowBox[{"FreeQ", "[", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{
       "F", ",", "b", ",", "c", ",", "d", ",", "e", ",", "f", ",", "g", ",", 
        "n"}], "}"}], ",", "x"}], "]"}], " ", "&&", " ", 
    RowBox[{"LtQ", "[", 
     RowBox[{"m", ",", 
      RowBox[{"-", "1"}]}], "]"}], " ", "&&", " ", 
    RowBox[{"IntegerQ", "[", 
     RowBox[{"2", "*", "m"}], "]"}], " ", "&&", " ", 
    RowBox[{"Not", "[", 
     RowBox[{"TrueQ", "[", "$UseGamma", "]"}], "]"}]}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, 
   3.4953222454097147`*^9, 3.4953324633733473`*^9, 3.4953326632436275`*^9, 
   3.4953341350856876`*^9, 3.4953343185259447`*^9, {3.5488167047575073`*^9, 
   3.548816711757517*^9}, {3.5488168502277107`*^9, 3.5488168508877115`*^9}, 
   3.5488188828625574`*^9, 3.5488190000167217`*^9, {3.5488731856574736`*^9, 
   3.548873205897502*^9}, {3.5488849424787827`*^9, 3.548884951398795*^9}, {
   3.550091906722148*^9, 3.5500919285221786`*^9}, {3.5501071331471357`*^9, 
   3.5501071585471716`*^9}, {3.5501074443975716`*^9, 
   3.5501074489175777`*^9}, {3.5501075034776545`*^9, 3.5501075187176757`*^9}, 
   3.5501082148686504`*^9, {3.5528423444169464`*^9, 3.55284234677695*^9}, 
   3.5528424801471367`*^9, 3.552842528467204*^9, {3.5528427621175313`*^9, 
   3.55284276828754*^9}, {3.5528430438179255`*^9, 3.552843044347926*^9}, 
   3.5528430907879915`*^9, 3.55284316083809*^9, 3.5528434220984554`*^9, {
   3.5528435761286707`*^9, 3.552843576598672*^9}, 3.552843686138825*^9, 
   3.5528442191495714`*^9, 3.5528442865096655`*^9, {3.5528448756904907`*^9, 
   3.5528448779104934`*^9}, 3.55284598967205*^9, {3.552847776544552*^9, 
   3.5528477893045692`*^9}, {3.5529262526099634`*^9, 3.552926253374365*^9}, {
   3.5529277273389516`*^9, 3.552927731738159*^9}, {3.552928881556175*^9, 
   3.552928916188236*^9}, {3.572925938770609*^9, 3.5729259395506105`*^9}, {
   3.5729259904106817`*^9, 3.5729259922906837`*^9}, 3.579549402055889*^9, 
   3.5795494809459996`*^9, 3.5795495364460773`*^9, {3.579549754426382*^9, 
   3.579549755596384*^9}, {3.579550262647094*^9, 3.5795502643270965`*^9}, {
   3.5795531305911093`*^9, 3.579553130791109*^9}, {3.579553646264331*^9, 
   3.579553649154335*^9}, {3.5795586272163763`*^9, 3.5795586279463773`*^9}, {
   3.5795586734464407`*^9, 3.579558703256483*^9}, 3.5795587434665394`*^9, {
   3.5795590509269695`*^9, 3.5795590568069773`*^9}, {3.57955990927069*^9, 
   3.579559914450697*^9}, {3.614438099476639*^9, 3.6144381017457685`*^9}, {
   3.614438230183115*^9, 3.6144383163410425`*^9}, {3.614439628240079*^9, 
   3.6144396371735897`*^9}, {3.614462505726565*^9, 3.6144625156891346`*^9}, {
   3.6144635443709717`*^9, 3.6144635566966767`*^9}, {3.614467622527229*^9, 
   3.61446762499337*^9}, {3.6144680911270313`*^9, 3.614468097118374*^9}, {
   3.614556694322298*^9, 3.614556721506853*^9}, {3.6145567931559515`*^9, 
   3.6145567958911076`*^9}, {3.614651075940565*^9, 3.6146510816148896`*^9}, {
   3.6738119853294597`*^9, 3.6738120249867277`*^9}, {3.6738153011491137`*^9, 
   3.6738153055023623`*^9}, {3.674583662950816*^9, 3.6745836775726523`*^9}, 
   3.674583730735693*^9, {3.674584420162126*^9, 3.6745844313037634`*^9}, {
   3.674584465479718*^9, 3.6745844822166753`*^9}, {3.7151093323419924`*^9, 
   3.7151093400404325`*^9}, 3.747859852571845*^9},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9}]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t", 
  RowBox[{Cell[TextData[StyleBox["3.",
    FontFamily->"Arial"]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
     SuperscriptBox["F", 
      RowBox[{"g", " ", 
       RowBox[{"(", 
        RowBox[{"e", "+", 
         RowBox[{"f", " ", "x"}]}], ")"}]}]], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193415004828687`*^9, 3.5193415113404875`*^9}, {
   3.5193513965602303`*^9, 3.519351397420232*^9}, {3.5194037839416766`*^9, 
   3.5194037847117205`*^9}, 3.52105258275843*^9, 3.5210531741726685`*^9, {
   3.521057461672199*^9, 3.5210574760242243`*^9}, {3.521059881564049*^9, 
   3.521059886088057*^9}, {3.5213902158895197`*^9, 3.521390217543123*^9}, {
   3.521390827878595*^9, 3.521390829376197*^9}, {3.5214688792561245`*^9, 
   3.521468879443325*^9}, {3.523320201951167*^9, 3.523320202247567*^9}, {
   3.5288596764894824`*^9, 3.5288596768094826`*^9}, {3.529196266260353*^9, 
   3.5291962665099535`*^9}, {3.5291997050223927`*^9, 
   3.5291997052563934`*^9}, {3.529201346457276*^9, 3.529201347252877*^9}, {
   3.529285041073242*^9, 3.5292850412632427`*^9}, {3.529285215443486*^9, 
   3.529285215443486*^9}, {3.5295115356999426`*^9, 3.5295115433443794`*^9}, {
   3.529521412127842*^9, 3.529521414647986*^9}, {3.529529429975619*^9, 
   3.529529432955224*^9}, 3.5340074894011116`*^9, 3.5415537711400824`*^9, {
   3.5415538454742126`*^9, 3.54155384951462*^9}, {3.5417297197063265`*^9, 
   3.5417297252443366`*^9}, {3.546297494083555*^9, 3.5462974982435613`*^9}, {
   3.5462989134855423`*^9, 3.5462989153655453`*^9}, 3.548544146508506*^9, 
   3.5488738855784535`*^9, 3.548877085095433*^9, {3.548885157919084*^9, 
   3.5488851581390843`*^9}, {3.5501050720442505`*^9, 3.550105091724278*^9}, 
   3.552842303236889*^9, {3.5528426826774197`*^9, 3.552842683117421*^9}, 
   3.552842964957815*^9, 3.5528435586686463`*^9, {3.5528438442590466`*^9, 
   3.5528438442590466`*^9}, {3.5528470750635695`*^9, 
   3.5528470992336035`*^9}, {3.552925992292306*^9, 3.5529260013247223`*^9}, 
   3.553222405588056*^9, 3.5795486748848715`*^9, {3.579548905405194*^9, 
   3.579548907775197*^9}, 3.5795492772757144`*^9, 3.579549350275817*^9, 
   3.5795585715262985`*^9, 3.5795588167566414`*^9, {3.6738105630461097`*^9, 
   3.67381058140716*^9}, {3.6738146482907724`*^9, 3.6738146592433987`*^9}, 
   3.6745830201180477`*^9, 3.6745848362409244`*^9, 3.6745849386167803`*^9},
 FontSize->12,
 FontWeight->"Bold"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t\t", 
  RowBox[{Cell[TextData[StyleBox["1.",
    FontFamily->"Arial"]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
     SuperscriptBox["F", 
      RowBox[{"g", " ", 
       RowBox[{"(", 
        RowBox[{"e", "+", 
         RowBox[{"f", " ", "x"}]}], ")"}]}]], 
     RowBox[{"\[DifferentialD]", "x"}], "  ", 
     StyleBox["when",
      FontFamily->"Arial",
      FontWeight->"Plain"], 
     StyleBox["  ",
      FontFamily->"Arial",
      FontWeight->"Plain"], Cell[TextData[Cell[BoxData[
      RowBox[{"m", "\[Element]", "\[DoubleStruckCapitalZ]"}]]]], 
      "None"]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193415004828687`*^9, 3.5193415113404875`*^9}, {
   3.5193513965602303`*^9, 3.519351397420232*^9}, {3.5194037839416766`*^9, 
   3.5194037847117205`*^9}, 3.52105258275843*^9, 3.5210531741726685`*^9, {
   3.521057461672199*^9, 3.5210574760242243`*^9}, {3.521059881564049*^9, 
   3.521059886088057*^9}, {3.5213902158895197`*^9, 3.521390217543123*^9}, {
   3.521390827878595*^9, 3.521390829376197*^9}, {3.5214688792561245`*^9, 
   3.521468879443325*^9}, {3.523320201951167*^9, 3.523320202247567*^9}, {
   3.5288596764894824`*^9, 3.5288596768094826`*^9}, {3.529196266260353*^9, 
   3.5291962665099535`*^9}, {3.5291997050223927`*^9, 
   3.5291997052563934`*^9}, {3.529201346457276*^9, 3.529201347252877*^9}, {
   3.529285041073242*^9, 3.5292850412632427`*^9}, {3.529285215443486*^9, 
   3.529285215443486*^9}, {3.5295115356999426`*^9, 3.5295115433443794`*^9}, {
   3.529521412127842*^9, 3.529521414647986*^9}, {3.529529429975619*^9, 
   3.529529432955224*^9}, 3.5340074894011116`*^9, 3.5415537711400824`*^9, {
   3.5415538454742126`*^9, 3.54155384951462*^9}, {3.5417297197063265`*^9, 
   3.5417297252443366`*^9}, {3.546297494083555*^9, 3.5462974982435613`*^9}, {
   3.5462989134855423`*^9, 3.5462989153655453`*^9}, 3.548544146508506*^9, 
   3.5488738855784535`*^9, 3.548877085095433*^9, {3.548885157919084*^9, 
   3.5488851581390843`*^9}, {3.5501050720442505`*^9, 3.550105091724278*^9}, 
   3.552842303236889*^9, {3.5528426826774197`*^9, 3.552842683117421*^9}, 
   3.552842964957815*^9, 3.5528435586686463`*^9, {3.5528438442590466`*^9, 
   3.5528438442590466`*^9}, {3.5528470750635695`*^9, 
   3.5528470992336035`*^9}, {3.552925992292306*^9, 3.5529260013247223`*^9}, 
   3.553222405588056*^9, 3.5795486748848715`*^9, {3.579548905405194*^9, 
   3.579548907775197*^9}, 3.5795492772757144`*^9, 3.579549350275817*^9, 
   3.5795585715262985`*^9, 3.5795588167566414`*^9, {3.6738105630461097`*^9, 
   3.67381058140716*^9}, {3.6738146482907724`*^9, 3.6738146592433987`*^9}, {
   3.673814995511632*^9, 3.673814997280733*^9}, 3.6738150324417443`*^9, 
   3.674583020138049*^9, 3.6745848369529653`*^9, 3.674584939584835*^9},
 FontSize->12,
 FontWeight->"Bold"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t\t\t", 
  RowBox[{Cell[TextData[StyleBox["1:",
    FontFamily->"Arial",
    FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     FractionBox[
      SuperscriptBox["F", 
       RowBox[{"g", " ", 
        RowBox[{"(", 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], ")"}]}]], 
      RowBox[{"c", "+", 
       RowBox[{"d", " ", "x"}]}]], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.4964645213514385`*^9, 
   3.5192470594155855`*^9, {3.519247288125906*^9, 3.5192473207459517`*^9}, {
   3.5192504337891226`*^9, 3.5192504436696877`*^9}, {3.519320736057986*^9, 
   3.5193207456364026`*^9}, {3.5193282921909866`*^9, 
   3.5193283252786446`*^9}, {3.5193283634207115`*^9, 
   3.5193283676639194`*^9}, {3.5193286295571795`*^9, 
   3.5193286611160345`*^9}, {3.519341560355774*^9, 3.5193415713849936`*^9}, {
   3.5193514022902384`*^9, 3.5193514024302387`*^9}, {3.5194037903190413`*^9, 
   3.519403790599057*^9}, 3.51941073361316*^9, 3.5196771044605494`*^9, {
   3.5233155369029784`*^9, 3.5233155369029784`*^9}, 3.5239204146855583`*^9, 
   3.5239297857543354`*^9, 3.5239298325544176`*^9, {3.5295294635000772`*^9, 
   3.5295294742328963`*^9}, 3.5415539341135683`*^9, 3.5417297751644244`*^9, 
   3.5472297585730677`*^9, 3.547229903793271*^9, 3.5488746103394685`*^9, {
   3.5488773053657417`*^9, 3.54887731152575*^9}, 3.5488843073378935`*^9, 
   3.548884449468092*^9, {3.548884523728196*^9, 3.5488845252481985`*^9}, {
   3.5500925158530006`*^9, 3.5500925216230087`*^9}, {3.5500925554130564`*^9, 
   3.550092557073058*^9}, {3.550105202564433*^9, 3.550105202804434*^9}, {
   3.550105278274539*^9, 3.5501052972045655`*^9}, {3.5501056179250145`*^9, 
   3.550105618185015*^9}, 3.550108331138813*^9, 3.5501084509589806`*^9, 
   3.5528423195669117`*^9, {3.5528427060274525`*^9, 3.552842707427455*^9}, 
   3.5528429709478235`*^9, 3.5528435657286563`*^9, {3.552843847579051*^9, 
   3.552843847579051*^9}, {3.552847237063796*^9, 3.5528472380437975`*^9}, 
   3.552926070947644*^9, 3.552926198103468*^9, 3.5532224111003714`*^9, 
   3.579548747564973*^9, 3.5795489530552607`*^9, 3.579549282045721*^9, 
   3.579549353365821*^9, 3.5795585752263036`*^9, 3.5795588252066536`*^9, {
   3.6144622338140125`*^9, 3.6144622409034176`*^9}, 3.6144633197881265`*^9, {
   3.6144635055527515`*^9, 3.6144635172474203`*^9}, {3.6144667877324815`*^9, 
   3.6144667973330307`*^9}, 3.673811477865434*^9, {3.673812069655283*^9, 
   3.6738120724794445`*^9}, {3.6738121250724525`*^9, 3.673812127528593*^9}, {
   3.673814896377962*^9, 3.673814897059001*^9}, 3.6738149983047915`*^9, {
   3.673815346650716*^9, 3.6738153468237257`*^9}, 3.6745848651455774`*^9, 
   3.674585040265594*^9},
 FontSize->12,
 FontWeight->"Bold"],

Cell[TextData[{
 "Basis: ",
 Cell[BoxData[
  RowBox[{
   RowBox[{
    SuperscriptBox["ExpIntegralEi", "\[Prime]",
     MultilineFunction->None], "[", "z", "]"}], "\[Equal]", 
   FractionBox[
    SuperscriptBox["\[ExponentialE]", "z"], "z"]}]]]
}], "Subsubsection",
 CellChangeTimes->{{3.495321995009364*^9, 3.4953220146593914`*^9}, {
   3.495322060049455*^9, 3.4953220695394683`*^9}, 3.4953339702454567`*^9, {
   3.5488752946004267`*^9, 3.5488752946004267`*^9}}],

Cell["Rule:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.495322033659418*^9, 3.495322068589467*^9}, {
   3.495332560963484*^9, 3.4953325798335104`*^9}, {3.495332835663869*^9, 
   3.4953328392438736`*^9}, {3.5488728293569746`*^9, 
   3.5488728860770545`*^9}, {3.54887301891724*^9, 3.548873026407251*^9}, {
   3.550107662367877*^9, 3.5501076662278824`*^9}, {3.5501082976187663`*^9, 
   3.5501082976187663`*^9}, {3.552842770737543*^9, 3.552842770737543*^9}, {
   3.55284386845908*^9, 3.55284386845908*^9}, {3.5529261510069847`*^9, 
   3.552926151646586*^9}, {3.6144551006191587`*^9, 3.614455100620159*^9}, {
   3.614462269426049*^9, 3.614462269426049*^9}, 3.6738153498368983`*^9}],

Cell[BoxData[
 RowBox[{"\[Integral]", 
  RowBox[{
   FractionBox[
    SuperscriptBox["F", 
     RowBox[{"g", " ", 
      RowBox[{"(", 
       RowBox[{"e", "+", 
        RowBox[{"f", " ", "x"}]}], ")"}]}]], 
    RowBox[{"c", "+", 
     RowBox[{"d", " ", "x"}]}]], 
   RowBox[{"\[DifferentialD]", 
    RowBox[{"x", "  ", "\[LongRightArrow]", "  ", 
     FractionBox["1", "d"]}]}], 
   SuperscriptBox["F", 
    RowBox[{"g", " ", 
     RowBox[{"(", 
      RowBox[{"e", "-", 
       FractionBox[
        RowBox[{"c", " ", "f"}], "d"]}], ")"}]}]], 
   RowBox[{"ExpIntegralEi", "[", 
    FractionBox[
     RowBox[{"f", " ", "g", 
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", " ", "x"}]}], ")"}], " ", 
      RowBox[{"Log", "[", "F", "]"}]}], "d"], "]"}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579723816*^9, 3.4796580261689425`*^9}, 
   3.479661191290163*^9, {3.479686720164157*^9, 3.479686720204214*^9}, {
   3.479686831554328*^9, 3.479686834819022*^9}, {3.4940970712485504`*^9, 
   3.4940971130286083`*^9}, 3.4940971455886545`*^9, 3.4953221152495327`*^9, 
   3.495322146199576*^9, {3.495322261579737*^9, 3.4953222704097495`*^9}, {
   3.495334005335506*^9, 3.4953340139555182`*^9}, 3.550105346464635*^9, {
   3.550105445094773*^9, 3.5501054639847994`*^9}, {3.550105502604853*^9, 
   3.5501055057548575`*^9}, {3.552842323176917*^9, 3.552842324766919*^9}, {
   3.552842709067457*^9, 3.5528427128074627`*^9}, 3.552842971637825*^9, {
   3.5528433599583683`*^9, 3.5528433629083724`*^9}, {3.552843566398657*^9, 
   3.5528435670986586`*^9}, 3.552843849149053*^9, 3.552843918859151*^9, {
   3.5528440237692976`*^9, 3.552844032069309*^9}, {3.552844727530283*^9, 
   3.5528447317202888`*^9}, {3.552847243993806*^9, 3.5528472537638197`*^9}, 
   3.5795487931050367`*^9, {3.579549282735722*^9, 3.5795492833257227`*^9}, {
   3.5795493537958217`*^9, 3.579549354215822*^9}, 3.579549683396283*^9, {
   3.5795498116064625`*^9, 3.579549814596467*^9}, 3.5795499247066207`*^9, 
   3.579549997766723*^9, {3.579558575466304*^9, 3.5795585758363047`*^9}, 
   3.5795588258066545`*^9, {3.579559577750226*^9, 3.579559635570307*^9}, 
   3.6745848659076214`*^9, {3.6745850408116255`*^9, 3.6745850413316545`*^9}, {
   3.674585102370146*^9, 3.6745851425284433`*^9}, {3.674585181566676*^9, 
   3.6745851989506702`*^9}},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{"F_", "^", 
      RowBox[{"(", 
       RowBox[{"g_.", "*", 
        RowBox[{"(", 
         RowBox[{"e_.", "+", 
          RowBox[{"f_.", "*", "x_"}]}], ")"}]}], ")"}]}], "/", 
     RowBox[{"(", 
      RowBox[{"c_.", "+", 
       RowBox[{"d_.", "*", "x_"}]}], ")"}]}], ",", "x_Symbol"}], "]"}], " ", ":=",
   "\n", "  ", 
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{"F", "^", 
      RowBox[{"(", 
       RowBox[{"g", "*", 
        RowBox[{"(", 
         RowBox[{"e", "-", 
          RowBox[{"c", "*", 
           RowBox[{"f", "/", "d"}]}]}], ")"}]}], ")"}]}], "/", "d"}], "*", 
    RowBox[{"ExpIntegralEi", "[", 
     RowBox[{"f", "*", "g", "*", 
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", "*", "x"}]}], ")"}], "*", 
      RowBox[{
       RowBox[{"Log", "[", "F", "]"}], "/", "d"}]}], "]"}]}], " ", "/;", "\n", 
   RowBox[{
    RowBox[{"FreeQ", "[", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{"F", ",", "c", ",", "d", ",", "e", ",", "f", ",", "g"}], "}"}],
       ",", "x"}], "]"}], " ", "&&", " ", 
    RowBox[{"Not", "[", 
     RowBox[{"TrueQ", "[", "$UseGamma", "]"}], "]"}]}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, 
   3.4953222454097147`*^9, 3.4953339911454864`*^9, 3.550105533144896*^9, {
   3.550105568824946*^9, 3.550105588814974*^9}, {3.5501061488157578`*^9, 
   3.55010615022576*^9}, 3.5501081766885967`*^9, {3.550119158458813*^9, 
   3.55011916332882*^9}, {3.5528423255269203`*^9, 3.552842327626923*^9}, 
   3.552842478347134*^9, 3.5528425248471994`*^9, {3.5528427144374647`*^9, 
   3.552842720677473*^9}, 3.55284308956799*^9, 3.552843158838087*^9, {
   3.552843367428379*^9, 3.552843374158388*^9}, 3.552843567748659*^9, 
   3.5528436844988227`*^9, 3.552844217039568*^9, 3.5528442850496635`*^9, 
   3.5528443721397853`*^9, {3.552844416069847*^9, 3.5528444215698547`*^9}, {
   3.552844736730296*^9, 3.5528447392002993`*^9}, 3.552845987582047*^9, {
   3.5528472727238464`*^9, 3.552847283983862*^9}, 3.5529262049518795`*^9, 
   3.5795494007358875`*^9, 3.579549439885942*^9, 3.579549477845995*^9, 
   3.5795497511763783`*^9, 3.579549815856468*^9, {3.5795500144667463`*^9, 
   3.57955002416676*^9}, 3.5795586255463743`*^9, {3.579558672046439*^9, 
   3.579558702006481*^9}, 3.5795587408065352`*^9, {3.579559646230322*^9, 
   3.5795596674203515`*^9}, {3.614462272376218*^9, 3.6144622948785048`*^9}, {
   3.614468102097659*^9, 3.6144681072689543`*^9}, {3.6144763347651014`*^9, 
   3.6144763442406435`*^9}, {3.673811373516466*^9, 3.6738113775306954`*^9}, {
   3.6738118050231466`*^9, 3.6738118074162836`*^9}, {3.674585226152226*^9, 
   3.674585249522563*^9}, 3.674585390788643*^9, 3.74785985388492*^9},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9}]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t\t\t", 
  RowBox[{Cell[TextData[StyleBox["2:",
    FontFamily->"Arial",
    FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", " ", "x"}]}], ")"}], "m"], 
     SuperscriptBox["F", 
      RowBox[{"g", " ", 
       RowBox[{"(", 
        RowBox[{"e", "+", 
         RowBox[{"f", " ", "x"}]}], ")"}]}]], 
     RowBox[{"\[DifferentialD]", "x"}], "  ", 
     StyleBox["when",
      FontFamily->"Arial",
      FontWeight->"Plain"], 
     StyleBox["  ",
      FontFamily->"Arial",
      FontWeight->"Plain"], Cell[TextData[Cell[BoxData[
      RowBox[{"m", "\[Element]", "\[DoubleStruckCapitalZ]"}]]]], 
      "None"]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193422443984776`*^9, 3.5193422447260776`*^9}, 
   3.5193424488212357`*^9, {3.5193434708254333`*^9, 3.519343473103037*^9}, {
   3.5193435097631016`*^9, 3.51934353169674*^9}, {3.519345739319023*^9, 
   3.519345739599823*^9}, 3.5193462601259375`*^9, {3.5193475061060295`*^9, 
   3.5193475233908596`*^9}, {3.519351391880224*^9, 3.5193513931402254`*^9}, {
   3.519403773021052*^9, 3.5194037798564425`*^9}, 3.5194106793250647`*^9, {
   3.519537578341393*^9, 3.5195375792305946`*^9}, {3.5195376277778797`*^9, 
   3.51953767927357*^9}, {3.5197586260636144`*^9, 3.5197586260636144`*^9}, {
   3.51978053295651*^9, 3.519780542862528*^9}, {3.5197806526087203`*^9, 
   3.5197806657595434`*^9}, {3.5197807490168896`*^9, 3.5197807642581167`*^9}, 
   3.5197809004619555`*^9, {3.519782653514635*^9, 3.5197826623286505`*^9}, {
   3.5200207810164795`*^9, 3.520020785046485*^9}, {3.5213126833374643`*^9, 
   3.5213126833374643`*^9}, {3.521328235411002*^9, 3.521328235411002*^9}, {
   3.523315567893022*^9, 3.523315567893022*^9}, {3.5233157569332867`*^9, 
   3.5233157613032923`*^9}, {3.5323023198133526`*^9, 3.532302355723403*^9}, {
   3.5323026417638035`*^9, 3.5323026417638035`*^9}, {3.5323026980138817`*^9, 
   3.5323026980138817`*^9}, {3.532636287104328*^9, 3.532636287104328*^9}, {
   3.5330803686035347`*^9, 3.533080368863535*^9}, {3.533248909289961*^9, 
   3.5332489110299635`*^9}, {3.534962469565694*^9, 3.5349624910757236`*^9}, {
   3.5349625224057674`*^9, 3.534962522655768*^9}, {3.534965265519608*^9, 
   3.534965265519608*^9}, {3.5349668288168063`*^9, 3.5349668292668066`*^9}, {
   3.5349747358481646`*^9, 3.5349747358481646`*^9}, {3.534975457949176*^9, 
   3.534975457949176*^9}, {3.5368633963532934`*^9, 3.536863396571694*^9}, {
   3.541644322511304*^9, 3.5416443577513533`*^9}, {3.5416458269234104`*^9, 
   3.541645839633428*^9}, {3.5416460312236967`*^9, 3.541646031483697*^9}, 
   3.541646227783972*^9, {3.541647080335165*^9, 3.5416471077052035`*^9}, {
   3.541648997607849*^9, 3.5416489978278494`*^9}, 3.541649088407976*^9, {
   3.5417848711596775`*^9, 3.541784902429721*^9}, {3.541786145331461*^9, 
   3.5417861455914617`*^9}, {3.5417870037166634`*^9, 3.5417870039366636`*^9}, 
   3.5462979620242105`*^9, {3.546298264094633*^9, 3.546298265344635*^9}, {
   3.546299000235664*^9, 3.546299001505666*^9}, {3.546315119909564*^9, 
   3.5463151267659564`*^9}, {3.547227356319705*^9, 3.547227356319705*^9}, 
   3.547918559331044*^9, {3.547918610170952*^9, 3.5479186304411116`*^9}, 
   3.548537755517557*^9, 3.5485441629685287`*^9, 3.5485517520154133`*^9, {
   3.5488740727887154`*^9, 3.5488740727887154`*^9}, {3.548874206618903*^9, 
   3.548874206808903*^9}, {3.5488771226154857`*^9, 3.5488771332055006`*^9}, {
   3.548878208637006*^9, 3.548878208777006*^9}, {3.5488822719350433`*^9, 
   3.5488822719350433`*^9}, {3.5488837630471315`*^9, 
   3.5488837632171316`*^9}, {3.5488838279872217`*^9, 
   3.5488838281672225`*^9}, {3.548884313497902*^9, 3.548884313657902*^9}, {
   3.5501075528977237`*^9, 3.5501075530877237`*^9}, {3.5501076445378523`*^9, 
   3.5501076543578653`*^9}, {3.550108292398759*^9, 3.550108292398759*^9}, 
   3.552842347556951*^9, {3.5528427692075415`*^9, 3.552842769987542*^9}, 
   3.5528429787078342`*^9, 3.552843577428673*^9, {3.552843867749079*^9, 
   3.552843867749079*^9}, {3.552847864124674*^9, 3.5528478651946754`*^9}, 
   3.5529260847224684`*^9, {3.5529261552657924`*^9, 3.5529261558429937`*^9}, 
   3.5532224170147095`*^9, {3.573076016280301*^9, 3.573076018150304*^9}, 
   3.57307641528086*^9, 3.579548684724885*^9, 3.579548955485264*^9, 
   3.5795492995557456`*^9, 3.5795493587558284`*^9, 3.5795505718875265`*^9, 
   3.5795585823263135`*^9, 3.579558847036684*^9, {3.6144548216682034`*^9, 
   3.6144548435994577`*^9}, {3.614455056887657*^9, 3.614455082664132*^9}, 
   3.6144629875481234`*^9, 3.6144636931114793`*^9, {3.614463740399184*^9, 
   3.614463749313694*^9}, {3.61447150421881*^9, 3.614471504903849*^9}, {
   3.6144719762008057`*^9, 3.6144719768558435`*^9}, 3.6738114922772584`*^9, 
   3.6738141347573996`*^9, {3.673814193436756*^9, 3.673814193436756*^9}, {
   3.6738150026430397`*^9, 3.6738150042411313`*^9}, 3.67458302014705*^9, 
   3.6745848667296686`*^9},
 FontSize->12,
 FontWeight->"Bold"],

Cell[TextData[{
 "Rule: If  ",
 Cell[BoxData[
  RowBox[{"m", "\[Element]", "\[DoubleStruckCapitalZ]"}]]],
 ", then"
}], "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.495322033659418*^9, 3.495322068589467*^9}, {
   3.495332560963484*^9, 3.4953325798335104`*^9}, {3.495332835663869*^9, 
   3.4953328392438736`*^9}, {3.5488728293569746`*^9, 
   3.5488728860770545`*^9}, {3.54887301891724*^9, 3.548873026407251*^9}, {
   3.550107662367877*^9, 3.5501076662278824`*^9}, {3.5501082976187663`*^9, 
   3.5501082976187663`*^9}, {3.552842770737543*^9, 3.552842770737543*^9}, {
   3.55284386845908*^9, 3.55284386845908*^9}, {3.5529261510069847`*^9, 
   3.552926151646586*^9}, {3.6144551006191587`*^9, 3.614455100620159*^9}, {
   3.6144637590452504`*^9, 3.6144637590452504`*^9}, {3.6745791410211763`*^9, 
   3.674579142155241*^9}}],

Cell[BoxData[
 RowBox[{"\[Integral]", 
  RowBox[{
   SuperscriptBox[
    RowBox[{"(", 
     RowBox[{"c", "+", 
      RowBox[{"d", " ", "x"}]}], ")"}], "m"], 
   SuperscriptBox["F", 
    RowBox[{"g", " ", 
     RowBox[{"(", 
      RowBox[{"e", "+", 
       RowBox[{"f", " ", "x"}]}], ")"}]}]], 
   RowBox[{"\[DifferentialD]", 
    RowBox[{"x", "  ", "\[LongRightArrow]", "  ", 
     FractionBox[
      RowBox[{
       SuperscriptBox[
        RowBox[{"(", 
         RowBox[{"-", "d"}], ")"}], "m"], " ", 
       SuperscriptBox["F", 
        RowBox[{"g", " ", 
         RowBox[{"(", 
          RowBox[{"e", "-", 
           FractionBox[
            RowBox[{"c", " ", "f"}], "d"]}], ")"}]}]]}], 
      RowBox[{
       SuperscriptBox["f", 
        RowBox[{"m", "+", "1"}]], " ", 
       SuperscriptBox["g", 
        RowBox[{"m", "+", "1"}]], 
       SuperscriptBox[
        RowBox[{"Log", "[", "F", "]"}], 
        RowBox[{"m", "+", "1"}]]}]]}]}], 
   RowBox[{"Gamma", "[", 
    RowBox[{
     RowBox[{"m", "+", "1"}], ",", 
     RowBox[{
      RowBox[{"-", 
       FractionBox[
        RowBox[{"f", " ", "g", " ", 
         RowBox[{"Log", "[", "F", "]"}]}], "d"]}], " ", 
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", " ", "x"}]}], ")"}]}]}], "]"}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579723816*^9, 3.4796580261689425`*^9}, 
   3.479661191290163*^9, {3.479686720164157*^9, 3.479686720204214*^9}, {
   3.479686831554328*^9, 3.479686834819022*^9}, {3.4940970712485504`*^9, 
   3.4940971130286083`*^9}, 3.4940971455886545`*^9, 3.4953221152495327`*^9, 
   3.495322146199576*^9, {3.495322261579737*^9, 3.4953222704097495`*^9}, 
   3.4953325041434045`*^9, 3.4953325401034546`*^9, {3.4953327138936977`*^9, 
   3.4953327232437115`*^9}, {3.4953341614457245`*^9, 3.4953341867757597`*^9}, 
   3.4953344255060945`*^9, 3.4953345134762173`*^9, {3.5487965814803677`*^9, 
   3.548796592310383*^9}, 3.5501076914179173`*^9, {3.5501077413979874`*^9, 
   3.5501077435379906`*^9}, {3.5501077741680336`*^9, 
   3.5501078383981233`*^9}, {3.552842351006956*^9, 3.5528423527969584`*^9}, {
   3.5528427715475445`*^9, 3.5528427773575525`*^9}, 3.552842979787836*^9, {
   3.5528434309484673`*^9, 3.552843433878472*^9}, {3.5528435781286736`*^9, 
   3.5528435794086757`*^9}, 3.5528438716190844`*^9, {3.552844066309357*^9, 
   3.5528440891993895`*^9}, {3.55284488986051*^9, 3.5528448986205225`*^9}, {
   3.55284786829468*^9, 3.552847872034685*^9}, 3.573074904324668*^9, {
   3.573076093080408*^9, 3.573076112960436*^9}, 3.5730761549004946`*^9, {
   3.5795493001757464`*^9, 3.5795493017857485`*^9}, {3.57954935950583*^9, 
   3.5795493609858313`*^9}, {3.57954968853629*^9, 3.579549690576293*^9}, {
   3.579550368627242*^9, 3.579550421737317*^9}, {3.5795585829763145`*^9, 
   3.5795585851163177`*^9}, 3.5795588479766855`*^9, {3.579559962710765*^9, 
   3.5795599941608086`*^9}, 3.6144555141108093`*^9, {3.6144626260764484`*^9, 
   3.614462664097623*^9}, 3.6144628394586535`*^9, 3.6745830201550503`*^9, 
   3.674584867530714*^9, 3.674585042234707*^9, {3.67458542745774*^9, 
   3.6745854582054987`*^9}, 3.674585888351102*^9},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{"c_.", "+", 
        RowBox[{"d_.", "*", "x_"}]}], ")"}], "^", "m_."}], "*", 
     RowBox[{"F_", "^", 
      RowBox[{"(", 
       RowBox[{"g_.", "*", 
        RowBox[{"(", 
         RowBox[{"e_.", "+", 
          RowBox[{"f_.", "*", "x_"}]}], ")"}]}], ")"}]}]}], ",", "x_Symbol"}],
    "]"}], " ", ":=", "\n", "  ", 
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{"(", 
      RowBox[{"-", "d"}], ")"}], "^", "m"}], "*", 
    RowBox[{
     RowBox[{"F", "^", 
      RowBox[{"(", 
       RowBox[{"g", "*", 
        RowBox[{"(", 
         RowBox[{"e", "-", 
          RowBox[{"c", "*", 
           RowBox[{"f", "/", "d"}]}]}], ")"}]}], ")"}]}], "/", 
     RowBox[{"(", 
      RowBox[{
       RowBox[{"f", "^", 
        RowBox[{"(", 
         RowBox[{"m", "+", "1"}], ")"}]}], "*", 
       RowBox[{"g", "^", 
        RowBox[{"(", 
         RowBox[{"m", "+", "1"}], ")"}]}], "*", 
       RowBox[{
        RowBox[{"Log", "[", "F", "]"}], "^", 
        RowBox[{"(", 
         RowBox[{"m", "+", "1"}], ")"}]}]}], ")"}]}], "*", 
    RowBox[{"Gamma", "[", 
     RowBox[{
      RowBox[{"m", "+", "1"}], ",", 
      RowBox[{
       RowBox[{"-", "f"}], "*", "g", "*", 
       RowBox[{
        RowBox[{"Log", "[", "F", "]"}], "/", "d"}], "*", 
       RowBox[{"(", 
        RowBox[{"c", "+", 
         RowBox[{"d", "*", "x"}]}], ")"}]}]}], "]"}]}], " ", "/;", "\n", 
   RowBox[{
    RowBox[{"FreeQ", "[", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{"F", ",", "c", ",", "d", ",", "e", ",", "f", ",", "g"}], "}"}],
       ",", "x"}], "]"}], " ", "&&", " ", 
    RowBox[{"IntegerQ", "[", "m", "]"}]}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, 
   3.4953222454097147`*^9, 3.4953324633733473`*^9, 3.4953326632436275`*^9, 
   3.4953341350856876`*^9, 3.4953343185259447`*^9, 3.4953344851661777`*^9, {
   3.517189312453474*^9, 3.5171893153394785`*^9}, {3.5488177570189805`*^9, 
   3.5488178056090484`*^9}, {3.5488179603592653`*^9, 
   3.5488179667692738`*^9}, {3.548872978777184*^9, 3.5488729834871907`*^9}, {
   3.5501076015577917`*^9, 3.5501076155578117`*^9}, {3.55010805756843*^9, 
   3.5501080613384356`*^9}, {3.550108103288494*^9, 3.5501081340285373`*^9}, 
   3.550108317968795*^9, {3.5528423534869595`*^9, 3.5528423564869633`*^9}, 
   3.552842481077138*^9, 3.552842530397207*^9, {3.552842778357554*^9, 
   3.552842786147565*^9}, 3.5528430915179925`*^9, 3.5528431614780903`*^9, {
   3.5528434361084747`*^9, 3.5528434383484783`*^9}, {3.5528435800486765`*^9, 
   3.552843590448691*^9}, 3.5528437101288586`*^9, 3.5528437610289297`*^9, 
   3.5528442196895723`*^9, 3.552844287699667*^9, 3.5528443729997864`*^9, {
   3.5528444286798644`*^9, 3.5528444352498736`*^9}, {3.5528449057805324`*^9, 
   3.552844913520543*^9}, 3.5528459903920507`*^9, {3.552847915604746*^9, 
   3.5528479253747597`*^9}, {3.5529261308517494`*^9, 3.552926131662951*^9}, {
   3.5529277699426265`*^9, 3.552927773203032*^9}, {3.552928979508747*^9, 
   3.5529289963567767`*^9}, 3.57292916066512*^9, 3.5795494039558916`*^9, 
   3.579549485386006*^9, 3.579549540496083*^9, {3.5795497603363905`*^9, 
   3.5795497621463933`*^9}, {3.579550653637641*^9, 3.5795506902876925`*^9}, {
   3.5795586315263824`*^9, 3.5795586329263844`*^9}, {3.5795586757064443`*^9, 
   3.579558705096485*^9}, 3.579558746056543*^9, {3.57956011640098*^9, 
   3.579560179221068*^9}, 3.614463018916918*^9, 3.614463103708767*^9, {
   3.61446377058091*^9, 3.614463772952046*^9}, {3.614463944121836*^9, 
   3.6144639442878456`*^9}, {3.614471992841758*^9, 3.6144719971420035`*^9}, 
   3.6738141971479683`*^9, {3.674585497831765*^9, 3.6745855110725226`*^9}, 
   3.6745855720530105`*^9},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9}]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t\t", 
  RowBox[{Cell[TextData[StyleBox["2.",
    FontFamily->"Arial"]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
     SuperscriptBox["F", 
      RowBox[{"g", " ", 
       RowBox[{"(", 
        RowBox[{"e", "+", 
         RowBox[{"f", " ", "x"}]}], ")"}]}]], 
     RowBox[{"\[DifferentialD]", "x"}], "  ", 
     StyleBox["when",
      FontFamily->"Arial",
      FontWeight->"Plain"], 
     StyleBox["  ",
      FontFamily->"Arial",
      FontWeight->"Plain"], Cell[TextData[Cell[BoxData[
      RowBox[{"m", "\[NotElement]", "\[DoubleStruckCapitalZ]"}]]]], 
      "None"]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193415004828687`*^9, 3.5193415113404875`*^9}, {
   3.5193513965602303`*^9, 3.519351397420232*^9}, {3.5194037839416766`*^9, 
   3.5194037847117205`*^9}, 3.52105258275843*^9, 3.5210531741726685`*^9, {
   3.521057461672199*^9, 3.5210574760242243`*^9}, {3.521059881564049*^9, 
   3.521059886088057*^9}, {3.5213902158895197`*^9, 3.521390217543123*^9}, {
   3.521390827878595*^9, 3.521390829376197*^9}, {3.5214688792561245`*^9, 
   3.521468879443325*^9}, {3.523320201951167*^9, 3.523320202247567*^9}, {
   3.5288596764894824`*^9, 3.5288596768094826`*^9}, {3.529196266260353*^9, 
   3.5291962665099535`*^9}, {3.5291997050223927`*^9, 
   3.5291997052563934`*^9}, {3.529201346457276*^9, 3.529201347252877*^9}, {
   3.529285041073242*^9, 3.5292850412632427`*^9}, {3.529285215443486*^9, 
   3.529285215443486*^9}, {3.5295115356999426`*^9, 3.5295115433443794`*^9}, {
   3.529521412127842*^9, 3.529521414647986*^9}, {3.529529429975619*^9, 
   3.529529432955224*^9}, 3.5340074894011116`*^9, 3.5415537711400824`*^9, {
   3.5415538454742126`*^9, 3.54155384951462*^9}, {3.5417297197063265`*^9, 
   3.5417297252443366`*^9}, {3.546297494083555*^9, 3.5462974982435613`*^9}, {
   3.5462989134855423`*^9, 3.5462989153655453`*^9}, 3.548544146508506*^9, 
   3.5488738855784535`*^9, 3.548877085095433*^9, {3.548885157919084*^9, 
   3.5488851581390843`*^9}, {3.5501050720442505`*^9, 3.550105091724278*^9}, 
   3.552842303236889*^9, {3.5528426826774197`*^9, 3.552842683117421*^9}, 
   3.552842964957815*^9, 3.5528435586686463`*^9, {3.5528438442590466`*^9, 
   3.5528438442590466`*^9}, {3.5528470750635695`*^9, 
   3.5528470992336035`*^9}, {3.552925992292306*^9, 3.5529260013247223`*^9}, 
   3.553222405588056*^9, 3.5795486748848715`*^9, {3.579548905405194*^9, 
   3.579548907775197*^9}, 3.5795492772757144`*^9, 3.579549350275817*^9, 
   3.5795585715262985`*^9, 3.5795588167566414`*^9, {3.6738105630461097`*^9, 
   3.67381058140716*^9}, {3.6738146482907724`*^9, 3.6738146592433987`*^9}, {
   3.673814995511632*^9, 3.673814997280733*^9}, {3.6738150324417443`*^9, 
   3.6738150674267454`*^9}, 3.674583020164051*^9, 3.6745848390730867`*^9, 
   3.6745849406008935`*^9},
 FontSize->12,
 FontWeight->"Bold"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t\t\t", 
  RowBox[{Cell[TextData[StyleBox["1:",
    FontFamily->"Arial",
    FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     FractionBox[
      SuperscriptBox["F", 
       RowBox[{"g", " ", 
        RowBox[{"(", 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], ")"}]}]], 
      SqrtBox[
       RowBox[{"c", "+", 
        RowBox[{"d", " ", "x"}]}]]], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.4964645213514385`*^9, 
   3.5192470594155855`*^9, {3.519247288125906*^9, 3.5192473207459517`*^9}, {
   3.5192504337891226`*^9, 3.5192504436696877`*^9}, {3.519320736057986*^9, 
   3.5193207456364026`*^9}, {3.5193282921909866`*^9, 
   3.5193283252786446`*^9}, {3.5193283634207115`*^9, 
   3.5193283676639194`*^9}, {3.5193286295571795`*^9, 
   3.5193286611160345`*^9}, {3.519341560355774*^9, 3.5193415713849936`*^9}, {
   3.5193514022902384`*^9, 3.5193514024302387`*^9}, {3.5194037903190413`*^9, 
   3.519403790599057*^9}, 3.51941073361316*^9, 3.5196771044605494`*^9, {
   3.5233155369029784`*^9, 3.5233155369029784`*^9}, 3.5239204146855583`*^9, 
   3.5239297857543354`*^9, 3.5239298325544176`*^9, {3.5295294635000772`*^9, 
   3.5295294742328963`*^9}, 3.5415539341135683`*^9, 3.5417297751644244`*^9, 
   3.5472297585730677`*^9, 3.547229903793271*^9, 3.5488746103394685`*^9, {
   3.548875184400272*^9, 3.5488751845702724`*^9}, 3.548875455000651*^9, 
   3.548875553730789*^9, {3.5488756023708572`*^9, 3.5488756656309457`*^9}, {
   3.5488758328111796`*^9, 3.5488758338711815`*^9}, 3.548877640756211*^9, {
   3.5488823448151455`*^9, 3.5488823448151455`*^9}, {3.5488824670053167`*^9, 
   3.548882467155317*^9}, {3.5488838533072577`*^9, 3.5488838534972577`*^9}, {
   3.5488845420082216`*^9, 3.548884542168222*^9}, {3.5500924206128674`*^9, 
   3.55009242256287*^9}, {3.550096370849629*^9, 3.5500963834296465`*^9}, {
   3.550096732360135*^9, 3.550096749270159*^9}, {3.550105633415036*^9, 
   3.55010563565504*^9}, {3.5501057021951327`*^9, 3.5501057156951513`*^9}, 
   3.550108333068816*^9, 3.5501084525789833`*^9, 3.552842328336924*^9, {
   3.5528427213174744`*^9, 3.552842722047475*^9}, 3.552842972557826*^9, 
   3.5528435684686604`*^9, {3.552843862919072*^9, 3.552843862919072*^9}, 
   3.552847483954142*^9, 3.5529260737400494`*^9, 3.5529262264175177`*^9, 
   3.5532224130184813`*^9, 3.5795487495349755`*^9, 3.5795489536352615`*^9, 
   3.579549284015724*^9, 3.579549354625823*^9, 3.5795585762063046`*^9, 
   3.579558826636656*^9, 3.6144623529498262`*^9, 3.6144633214442215`*^9, {
   3.6144635080288935`*^9, 3.614463511716104*^9}, {3.6144667893265724`*^9, 
   3.6144667924887533`*^9}, 3.673811485027844*^9, {3.6738126073380365`*^9, 
   3.6738126074820447`*^9}, {3.673814898386077*^9, 3.6738149026283197`*^9}, {
   3.6738150553380537`*^9, 3.6738150585312366`*^9}, 3.673815376861444*^9, 
   3.674584868369762*^9, 3.674585043113757*^9},
 FontSize->12,
 FontWeight->"Bold"],

Cell["Derivation: Integration by substitution", "Subsubsection",
 CellChangeTimes->{
  3.49539151296132*^9, 3.495678787084699*^9, 3.4956789080048685`*^9, 
   3.4956790295250387`*^9, 3.495679129075178*^9, {3.495679335075466*^9, 
   3.4956793607555027`*^9}, {3.495679674485942*^9, 3.4956796884659615`*^9}, {
   3.5373086315425825`*^9, 3.537308666073558*^9}, {3.537308710886121*^9, 
   3.5373087144283233`*^9}}],

Cell[TextData[{
 "Basis: ",
 Cell[BoxData[
  RowBox[{
   FractionBox[
    SuperscriptBox["F", 
     RowBox[{"g", " ", 
      RowBox[{"(", 
       RowBox[{"e", "+", 
        RowBox[{"f", " ", "x"}]}], ")"}]}]], 
    SqrtBox[
     RowBox[{"c", "+", 
      RowBox[{"d", " ", "x"}]}]]], "\[Equal]", 
   RowBox[{
    FractionBox["2", "d"], " ", 
    RowBox[{"Subst", "[", 
     RowBox[{
      SuperscriptBox["F", 
       RowBox[{
        RowBox[{"g", " ", 
         RowBox[{"(", 
          RowBox[{"e", "-", 
           FractionBox[
            RowBox[{"c", " ", "f"}], "d"]}], ")"}]}], "+", 
        FractionBox[
         RowBox[{"f", " ", "g", " ", 
          SuperscriptBox["x", "2"]}], "d"]}]], ",", "x", ",", 
      SqrtBox[
       RowBox[{"c", "+", 
        RowBox[{"d", " ", "x"}]}]]}], "]"}], 
    RowBox[{
     SubscriptBox["\[PartialD]", "x"], 
     SqrtBox[
      RowBox[{"c", "+", 
       RowBox[{"d", " ", "x"}]}]]}]}]}]],
  CellChangeTimes->{{3.5373091114840336`*^9, 3.5373091693493433`*^9}}]
}], "Subsubsection",
 CellChangeTimes->{
  3.4953915070313115`*^9, 3.495840214754669*^9, {3.4958483177760134`*^9, 
   3.495848318886015*^9}, {3.495925911922002*^9, 3.495925920122013*^9}, 
   3.4961847242848682`*^9, 3.4961856693161907`*^9, 3.4961946839914446`*^9, {
   3.496625750314972*^9, 3.4966258381431265`*^9}, {3.496625899856835*^9, 
   3.496625904224842*^9}, 3.5373071703820086`*^9, {3.537307538265051*^9, 
   3.537307560169304*^9}, {3.537307696208084*^9, 3.537307705083592*^9}, 
   3.537308649487609*^9, {3.5373087432119694`*^9, 3.5373087449050665`*^9}, {
   3.537308778379981*^9, 3.537308783697285*^9}, {3.5373090613731675`*^9, 
   3.537309083016405*^9}, 3.5373091753356857`*^9, {3.537320793685435*^9, 
   3.5373208080942593`*^9}, {3.537320871459883*^9, 3.537320871459883*^9}, 
   3.537320903431712*^9, {3.537321013770023*^9, 3.5373210393994884`*^9}, {
   3.5373211211661654`*^9, 3.5373211211661654`*^9}, {3.548876327384372*^9, 
   3.5488763552444115`*^9}, {3.548876396394469*^9, 3.548876396394469*^9}, {
   3.5488766723548555`*^9, 3.548876731474938*^9}, {3.548876765184985*^9, 
   3.548876792755024*^9}, {3.5488768405350904`*^9, 3.548876842235093*^9}, {
   3.5488768746251383`*^9, 3.548876883725151*^9}, {3.5488776490862226`*^9, 
   3.5488776490862226`*^9}, {3.548877842946494*^9, 3.5488778468764997`*^9}, {
   3.5488779108665895`*^9, 3.5488779108665895`*^9}, {3.5500964027296734`*^9, 
   3.5500964035696745`*^9}, {3.550096463549759*^9, 3.550096479759782*^9}, {
   3.5500965547198863`*^9, 3.55009658576993*^9}, {3.5500967563501687`*^9, 
   3.5500967563501687`*^9}, {3.550105863205358*^9, 3.550105939865465*^9}, {
   3.550106019495577*^9, 3.5501060427056093`*^9}, {3.5528423293369255`*^9, 
   3.552842330246927*^9}, {3.5528427268474817`*^9, 3.552842729697486*^9}, {
   3.552842973517827*^9, 3.552842973517827*^9}, {3.552843392498414*^9, 
   3.552843398008422*^9}, {3.5528435691786613`*^9, 3.5528435700986624`*^9}, {
   3.5528446136901236`*^9, 3.5528446136901236`*^9}, {3.5528447732103467`*^9, 
   3.5528447894803696`*^9}, {3.5528475110141797`*^9, 
   3.5528475185941906`*^9}, {3.5528475675242586`*^9, 
   3.5528475675242586`*^9}, {3.552847627284343*^9, 3.552847627284343*^9}, {
   3.579549284935725*^9, 3.5795492861257267`*^9}, {3.5795493549358234`*^9, 
   3.579549355455824*^9}, {3.5795496847362847`*^9, 3.5795496847362847`*^9}, {
   3.5795501063568754`*^9, 3.5795501063568754`*^9}, {3.579550148956935*^9, 
   3.579550148956935*^9}, {3.5795585765363054`*^9, 3.5795585773663063`*^9}, {
   3.5795588389066725`*^9, 3.5795588389066725`*^9}, {3.579559739210452*^9, 
   3.579559763550486*^9}, {3.6745850438758*^9, 3.674585044900859*^9}, {
   3.6745856512755413`*^9, 3.674585657095875*^9}, {3.6745857432328014`*^9, 
   3.6745857432328014`*^9}}],

Cell["Rule:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.495322033659418*^9, 3.495322068589467*^9}, {
   3.495332560963484*^9, 3.4953325798335104`*^9}, {3.495332835663869*^9, 
   3.4953328392438736`*^9}, {3.5488728293569746`*^9, 
   3.5488728860770545`*^9}, {3.54887301891724*^9, 3.548873026407251*^9}, {
   3.550107662367877*^9, 3.5501076662278824`*^9}, {3.5501082976187663`*^9, 
   3.5501082976187663`*^9}, {3.552842770737543*^9, 3.552842770737543*^9}, {
   3.55284386845908*^9, 3.55284386845908*^9}, {3.5529261510069847`*^9, 
   3.552926151646586*^9}, {3.6144551006191587`*^9, 3.614455100620159*^9}, {
   3.614462269426049*^9, 3.614462269426049*^9}, 3.6738153799546213`*^9}],

Cell[BoxData[
 RowBox[{"\[Integral]", 
  RowBox[{
   FractionBox[
    SuperscriptBox["F", 
     RowBox[{"g", " ", 
      RowBox[{"(", 
       RowBox[{"e", "+", 
        RowBox[{"f", " ", "x"}]}], ")"}]}]], 
    SqrtBox[
     RowBox[{"c", "+", 
      RowBox[{"d", " ", "x"}]}]]], 
   RowBox[{"\[DifferentialD]", 
    RowBox[{"x", "  ", "\[LongRightArrow]", "  ", 
     FractionBox["2", "d"]}]}], " ", 
   RowBox[{"Subst", "[", 
    RowBox[{
     RowBox[{"\[Integral]", " ", 
      RowBox[{
       SuperscriptBox["F", 
        RowBox[{
         RowBox[{"g", " ", 
          RowBox[{"(", 
           RowBox[{"e", "-", 
            FractionBox[
             RowBox[{"c", " ", "f"}], "d"]}], ")"}]}], "+", 
         FractionBox[
          RowBox[{"f", " ", "g", " ", 
           SuperscriptBox["x", "2"]}], "d"]}]], 
       RowBox[{"\[DifferentialD]", "x"}]}]}], ",", "x", ",", 
     SqrtBox[
      RowBox[{"c", "+", 
       RowBox[{"d", " ", "x"}]}]]}], "]"}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579732027807`*^9, 3.479658002474872*^9}, {
   3.4796580599975853`*^9, 3.4796580609489536`*^9}, {3.479658295195784*^9, 
   3.47965829684816*^9}, 3.4796611938939075`*^9, 3.4796613924794593`*^9, {
   3.479661443342597*^9, 3.479661446647349*^9}, {3.479661576554146*^9, 
   3.479661589302477*^9}, 3.479661847553824*^9, {3.4796622294729967`*^9, 
   3.4796622450654173`*^9}, {3.479686720424531*^9, 3.4796867204345455`*^9}, {
   3.4796874646646957`*^9, 3.4796874745589232`*^9}, {3.492826573513727*^9, 
   3.492826578630536*^9}, {3.4928267070863624`*^9, 3.492826707460763*^9}, 
   3.4928267903497095`*^9, {3.492829799151373*^9, 3.4928299617216005`*^9}, {
   3.4940972077187414`*^9, 3.4940972347887793`*^9}, {3.496441851501421*^9, 
   3.496441872881451*^9}, {3.496539603658884*^9, 3.4965396173789034`*^9}, {
   3.4965397325490646`*^9, 3.496539746369084*^9}, 3.496540440080055*^9, 
   3.496540779670531*^9, {3.4965409101807137`*^9, 3.496540937820752*^9}, {
   3.5188417573250003`*^9, 3.518841761725007*^9}, {3.5188807967748365`*^9, 
   3.518880816014863*^9}, 3.518880850324911*^9, {3.5188808923449697`*^9, 
   3.51888094941505*^9}, {3.518881122135292*^9, 3.518881146375326*^9}, {
   3.518881187185383*^9, 3.5188811969353967`*^9}, {3.518881718766127*^9, 
   3.5188817492661695`*^9}, {3.521260910274908*^9, 3.5212609173593135`*^9}, {
   3.527385186264098*^9, 3.527385194266912*^9}, 3.528405628250476*^9, {
   3.5284060034610014`*^9, 3.528406005151004*^9}, {3.5355006982553825`*^9, 
   3.5355007023453884`*^9}, {3.535500739845441*^9, 3.535500757605466*^9}, 
   3.535501028975846*^9, {3.535509140667202*^9, 3.535509158837228*^9}, 
   3.536079676970666*^9, {3.536079753487042*^9, 3.5360797729031525`*^9}, {
   3.536079815099566*^9, 3.536079843946216*^9}, 3.5366885088761244`*^9, {
   3.536713447408527*^9, 3.5367134613263226`*^9}, {3.5373073596468344`*^9, 
   3.5373073995031137`*^9}, {3.5373074496549826`*^9, 3.537307491467374*^9}, 
   3.5373203808248205`*^9, {3.537320418457973*^9, 3.53732042085011*^9}, {
   3.5373207395503387`*^9, 3.5373207456366863`*^9}, 3.537320962173072*^9, {
   3.5373209992561927`*^9, 3.537321003418431*^9}, {3.5373210608827176`*^9, 
   3.5373210700082397`*^9}, {3.5488764455045376`*^9, 
   3.5488764765145807`*^9}, {3.548876508524626*^9, 3.5488765226146455`*^9}, 
   3.548876572294715*^9, {3.5488766198747816`*^9, 3.548876650804825*^9}, 
   3.5488776435662146`*^9, 3.5488779280766134`*^9, {3.550096390779657*^9, 
   3.5500964145596905`*^9}, {3.5500965205798388`*^9, 
   3.5500965255298457`*^9}, {3.5501057658552217`*^9, 3.5501057936752605`*^9}, 
   3.55010585027534*^9, {3.552842333916932*^9, 3.552842334576933*^9}, {
   3.552842731707489*^9, 3.5528427391974993`*^9}, 3.5528429742278285`*^9, {
   3.552843402738428*^9, 3.5528434056484327`*^9}, {3.5528435706986637`*^9, 
   3.552843571248664*^9}, 3.552844617470129*^9, {3.5528447466503096`*^9, 
   3.552844766740338*^9}, {3.552847486314145*^9, 3.552847497794161*^9}, {
   3.552847655624382*^9, 3.552847662524392*^9}, 3.5795487951850395`*^9, {
   3.579549286715728*^9, 3.5795492873957286`*^9}, {3.579549355715824*^9, 
   3.5795493560658245`*^9}, 3.579549686446287*^9, {3.579550157006946*^9, 
   3.5795501775169744`*^9}, {3.579558577936307*^9, 3.579558578396308*^9}, 
   3.5795588414966764`*^9, {3.579559695860391*^9, 3.579559706450406*^9}, {
   3.5795597828505135`*^9, 3.5795597886605215`*^9}, 3.6745848691078043`*^9, {
   3.674585045491893*^9, 3.674585045979921*^9}, {3.6745856222878838`*^9, 
   3.6745856393468595`*^9}, 3.6745857619628725`*^9},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{"F_", "^", 
      RowBox[{"(", 
       RowBox[{"g_.", "*", 
        RowBox[{"(", 
         RowBox[{"e_.", "+", 
          RowBox[{"f_.", "*", "x_"}]}], ")"}]}], ")"}]}], "/", 
     RowBox[{"Sqrt", "[", 
      RowBox[{"c_.", "+", 
       RowBox[{"d_.", "*", "x_"}]}], "]"}]}], ",", "x_Symbol"}], "]"}], " ", ":=",
   "\n", "  ", 
  RowBox[{
   RowBox[{
    RowBox[{"2", "/", "d"}], "*", 
    RowBox[{"Subst", "[", 
     RowBox[{
      RowBox[{"Int", "[", 
       RowBox[{
        RowBox[{"F", "^", 
         RowBox[{"(", 
          RowBox[{
           RowBox[{"g", "*", 
            RowBox[{"(", 
             RowBox[{"e", "-", 
              RowBox[{"c", "*", 
               RowBox[{"f", "/", "d"}]}]}], ")"}]}], "+", 
           RowBox[{"f", "*", "g", "*", 
            RowBox[{
             RowBox[{"x", "^", "2"}], "/", "d"}]}]}], ")"}]}], ",", "x"}], 
       "]"}], ",", "x", ",", 
      RowBox[{"Sqrt", "[", 
       RowBox[{"c", "+", 
        RowBox[{"d", "*", "x"}]}], "]"}]}], "]"}]}], " ", "/;", "\n", 
   RowBox[{
    RowBox[{"FreeQ", "[", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{"F", ",", "c", ",", "d", ",", "e", ",", "f", ",", "g"}], "}"}],
       ",", "x"}], "]"}], " ", "&&", " ", 
    RowBox[{"Not", "[", 
     RowBox[{"TrueQ", "[", "$UseGamma", "]"}], "]"}]}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, 
   3.4953222454097147`*^9, {3.548875878551244*^9, 3.548875923921307*^9}, {
   3.548876935125223*^9, 3.5488770026553173`*^9}, {3.548877965796666*^9, 
   3.5488779889566984`*^9}, 3.548882401405225*^9, {3.5500240168995347`*^9, 
   3.5500240186695375`*^9}, {3.5500242260398273`*^9, 
   3.5500242312998347`*^9}, {3.5500924517929115`*^9, 
   3.5500924560429173`*^9}, {3.550096676340057*^9, 3.550096707950101*^9}, {
   3.55009684294029*^9, 3.5500968710203295`*^9}, {3.5500974599911537`*^9, 
   3.5500974632511587`*^9}, {3.550106076205656*^9, 3.5501060856256695`*^9}, {
   3.5501061257357254`*^9, 3.5501061604957743`*^9}, 3.550108183178606*^9, {
   3.5528423356469345`*^9, 3.5528423369469357`*^9}, 3.5528424792771354`*^9, 
   3.552842527027202*^9, {3.5528427399175005`*^9, 3.5528427506575155`*^9}, 
   3.5528430901479907`*^9, 3.5528431601180887`*^9, {3.5528434093784375`*^9, 
   3.5528434123284416`*^9}, 3.5528435721286654`*^9, 3.5528436853988237`*^9, 
   3.55284421858957*^9, 3.5528442857696643`*^9, 3.552844620890134*^9, {
   3.5528447985503826`*^9, 3.5528448112004004`*^9}, 3.552845988872049*^9, {
   3.5528476818944187`*^9, 3.5528477179344697`*^9}, 3.5529262333595295`*^9, 
   3.5795494014058886`*^9, 3.5795494785859966`*^9, 3.5795495343660746`*^9, 
   3.5795497522463794`*^9, {3.5795501943069983`*^9, 3.579550211797023*^9}, 
   3.579558626206375*^9, {3.5795586727364397`*^9, 3.5795587025964823`*^9}, 
   3.579558741686537*^9, {3.579559810060551*^9, 3.5795598277205763`*^9}, {
   3.614462342840248*^9, 3.6144623453463917`*^9}, {3.614468130371276*^9, 
   3.6144681362066097`*^9}, {3.6144770745704155`*^9, 
   3.6144770830979033`*^9}, {3.674585779209859*^9, 3.674585790109483*^9}, 
   3.674585826943589*^9, 3.7478598562380548`*^9},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9}]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t\t\t", 
  RowBox[{Cell[TextData[StyleBox["2:",
    FontFamily->"Arial",
    FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", " ", "x"}]}], ")"}], "m"], 
     SuperscriptBox["F", 
      RowBox[{"g", " ", 
       RowBox[{"(", 
        RowBox[{"e", "+", 
         RowBox[{"f", " ", "x"}]}], ")"}]}]], 
     RowBox[{"\[DifferentialD]", "x"}], "  ", 
     StyleBox["when",
      FontFamily->"Arial",
      FontWeight->"Plain"], 
     StyleBox["  ",
      FontFamily->"Arial",
      FontWeight->"Plain"], Cell[TextData[Cell[BoxData[
      RowBox[{"m", "\[NotElement]", "\[DoubleStruckCapitalZ]"}]]]], 
      "None"]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193422443984776`*^9, 3.5193422447260776`*^9}, 
   3.5193424488212357`*^9, {3.5193434708254333`*^9, 3.519343473103037*^9}, {
   3.5193435097631016`*^9, 3.51934353169674*^9}, {3.519345739319023*^9, 
   3.519345739599823*^9}, 3.5193462601259375`*^9, {3.5193475061060295`*^9, 
   3.5193475233908596`*^9}, {3.519351391880224*^9, 3.5193513931402254`*^9}, {
   3.519403773021052*^9, 3.5194037798564425`*^9}, 3.5194106793250647`*^9, {
   3.519537578341393*^9, 3.5195375792305946`*^9}, {3.5195376277778797`*^9, 
   3.51953767927357*^9}, {3.5197586260636144`*^9, 3.5197586260636144`*^9}, {
   3.51978053295651*^9, 3.519780542862528*^9}, {3.5197806526087203`*^9, 
   3.5197806657595434`*^9}, {3.5197807490168896`*^9, 3.5197807642581167`*^9}, 
   3.5197809004619555`*^9, {3.519782653514635*^9, 3.5197826623286505`*^9}, {
   3.5200207810164795`*^9, 3.520020785046485*^9}, {3.5213126833374643`*^9, 
   3.5213126833374643`*^9}, {3.521328235411002*^9, 3.521328235411002*^9}, {
   3.523315567893022*^9, 3.523315567893022*^9}, {3.5233157569332867`*^9, 
   3.5233157613032923`*^9}, {3.5323023198133526`*^9, 3.532302355723403*^9}, {
   3.5323026417638035`*^9, 3.5323026417638035`*^9}, {3.5323026980138817`*^9, 
   3.5323026980138817`*^9}, {3.532636287104328*^9, 3.532636287104328*^9}, {
   3.5330803686035347`*^9, 3.533080368863535*^9}, {3.533248909289961*^9, 
   3.5332489110299635`*^9}, {3.534962469565694*^9, 3.5349624910757236`*^9}, {
   3.5349625224057674`*^9, 3.534962522655768*^9}, {3.534965265519608*^9, 
   3.534965265519608*^9}, {3.5349668288168063`*^9, 3.5349668292668066`*^9}, {
   3.5349747358481646`*^9, 3.5349747358481646`*^9}, {3.534975457949176*^9, 
   3.534975457949176*^9}, {3.5368633963532934`*^9, 3.536863396571694*^9}, {
   3.541644322511304*^9, 3.5416443577513533`*^9}, {3.5416458269234104`*^9, 
   3.541645839633428*^9}, {3.5416460312236967`*^9, 3.541646031483697*^9}, 
   3.541646227783972*^9, {3.541647080335165*^9, 3.5416471077052035`*^9}, {
   3.541648997607849*^9, 3.5416489978278494`*^9}, 3.541649088407976*^9, {
   3.5417848711596775`*^9, 3.541784902429721*^9}, {3.541786145331461*^9, 
   3.5417861455914617`*^9}, {3.5417870037166634`*^9, 3.5417870039366636`*^9}, 
   3.5462979620242105`*^9, {3.546298264094633*^9, 3.546298265344635*^9}, {
   3.546299000235664*^9, 3.546299001505666*^9}, {3.546315119909564*^9, 
   3.5463151267659564`*^9}, {3.547227356319705*^9, 3.547227356319705*^9}, 
   3.547918559331044*^9, {3.547918610170952*^9, 3.5479186304411116`*^9}, 
   3.548537755517557*^9, 3.5485441629685287`*^9, 3.5485517520154133`*^9, {
   3.5488740727887154`*^9, 3.5488740727887154`*^9}, {3.548874206618903*^9, 
   3.548874206808903*^9}, {3.5488771226154857`*^9, 3.5488771332055006`*^9}, {
   3.548878208637006*^9, 3.548878208777006*^9}, {3.5488822719350433`*^9, 
   3.5488822719350433`*^9}, {3.5488837630471315`*^9, 
   3.5488837632171316`*^9}, {3.5488838279872217`*^9, 
   3.5488838281672225`*^9}, {3.548884313497902*^9, 3.548884313657902*^9}, {
   3.5501075528977237`*^9, 3.5501075530877237`*^9}, {3.5501076445378523`*^9, 
   3.5501076543578653`*^9}, {3.550108292398759*^9, 3.550108292398759*^9}, 
   3.552842347556951*^9, {3.5528427692075415`*^9, 3.552842769987542*^9}, 
   3.5528429787078342`*^9, 3.552843577428673*^9, {3.552843867749079*^9, 
   3.552843867749079*^9}, {3.552847864124674*^9, 3.5528478651946754`*^9}, 
   3.5529260847224684`*^9, {3.5529261552657924`*^9, 3.5529261558429937`*^9}, 
   3.5532224170147095`*^9, 3.5730760207603073`*^9, 3.5795486867148876`*^9, 
   3.5795489561452646`*^9, 3.57954930251575*^9, 3.579549361785833*^9, 
   3.5795585861063185`*^9, 3.579558849036687*^9, {3.6144549903758535`*^9, 
   3.6144549903758535`*^9}, {3.614464077817483*^9, 3.6144640804116316`*^9}, 
   3.673811506147052*^9, {3.6738150564371166`*^9, 3.6738150704109163`*^9}, 
   3.6745830201720514`*^9, 3.674584869937852*^9},
 FontSize->12,
 FontWeight->"Bold"],

Cell[TextData[{
 "Rule: If  ",
 Cell[BoxData[
  RowBox[{
   RowBox[{"2", "m"}], "\[NotElement]", "\[DoubleStruckCapitalZ]"}]]],
 ", then"
}], "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.495322033659418*^9, 3.495322068589467*^9}, {
   3.495332560963484*^9, 3.4953325798335104`*^9}, {3.495332835663869*^9, 
   3.4953328392438736`*^9}, {3.5488728293569746`*^9, 
   3.5488728860770545`*^9}, {3.54887301891724*^9, 3.548873026407251*^9}, {
   3.550107662367877*^9, 3.5501076662278824`*^9}, {3.5501082976187663`*^9, 
   3.5501082976187663`*^9}, {3.552842770737543*^9, 3.552842770737543*^9}, {
   3.55284386845908*^9, 3.55284386845908*^9}, {3.5529261510069847`*^9, 
   3.552926151646586*^9}}],

Cell[BoxData[
 RowBox[{"\[Integral]", 
  RowBox[{
   SuperscriptBox[
    RowBox[{"(", 
     RowBox[{"c", "+", 
      RowBox[{"d", " ", "x"}]}], ")"}], "m"], 
   SuperscriptBox["F", 
    RowBox[{"g", " ", 
     RowBox[{"(", 
      RowBox[{"e", "+", 
       RowBox[{"f", " ", "x"}]}], ")"}]}]], 
   RowBox[{"\[DifferentialD]", 
    RowBox[{"x", "  ", "\[LongRightArrow]", "  ", 
     RowBox[{"-", 
      FractionBox[
       RowBox[{
        SuperscriptBox["F", 
         RowBox[{"g", " ", 
          RowBox[{"(", 
           RowBox[{"e", "-", 
            FractionBox[
             RowBox[{"c", " ", "f"}], "d"]}], ")"}]}]], " ", 
        SuperscriptBox[
         RowBox[{"(", 
          RowBox[{"c", "+", 
           RowBox[{"d", " ", "x"}]}], ")"}], 
         RowBox[{"FracPart", "[", "m", "]"}]]}], 
       RowBox[{"d", " ", 
        SuperscriptBox[
         RowBox[{"(", 
          RowBox[{"-", 
           FractionBox[
            RowBox[{"f", " ", "g", " ", 
             RowBox[{"Log", "[", "F", "]"}]}], "d"]}], ")"}], 
         RowBox[{
          RowBox[{"IntPart", "[", "m", "]"}], "+", "1"}]], " ", 
        SuperscriptBox[
         RowBox[{"(", 
          RowBox[{"-", 
           FractionBox[
            RowBox[{"f", " ", "g", " ", 
             RowBox[{"Log", "[", "F", "]"}], " ", 
             RowBox[{"(", 
              RowBox[{"c", "+", 
               RowBox[{"d", " ", "x"}]}], ")"}]}], "d"]}], ")"}], 
         RowBox[{"FracPart", "[", "m", "]"}]]}]]}]}]}], 
   RowBox[{"Gamma", "[", 
    RowBox[{
     RowBox[{"m", "+", "1"}], ",", 
     RowBox[{
      RowBox[{"-", 
       FractionBox[
        RowBox[{"f", " ", "g", " ", 
         RowBox[{"Log", "[", "F", "]"}]}], "d"]}], " ", 
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", " ", "x"}]}], ")"}]}]}], "]"}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579723816*^9, 3.4796580261689425`*^9}, 
   3.479661191290163*^9, {3.479686720164157*^9, 3.479686720204214*^9}, {
   3.479686831554328*^9, 3.479686834819022*^9}, {3.4940970712485504`*^9, 
   3.4940971130286083`*^9}, 3.4940971455886545`*^9, 3.4953221152495327`*^9, 
   3.495322146199576*^9, {3.495322261579737*^9, 3.4953222704097495`*^9}, 
   3.4953325041434045`*^9, 3.4953325401034546`*^9, {3.4953327138936977`*^9, 
   3.4953327232437115`*^9}, {3.4953341614457245`*^9, 3.4953341867757597`*^9}, 
   3.4953344255060945`*^9, 3.4953345134762173`*^9, {3.5487965814803677`*^9, 
   3.548796592310383*^9}, 3.5501076914179173`*^9, {3.5501077413979874`*^9, 
   3.5501077435379906`*^9}, {3.5501077741680336`*^9, 
   3.5501078383981233`*^9}, {3.552842351006956*^9, 3.5528423527969584`*^9}, {
   3.5528427715475445`*^9, 3.5528427773575525`*^9}, 3.552842979787836*^9, {
   3.5528434309484673`*^9, 3.552843433878472*^9}, {3.5528435781286736`*^9, 
   3.5528435794086757`*^9}, 3.5528438716190844`*^9, {3.552844066309357*^9, 
   3.5528440891993895`*^9}, {3.55284488986051*^9, 3.5528448986205225`*^9}, {
   3.55284786829468*^9, 3.552847872034685*^9}, {3.5730759480602055`*^9, 
   3.573075960960223*^9}, 3.57954880272505*^9, {3.5795493030857506`*^9, 
   3.579549304735753*^9}, {3.5795493621658335`*^9, 3.5795493645358367`*^9}, {
   3.5795496917362947`*^9, 3.5795496938362975`*^9}, {3.579550596527561*^9, 
   3.57955063804762*^9}, {3.5795585875363207`*^9, 3.579558591096326*^9}, 
   3.5795588496866875`*^9, {3.5795600871209393`*^9, 3.5795600989209557`*^9}, {
   3.673813331128435*^9, 3.6738134000943794`*^9}, {3.673813432173214*^9, 
   3.673813501766195*^9}, {3.673813552929121*^9, 3.6738135639387507`*^9}, 
   3.673813622477099*^9, 3.673813691449044*^9, 3.674583020180052*^9, 
   3.674584871628948*^9, {3.674585047099985*^9, 3.6745850481640453`*^9}, {
   3.6745858760143957`*^9, 3.674585918793843*^9}},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{"c_.", "+", 
        RowBox[{"d_.", "*", "x_"}]}], ")"}], "^", "m_"}], "*", 
     RowBox[{"F_", "^", 
      RowBox[{"(", 
       RowBox[{"g_.", "*", 
        RowBox[{"(", 
         RowBox[{"e_.", "+", 
          RowBox[{"f_.", "*", "x_"}]}], ")"}]}], ")"}]}]}], ",", "x_Symbol"}],
    "]"}], " ", ":=", "\n", "  ", 
  RowBox[{
   RowBox[{
    RowBox[{"-", 
     RowBox[{"F", "^", 
      RowBox[{"(", 
       RowBox[{"g", "*", 
        RowBox[{"(", 
         RowBox[{"e", "-", 
          RowBox[{"c", "*", 
           RowBox[{"f", "/", "d"}]}]}], ")"}]}], ")"}]}]}], "*", 
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", "*", "x"}]}], ")"}], "^", 
      RowBox[{"FracPart", "[", "m", "]"}]}], "/", 
     RowBox[{"(", 
      RowBox[{"d", "*", 
       RowBox[{
        RowBox[{"(", 
         RowBox[{
          RowBox[{"-", "f"}], "*", "g", "*", 
          RowBox[{
           RowBox[{"Log", "[", "F", "]"}], "/", "d"}]}], ")"}], "^", 
        RowBox[{"(", 
         RowBox[{
          RowBox[{"IntPart", "[", "m", "]"}], "+", "1"}], ")"}]}], "*", 
       RowBox[{
        RowBox[{"(", 
         RowBox[{
          RowBox[{"-", "f"}], "*", "g", "*", 
          RowBox[{"Log", "[", "F", "]"}], "*", 
          RowBox[{
           RowBox[{"(", 
            RowBox[{"c", "+", 
             RowBox[{"d", "*", "x"}]}], ")"}], "/", "d"}]}], ")"}], "^", 
        RowBox[{"FracPart", "[", "m", "]"}]}]}], ")"}]}], "*", "\n", "    ", 
    RowBox[{"Gamma", "[", 
     RowBox[{
      RowBox[{"m", "+", "1"}], ",", 
      RowBox[{
       RowBox[{"(", 
        RowBox[{
         RowBox[{"-", "f"}], "*", "g", "*", 
         RowBox[{
          RowBox[{"Log", "[", "F", "]"}], "/", "d"}]}], ")"}], "*", 
       RowBox[{"(", 
        RowBox[{"c", "+", 
         RowBox[{"d", "*", "x"}]}], ")"}]}]}], "]"}]}], " ", "/;", "\n", 
   RowBox[{
    RowBox[{"FreeQ", "[", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{
       "F", ",", "c", ",", "d", ",", "e", ",", "f", ",", "g", ",", "m"}], 
       "}"}], ",", "x"}], "]"}], " ", "&&", " ", 
    RowBox[{"Not", "[", 
     RowBox[{"IntegerQ", "[", "m", "]"}], "]"}]}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, 
   3.4953222454097147`*^9, 3.4953324633733473`*^9, 3.4953326632436275`*^9, 
   3.4953341350856876`*^9, 3.4953343185259447`*^9, 3.4953344851661777`*^9, {
   3.517189312453474*^9, 3.5171893153394785`*^9}, {3.5488177570189805`*^9, 
   3.5488178056090484`*^9}, {3.5488179603592653`*^9, 
   3.5488179667692738`*^9}, {3.548872978777184*^9, 3.5488729834871907`*^9}, {
   3.5501076015577917`*^9, 3.5501076155578117`*^9}, {3.55010805756843*^9, 
   3.5501080613384356`*^9}, {3.550108103288494*^9, 3.5501081340285373`*^9}, 
   3.550108317968795*^9, {3.5528423534869595`*^9, 3.5528423564869633`*^9}, 
   3.552842481077138*^9, 3.552842530397207*^9, {3.552842778357554*^9, 
   3.552842786147565*^9}, 3.5528430915179925`*^9, 3.5528431614780903`*^9, {
   3.5528434361084747`*^9, 3.5528434383484783`*^9}, {3.5528435800486765`*^9, 
   3.552843590448691*^9}, 3.5528437101288586`*^9, 3.5528437610289297`*^9, 
   3.5528442196895723`*^9, 3.552844287699667*^9, 3.5528443729997864`*^9, {
   3.5528444286798644`*^9, 3.5528444352498736`*^9}, {3.5528449057805324`*^9, 
   3.552844913520543*^9}, 3.5528459903920507`*^9, {3.552847915604746*^9, 
   3.5528479253747597`*^9}, {3.5529261308517494`*^9, 3.552926131662951*^9}, {
   3.5529277699426265`*^9, 3.552927773203032*^9}, {3.552928979508747*^9, 
   3.5529289963567767`*^9}, 3.57292916066512*^9, 3.5795494039558916`*^9, 
   3.579549485386006*^9, 3.579549540496083*^9, {3.5795497603363905`*^9, 
   3.5795497621463933`*^9}, {3.579550653637641*^9, 3.5795506902876925`*^9}, {
   3.5795586315263824`*^9, 3.5795586329263844`*^9}, {3.5795586757064443`*^9, 
   3.579558705096485*^9}, 3.579558746056543*^9, {3.57956011640098*^9, 
   3.579560179221068*^9}, {3.61446408702901*^9, 3.614464100525782*^9}, 
   3.6144666073451643`*^9, {3.614467990594281*^9, 3.6144680000928245`*^9}, 
   3.6738131945946255`*^9, {3.6738150910840983`*^9, 3.673815095968378*^9}, 
   3.673818202724074*^9, {3.67381825754921*^9, 3.673818261208419*^9}, {
   3.674586025861967*^9, 3.6745860389097133`*^9}, {3.6745861652329383`*^9, 
   3.6745861697581973`*^9}},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9}]
}, Closed]]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t", 
  RowBox[{Cell[TextData[StyleBox["4:",
    FontFamily->"Arial",
    FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"b", " ", 
        SuperscriptBox["F", 
         RowBox[{"g", " ", 
          RowBox[{"(", 
           RowBox[{"e", "+", 
            RowBox[{"f", " ", "x"}]}], ")"}]}]]}], ")"}], "n"], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.4964645213514385`*^9, 
   3.5192470594155855`*^9, {3.519247288125906*^9, 3.5192473207459517`*^9}, {
   3.5192504337891226`*^9, 3.5192504436696877`*^9}, {3.519320736057986*^9, 
   3.5193207456364026`*^9}, {3.5193282921909866`*^9, 
   3.5193283252786446`*^9}, {3.5193283634207115`*^9, 
   3.5193283676639194`*^9}, {3.5193286295571795`*^9, 
   3.5193286611160345`*^9}, {3.519341560355774*^9, 3.5193415713849936`*^9}, {
   3.5193514022902384`*^9, 3.5193514024302387`*^9}, {3.5194037903190413`*^9, 
   3.519403790599057*^9}, 3.51941073361316*^9, 3.5196771044605494`*^9, {
   3.5233155369029784`*^9, 3.5233155369029784`*^9}, 3.5239204146855583`*^9, 
   3.5239297857543354`*^9, 3.5239298325544176`*^9, {3.5295294635000772`*^9, 
   3.5295294742328963`*^9}, 3.5415539341135683`*^9, 3.5417297751644244`*^9, 
   3.5472297585730677`*^9, 3.547229903793271*^9, 3.5488746103394685`*^9, {
   3.5488773053657417`*^9, 3.54887731152575*^9}, 3.5488843073378935`*^9, 
   3.548884449468092*^9, {3.548884523728196*^9, 3.5488845252481985`*^9}, {
   3.5500925158530006`*^9, 3.5500925216230087`*^9}, {3.5500925554130564`*^9, 
   3.550092557073058*^9}, {3.550105202564433*^9, 3.550105202804434*^9}, {
   3.550105278274539*^9, 3.5501052972045655`*^9}, {3.5501056179250145`*^9, 
   3.550105618185015*^9}, 3.550108331138813*^9, 3.5501084509589806`*^9, 
   3.5528423195669117`*^9, {3.5528427060274525`*^9, 3.552842707427455*^9}, 
   3.5528429709478235`*^9, 3.5528435657286563`*^9, {3.552843847579051*^9, 
   3.552843847579051*^9}, {3.552847237063796*^9, 3.5528472380437975`*^9}, 
   3.552926070947644*^9, 3.552926198103468*^9, 3.5532224111003714`*^9, 
   3.579548747564973*^9, 3.5795489530552607`*^9, 3.579549282045721*^9, 
   3.579549353365821*^9, 3.5795585752263036`*^9, 3.5795588252066536`*^9, {
   3.6144622338140125`*^9, 3.6144622409034176`*^9}, 3.6144633197881265`*^9, {
   3.6144635055527515`*^9, 3.6144635172474203`*^9}, {3.6144667877324815`*^9, 
   3.6144667973330307`*^9}, 3.673811477865434*^9, {3.673812069655283*^9, 
   3.6738120724794445`*^9}, {3.6738121250724525`*^9, 3.673812161522537*^9}, {
   3.6738146847528577`*^9, 3.6738147093652654`*^9}, {3.673814855095601*^9, 
   3.673814857336729*^9}, 3.6745830201910524`*^9, 3.6745845122943954`*^9, 
   3.6745849428890247`*^9},
 FontSize->12,
 FontWeight->"Bold"],

Cell["Derivation: Piecewise constant extraction", "Subsubsection",
 CellChangeTimes->{
  3.479686820458373*^9, {3.490657960122612*^9, 3.49065796095366*^9}, 
   3.495321989219356*^9, 3.4953835047314157`*^9, 3.4953841387223034`*^9, 
   3.495384868433325*^9, {3.55028273733331*^9, 3.5502827388777127`*^9}, {
   3.592588715691258*^9, 3.592588723521269*^9}}],

Cell[TextData[{
 "Basis: ",
 Cell[BoxData[
  RowBox[{
   RowBox[{
    SubscriptBox["\[PartialD]", "x"], 
    FractionBox[
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"b", " ", 
        SuperscriptBox["F", 
         RowBox[{"g", " ", 
          RowBox[{"(", 
           RowBox[{"e", "+", 
            RowBox[{"f", " ", "x"}]}], ")"}]}]]}], ")"}], "n"], 
     SuperscriptBox["F", 
      RowBox[{"g", " ", "n", " ", 
       RowBox[{"(", 
        RowBox[{"e", "+", 
         RowBox[{"f", " ", "x"}]}], ")"}]}]]]}], "\[Equal]", "0"}]],
  CellChangeTimes->{{3.592588842311435*^9, 3.592588846531441*^9}}]
}], "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.495322033659418*^9, 3.495322068589467*^9}, {
   3.495332560963484*^9, 3.4953325798335104`*^9}, {3.4953342240158124`*^9, 
   3.4953342584258604`*^9}, 3.495335158717121*^9, {3.495341957376853*^9, 
   3.495341983631699*^9}, {3.4953421021607075`*^9, 3.495342121239541*^9}, {
   3.4953424507176905`*^9, 3.4953424521876926`*^9}, {3.4953424887677436`*^9, 
   3.4953424992877584`*^9}, {3.4953425925878887`*^9, 
   3.4953425939178905`*^9}, {3.495342746828105*^9, 3.495342777558148*^9}, {
   3.495383656421628*^9, 3.495383657121629*^9}, {3.4953841859423695`*^9, 
   3.495384187672372*^9}, 3.4953845466828747`*^9, {3.5488882363533936`*^9, 
   3.5488882363533936`*^9}, {3.5502001674831877`*^9, 
   3.5502001674831877`*^9}, {3.5502804529904976`*^9, 
   3.5502804529904976`*^9}, {3.550281661929821*^9, 3.550281661929821*^9}, 
   3.5742737500605345`*^9, {3.592588872201477*^9, 3.592588879181487*^9}, {
   3.592589838147832*^9, 3.592589838567832*^9}, {3.6738124091597013`*^9, 
   3.6738124091597013`*^9}, {3.6745845702177086`*^9, 
   3.6745845702177086`*^9}, {3.674584945153154*^9, 3.674584945154154*^9}, {
   3.6745849788170795`*^9, 3.6745849788170795`*^9}}],

Cell["Rule:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.495322033659418*^9, 3.495322068589467*^9}, {
   3.495332560963484*^9, 3.4953325798335104`*^9}, {3.495332835663869*^9, 
   3.4953328392438736`*^9}, {3.5488728293569746`*^9, 
   3.5488728860770545`*^9}, {3.54887301891724*^9, 3.548873026407251*^9}, {
   3.550107662367877*^9, 3.5501076662278824`*^9}, {3.5501082976187663`*^9, 
   3.5501082976187663`*^9}, {3.552842770737543*^9, 3.552842770737543*^9}, {
   3.55284386845908*^9, 3.55284386845908*^9}, {3.5529261510069847`*^9, 
   3.552926151646586*^9}, {3.6144551006191587`*^9, 3.614455100620159*^9}, {
   3.614462269426049*^9, 3.614462269426049*^9}, 3.6738147814503884`*^9}],

Cell[BoxData[
 RowBox[{"\[Integral]", 
  RowBox[{
   SuperscriptBox[
    RowBox[{"(", 
     RowBox[{"c", "+", 
      RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
   SuperscriptBox[
    RowBox[{"(", 
     RowBox[{"b", " ", 
      SuperscriptBox["F", 
       RowBox[{"g", " ", 
        RowBox[{"(", 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], ")"}]}]]}], ")"}], "n"], 
   RowBox[{"\[DifferentialD]", 
    RowBox[{"x", "  ", "\[LongRightArrow]", "  ", 
     FractionBox[
      SuperscriptBox[
       RowBox[{"(", 
        RowBox[{"b", " ", 
         SuperscriptBox["F", 
          RowBox[{"g", " ", 
           RowBox[{"(", 
            RowBox[{"e", "+", 
             RowBox[{"f", " ", "x"}]}], ")"}]}]]}], ")"}], "n"], 
      SuperscriptBox["F", 
       RowBox[{"g", " ", "n", " ", 
        RowBox[{"(", 
         RowBox[{"e", "+", 
          RowBox[{"f", " ", "x"}]}], ")"}]}]]]}]}], " ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
     SuperscriptBox["F", 
      RowBox[{"g", " ", "n", " ", 
       RowBox[{"(", 
        RowBox[{"e", "+", 
         RowBox[{"f", " ", "x"}]}], ")"}]}]], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579723816*^9, 3.4796580261689425`*^9}, 
   3.479661191290163*^9, {3.479686720164157*^9, 3.479686720204214*^9}, {
   3.479686831554328*^9, 3.479686834819022*^9}, {3.4940970712485504`*^9, 
   3.4940971130286083`*^9}, 3.4940971455886545`*^9, 3.4953221152495327`*^9, 
   3.495322146199576*^9, {3.495322261579737*^9, 3.4953222704097495`*^9}, {
   3.495334005335506*^9, 3.4953340139555182`*^9}, 3.550105346464635*^9, {
   3.550105445094773*^9, 3.5501054639847994`*^9}, {3.550105502604853*^9, 
   3.5501055057548575`*^9}, {3.552842323176917*^9, 3.552842324766919*^9}, {
   3.552842709067457*^9, 3.5528427128074627`*^9}, 3.552842971637825*^9, {
   3.5528433599583683`*^9, 3.5528433629083724`*^9}, {3.552843566398657*^9, 
   3.5528435670986586`*^9}, 3.552843849149053*^9, 3.552843918859151*^9, {
   3.5528440237692976`*^9, 3.552844032069309*^9}, {3.552844727530283*^9, 
   3.5528447317202888`*^9}, {3.552847243993806*^9, 3.5528472537638197`*^9}, 
   3.5795487931050367`*^9, {3.579549282735722*^9, 3.5795492833257227`*^9}, {
   3.5795493537958217`*^9, 3.579549354215822*^9}, 3.579549683396283*^9, {
   3.5795498116064625`*^9, 3.579549814596467*^9}, 3.5795499247066207`*^9, 
   3.579549997766723*^9, {3.579558575466304*^9, 3.5795585758363047`*^9}, 
   3.5795588258066545`*^9, {3.579559577750226*^9, 3.579559635570307*^9}, {
   3.6738123335813785`*^9, 3.6738123845452933`*^9}, 3.6738147145785637`*^9, {
   3.673814746872411*^9, 3.673814756076937*^9}, {3.6745830202110534`*^9, 
   3.674583020218054*^9}, {3.674584525291139*^9, 3.6745845460723276`*^9}, 
   3.6745845782881703`*^9, {3.674584946274218*^9, 3.674584983813365*^9}},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{"c_.", "+", 
        RowBox[{"d_.", "*", "x_"}]}], ")"}], "^", "m_."}], "*", 
     RowBox[{
      RowBox[{"(", 
       RowBox[{"b_.", "*", 
        RowBox[{"F_", "^", 
         RowBox[{"(", 
          RowBox[{"g_.", "*", 
           RowBox[{"(", 
            RowBox[{"e_.", "+", 
             RowBox[{"f_.", "*", "x_"}]}], ")"}]}], ")"}]}]}], ")"}], "^", 
      "n_"}]}], ",", "x_Symbol"}], "]"}], " ", ":=", "\n", "  ", 
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{"b", "*", 
        RowBox[{"F", "^", 
         RowBox[{"(", 
          RowBox[{"g", "*", 
           RowBox[{"(", 
            RowBox[{"e", "+", 
             RowBox[{"f", "*", "x"}]}], ")"}]}], ")"}]}]}], ")"}], "^", "n"}],
      "/", 
     RowBox[{"F", "^", 
      RowBox[{"(", 
       RowBox[{"g", "*", "n", "*", 
        RowBox[{"(", 
         RowBox[{"e", "+", 
          RowBox[{"f", "*", "x"}]}], ")"}]}], ")"}]}]}], "*", 
    RowBox[{"Int", "[", 
     RowBox[{
      RowBox[{
       RowBox[{
        RowBox[{"(", 
         RowBox[{"c", "+", 
          RowBox[{"d", "*", "x"}]}], ")"}], "^", "m"}], "*", 
       RowBox[{"F", "^", 
        RowBox[{"(", 
         RowBox[{"g", "*", "n", "*", 
          RowBox[{"(", 
           RowBox[{"e", "+", 
            RowBox[{"f", "*", "x"}]}], ")"}]}], ")"}]}]}], ",", "x"}], 
     "]"}]}], " ", "/;", "\n", 
   RowBox[{"FreeQ", "[", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{
      "F", ",", "b", ",", "c", ",", "d", ",", "e", ",", "f", ",", "g", ",", 
       "m", ",", "n"}], "}"}], ",", "x"}], "]"}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, 
   3.4953222454097147`*^9, 3.4953339911454864`*^9, 3.550105533144896*^9, {
   3.550105568824946*^9, 3.550105588814974*^9}, {3.5501061488157578`*^9, 
   3.55010615022576*^9}, 3.5501081766885967`*^9, {3.550119158458813*^9, 
   3.55011916332882*^9}, {3.5528423255269203`*^9, 3.552842327626923*^9}, 
   3.552842478347134*^9, 3.5528425248471994`*^9, {3.5528427144374647`*^9, 
   3.552842720677473*^9}, 3.55284308956799*^9, 3.552843158838087*^9, {
   3.552843367428379*^9, 3.552843374158388*^9}, 3.552843567748659*^9, 
   3.5528436844988227`*^9, 3.552844217039568*^9, 3.5528442850496635`*^9, 
   3.5528443721397853`*^9, {3.552844416069847*^9, 3.5528444215698547`*^9}, {
   3.552844736730296*^9, 3.5528447392002993`*^9}, 3.552845987582047*^9, {
   3.5528472727238464`*^9, 3.552847283983862*^9}, 3.5529262049518795`*^9, 
   3.5795494007358875`*^9, 3.579549439885942*^9, 3.579549477845995*^9, 
   3.5795497511763783`*^9, 3.579549815856468*^9, {3.5795500144667463`*^9, 
   3.57955002416676*^9}, 3.5795586255463743`*^9, {3.579558672046439*^9, 
   3.579558702006481*^9}, 3.5795587408065352`*^9, {3.579559646230322*^9, 
   3.5795596674203515`*^9}, {3.614462272376218*^9, 3.6144622948785048`*^9}, {
   3.614468102097659*^9, 3.6144681072689543`*^9}, {3.6144763347651014`*^9, 
   3.6144763442406435`*^9}, {3.673811373516466*^9, 3.6738113775306954`*^9}, {
   3.6738118050231466`*^9, 3.6738118074162836`*^9}, 3.6738124530662127`*^9, 
   3.6738124870751576`*^9, {3.673812520914093*^9, 3.6738125492567143`*^9}, {
   3.673814784719576*^9, 3.6738148176734605`*^9}, {3.674584612823146*^9, 
   3.6745846412037687`*^9}, 3.674584674219657*^9},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9}]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{Cell[TextData[StyleBox["2:",
   FontFamily->"Arial",
   FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
  RowBox[{"\[Integral]", 
   RowBox[{
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"c", "+", 
       RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"a", "+", 
       RowBox[{"b", " ", 
        SuperscriptBox[
         RowBox[{"(", 
          SuperscriptBox["F", 
           RowBox[{"g", 
            RowBox[{"(", 
             RowBox[{"e", "+", 
              RowBox[{"f", " ", "x"}]}], ")"}]}]], ")"}], "n"]}]}], ")"}], 
     "p"], 
    RowBox[{"\[DifferentialD]", "x"}], "  ", 
    StyleBox["when",
     FontFamily->"Arial",
     FontWeight->"Plain"], 
    StyleBox["  ",
     FontFamily->"Arial",
     FontWeight->"Plain"], Cell[TextData[Cell[BoxData[
     RowBox[{"p", "\[Element]", 
      SuperscriptBox["\[DoubleStruckCapitalZ]", "+"]}]]]], 
     "None"]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193422443984776`*^9, 3.5193422447260776`*^9}, 
   3.5193424488212357`*^9, {3.5193434708254333`*^9, 3.519343473103037*^9}, {
   3.5193435097631016`*^9, 3.51934353169674*^9}, {3.519345739319023*^9, 
   3.519345739599823*^9}, 3.5193462601259375`*^9, {3.5193475061060295`*^9, 
   3.5193475233908596`*^9}, {3.519351391880224*^9, 3.5193513931402254`*^9}, {
   3.519403773021052*^9, 3.5194037798564425`*^9}, 3.5194106793250647`*^9, {
   3.519537578341393*^9, 3.5195375792305946`*^9}, {3.5195376277778797`*^9, 
   3.51953767927357*^9}, {3.5197586260636144`*^9, 3.5197586260636144`*^9}, {
   3.51978053295651*^9, 3.519780542862528*^9}, {3.5197806526087203`*^9, 
   3.5197806657595434`*^9}, {3.5197807490168896`*^9, 3.5197807642581167`*^9}, 
   3.5197809004619555`*^9, {3.519782653514635*^9, 3.5197826623286505`*^9}, {
   3.5200207810164795`*^9, 3.520020785046485*^9}, {3.5213126833374643`*^9, 
   3.5213126833374643`*^9}, {3.521328235411002*^9, 3.521328235411002*^9}, {
   3.523315567893022*^9, 3.523315567893022*^9}, {3.5233157569332867`*^9, 
   3.5233157613032923`*^9}, {3.5323023198133526`*^9, 3.532302355723403*^9}, {
   3.5323026417638035`*^9, 3.5323026417638035`*^9}, {3.5323026980138817`*^9, 
   3.5323026980138817`*^9}, {3.532636287104328*^9, 3.532636287104328*^9}, {
   3.5330803686035347`*^9, 3.533080368863535*^9}, {3.533248909289961*^9, 
   3.5332489110299635`*^9}, {3.534962469565694*^9, 3.5349624910757236`*^9}, {
   3.5349625224057674`*^9, 3.534962522655768*^9}, {3.534965265519608*^9, 
   3.534965265519608*^9}, {3.5349668288168063`*^9, 3.5349668292668066`*^9}, {
   3.5349747358481646`*^9, 3.5349747358481646`*^9}, {3.534975457949176*^9, 
   3.534975457949176*^9}, {3.5368633963532934`*^9, 3.536863396571694*^9}, {
   3.541644322511304*^9, 3.5416443577513533`*^9}, {3.5416458269234104`*^9, 
   3.541645839633428*^9}, {3.5416460312236967`*^9, 3.541646031483697*^9}, 
   3.541646227783972*^9, {3.541647080335165*^9, 3.5416471077052035`*^9}, {
   3.541648997607849*^9, 3.5416489978278494`*^9}, 3.541649088407976*^9, {
   3.5417848711596775`*^9, 3.541784902429721*^9}, {3.541786145331461*^9, 
   3.5417861455914617`*^9}, {3.5417870037166634`*^9, 3.5417870039366636`*^9}, 
   3.5462979620242105`*^9, {3.546298264094633*^9, 3.546298265344635*^9}, {
   3.546299000235664*^9, 3.546299001505666*^9}, {3.546315119909564*^9, 
   3.5463151267659564`*^9}, {3.547227356319705*^9, 3.547227356319705*^9}, 
   3.547918559331044*^9, {3.547918610170952*^9, 3.5479186304411116`*^9}, 
   3.548537755517557*^9, 3.5485441629685287`*^9, 3.5485517520154133`*^9, {
   3.5488740727887154`*^9, 3.5488740727887154`*^9}, {3.548874206618903*^9, 
   3.548874206808903*^9}, {3.5488771226154857`*^9, 3.5488771332055006`*^9}, {
   3.548878208637006*^9, 3.548878208777006*^9}, {3.5488822719350433`*^9, 
   3.5488822719350433`*^9}, {3.5488837630471315`*^9, 
   3.5488837632171316`*^9}, {3.5488838279872217`*^9, 
   3.5488838281672225`*^9}, {3.548884313497902*^9, 3.548884313657902*^9}, {
   3.5488863124207*^9, 3.5488863214107127`*^9}, {3.548886881361497*^9, 
   3.5488868816714973`*^9}, 3.5488873559521613`*^9, {3.548887457282303*^9, 
   3.548887457282303*^9}, 3.5494981973468494`*^9, {3.5494983516935205`*^9, 
   3.549498353986725*^9}, {3.5494983855299797`*^9, 3.5494983862787814`*^9}, {
   3.549498568892702*^9, 3.54949857343231*^9}, 3.553223130431923*^9, 
   3.5551170302883973`*^9, {3.5589039463428574`*^9, 3.5589039479428596`*^9}, 
   3.5742077541576214`*^9, {3.5743644486996126`*^9, 3.574364448959613*^9}, {
   3.6730611092325025`*^9, 3.6730611160908947`*^9}, 3.6730611651697016`*^9, 
   3.674580267124586*^9, 3.6745804858200946`*^9, {3.67458184315973*^9, 
   3.6745818580135794`*^9}, {3.6745824917508273`*^9, 3.6745824919778404`*^9}, 
   3.674582610987647*^9},
 FontSize->12,
 FontWeight->"Bold"],

Cell["\<\
Derivation: Algebraic expansion\
\>", "Subsubsection",
 CellChangeTimes->{
  3.479686820458373*^9, {3.490657960122612*^9, 3.49065796095366*^9}, 
   3.495321989219356*^9, 3.4953835047314157`*^9, 3.4953842443724513`*^9, {
   3.6971421582683487`*^9, 3.697142163416643*^9}}],

Cell[TextData[{
 "Rule: If  ",
 Cell[BoxData[
  RowBox[{"p", "\[Element]", 
   SuperscriptBox["\[DoubleStruckCapitalZ]", "+"]}]]],
 ", then"
}], "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.495322033659418*^9, 3.495322068589467*^9}, {
   3.495332560963484*^9, 3.4953325798335104`*^9}, {3.4953342240158124`*^9, 
   3.4953342584258604`*^9}, 3.495335158717121*^9, {3.495341957376853*^9, 
   3.495341983631699*^9}, {3.4953421021607075`*^9, 3.495342121239541*^9}, {
   3.4953424507176905`*^9, 3.4953424521876926`*^9}, {3.4953424887677436`*^9, 
   3.4953424992877584`*^9}, {3.4953425925878887`*^9, 
   3.4953425939178905`*^9}, {3.495342746828105*^9, 3.495342777558148*^9}, {
   3.495383656421628*^9, 3.495383657121629*^9}, {3.4953837077317*^9, 
   3.495383711161705*^9}, {3.495383774431793*^9, 3.4953837774217978`*^9}, {
   3.4953838495818987`*^9, 3.4953838560319076`*^9}, {3.4953839845720873`*^9, 
   3.4953839850920887`*^9}, {3.5494983915203905`*^9, 
   3.5494983915203905`*^9}, {3.673061123204302*^9, 3.673061123204302*^9}}],

Cell[BoxData[
 RowBox[{"\[Integral]", 
  RowBox[{
   SuperscriptBox[
    RowBox[{"(", 
     RowBox[{"c", "+", 
      RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
   SuperscriptBox[
    RowBox[{"(", 
     RowBox[{"a", "+", 
      RowBox[{"b", " ", 
       SuperscriptBox[
        RowBox[{"(", 
         SuperscriptBox["F", 
          RowBox[{"g", 
           RowBox[{"(", 
            RowBox[{"e", "+", 
             RowBox[{"f", " ", "x"}]}], ")"}]}]], ")"}], "n"]}]}], ")"}], 
    "p"], 
   RowBox[{"\[DifferentialD]", 
    RowBox[{"x", "  ", "\[LongRightArrow]", "  ", 
     RowBox[{"\[Integral]", 
      RowBox[{
       SuperscriptBox[
        RowBox[{"(", 
         RowBox[{"c", "+", 
          RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
       RowBox[{"ExpandIntegrand", "[", 
        RowBox[{
         SuperscriptBox[
          RowBox[{"(", 
           RowBox[{"a", "+", 
            RowBox[{"b", " ", 
             SuperscriptBox[
              RowBox[{"(", 
               SuperscriptBox["F", 
                RowBox[{"g", 
                 RowBox[{"(", 
                  RowBox[{"e", "+", 
                   RowBox[{"f", " ", "x"}]}], ")"}]}]], ")"}], "n"]}]}], 
           ")"}], "p"], ",", "x"}], "]"}], 
       RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579723816*^9, 3.4796580261689425`*^9}, 
   3.479661191290163*^9, {3.479686720164157*^9, 3.479686720204214*^9}, {
   3.479686831554328*^9, 3.479686834819022*^9}, {3.4940970712485504`*^9, 
   3.4940971130286083`*^9}, 3.4940971455886545`*^9, 3.4953221152495327`*^9, 
   3.495322146199576*^9, {3.495322261579737*^9, 3.4953222704097495`*^9}, 
   3.4953325041434045`*^9, 3.4953325401034546`*^9, {3.4953327138936977`*^9, 
   3.4953327232437115`*^9}, {3.4953341614457245`*^9, 
   3.4953341867757597`*^9}, {3.4953351287770786`*^9, 3.4953351438071003`*^9}, 
   3.4953417122068224`*^9, {3.4953420082485423`*^9, 3.4953420357669907`*^9}, 
   3.49534216649522*^9, {3.4953422416249523`*^9, 3.495342254650975*^9}, {
   3.4953424044876256`*^9, 3.4953424293576603`*^9}, 3.4953425414678173`*^9, {
   3.4953426292179403`*^9, 3.4953426498179693`*^9}, 3.4953427170580635`*^9, {
   3.495342827258218*^9, 3.4953428329682255`*^9}, 3.495383554791486*^9, 
   3.495383593001539*^9, 3.495383741531747*^9, {3.4953838030218334`*^9, 
   3.495383812201846*^9}, 3.495383876631936*^9, {3.4953839464920344`*^9, 
   3.495383955552047*^9}, {3.549498408976821*^9, 3.5494984286484556`*^9}, {
   3.549498583541128*^9, 3.5494985849763308`*^9}, {3.555117033538402*^9, 
   3.555117036978407*^9}, {3.574207756977625*^9, 3.5742077593776283`*^9}, {
   3.673061126608496*^9, 3.6730611321118107`*^9}, {3.6730611697039614`*^9, 
   3.673061173998207*^9}, {3.6745804921814585`*^9, 3.6745804956606574`*^9}, {
   3.674581924027355*^9, 3.6745819381811647`*^9}, 3.674582617718032*^9},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{"c_.", "+", 
        RowBox[{"d_.", "*", "x_"}]}], ")"}], "^", "m_."}], "*", 
     RowBox[{
      RowBox[{"(", 
       RowBox[{"a_", "+", 
        RowBox[{"b_.", "*", 
         RowBox[{
          RowBox[{"(", 
           RowBox[{"F_", "^", 
            RowBox[{"(", 
             RowBox[{"g_.", "*", 
              RowBox[{"(", 
               RowBox[{"e_.", "+", 
                RowBox[{"f_.", "*", "x_"}]}], ")"}]}], ")"}]}], ")"}], "^", 
          "n_."}]}]}], ")"}], "^", "p_."}]}], ",", "x_Symbol"}], "]"}], " ", ":=",
   "\n", "  ", 
  RowBox[{
   RowBox[{"Int", "[", 
    RowBox[{
     RowBox[{"ExpandIntegrand", "[", 
      RowBox[{
       RowBox[{
        RowBox[{"(", 
         RowBox[{"c", "+", 
          RowBox[{"d", "*", "x"}]}], ")"}], "^", "m"}], ",", 
       RowBox[{
        RowBox[{"(", 
         RowBox[{"a", "+", 
          RowBox[{"b", "*", 
           RowBox[{
            RowBox[{"(", 
             RowBox[{"F", "^", 
              RowBox[{"(", 
               RowBox[{"g", "*", 
                RowBox[{"(", 
                 RowBox[{"e", "+", 
                  RowBox[{"f", "*", "x"}]}], ")"}]}], ")"}]}], ")"}], "^", 
            "n"}]}]}], ")"}], "^", "p"}], ",", "x"}], "]"}], ",", "x"}], 
    "]"}], " ", "/;", "\n", 
   RowBox[{
    RowBox[{"FreeQ", "[", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{
       "F", ",", "a", ",", "b", ",", "c", ",", "d", ",", "e", ",", "f", ",", 
        "g", ",", "m", ",", "n"}], "}"}], ",", "x"}], "]"}], " ", "&&", " ", 
    RowBox[{"IGtQ", "[", 
     RowBox[{"p", ",", "0"}], "]"}]}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, 
   3.4953222454097147`*^9, 3.4953324633733473`*^9, 3.4953326632436275`*^9, 
   3.4953341350856876`*^9, 3.4953350962570333`*^9, 3.495341936441616*^9, 
   3.495342089821086*^9, 3.495342306911067*^9, 3.4953424785377293`*^9, 
   3.4953425860178795`*^9, 3.495342687468022*^9, 3.4953835379514627`*^9, 
   3.495383725451725*^9, 3.495383788381813*^9, 3.4953838337518764`*^9, 
   3.495383924742004*^9, 3.5488189132125998`*^9, 3.548819042776781*^9, {
   3.548887694932636*^9, 3.5488876962026377`*^9}, {3.5494983742511606`*^9, 
   3.549498381957574*^9}, {3.5494984415496783`*^9, 3.5494984587877088`*^9}, {
   3.5494985906079407`*^9, 3.5494986038367634`*^9}, {3.555117084148473*^9, 
   3.5551170938084865`*^9}, {3.5742077842576637`*^9, 
   3.5742077979476824`*^9}, {3.6730611362840495`*^9, 3.673061146553637*^9}, {
   3.673061190537153*^9, 3.6730612108183126`*^9}, {3.674580503211089*^9, 
   3.6745805368530135`*^9}, {3.6745819488317738`*^9, 3.674581981350634*^9}, {
   3.7151094366859603`*^9, 3.7151094370299797`*^9}, {3.7151094874268627`*^9, 
   3.715109488505924*^9}},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9}]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{Cell[TextData[StyleBox["3:",
   FontFamily->"Arial",
   FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
  RowBox[{"\[Integral]", 
   RowBox[{
    FractionBox[
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", " ", "x"}]}], ")"}], "m"], 
     RowBox[{"a", "+", 
      RowBox[{"b", " ", 
       SuperscriptBox[
        RowBox[{"(", 
         SuperscriptBox["F", 
          RowBox[{"g", 
           RowBox[{"(", 
            RowBox[{"e", "+", 
             RowBox[{"f", " ", "x"}]}], ")"}]}]], ")"}], "n"]}]}]], 
    RowBox[{"\[DifferentialD]", "x"}], "  ", 
    StyleBox["when",
     FontFamily->"Arial",
     FontWeight->"Plain"], 
    StyleBox["  ",
     FontFamily->"Arial",
     FontWeight->"Plain"], Cell[TextData[Cell[BoxData[
     RowBox[{"m", "\[Element]", 
      SuperscriptBox["\[DoubleStruckCapitalZ]", "+"]}]]]], 
     "None"]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193422443984776`*^9, 3.5193422447260776`*^9}, 
   3.5193424488212357`*^9, {3.5193434708254333`*^9, 3.519343473103037*^9}, {
   3.5193435097631016`*^9, 3.51934353169674*^9}, {3.519345739319023*^9, 
   3.519345739599823*^9}, 3.5193462601259375`*^9, {3.5193475061060295`*^9, 
   3.5193475233908596`*^9}, {3.519351391880224*^9, 3.5193513931402254`*^9}, {
   3.519403773021052*^9, 3.5194037798564425`*^9}, 3.5194106793250647`*^9, {
   3.519537578341393*^9, 3.5195375792305946`*^9}, {3.5195376277778797`*^9, 
   3.51953767927357*^9}, {3.5197586260636144`*^9, 3.5197586260636144`*^9}, {
   3.51978053295651*^9, 3.519780542862528*^9}, {3.5197806526087203`*^9, 
   3.5197806657595434`*^9}, {3.5197807490168896`*^9, 3.5197807642581167`*^9}, 
   3.5197809004619555`*^9, {3.519782653514635*^9, 3.5197826623286505`*^9}, {
   3.5200207810164795`*^9, 3.520020785046485*^9}, {3.5213126833374643`*^9, 
   3.5213126833374643`*^9}, {3.521328235411002*^9, 3.521328235411002*^9}, {
   3.523315567893022*^9, 3.523315567893022*^9}, {3.5233157569332867`*^9, 
   3.5233157613032923`*^9}, {3.5323023198133526`*^9, 3.532302355723403*^9}, {
   3.5323026417638035`*^9, 3.5323026417638035`*^9}, {3.5323026980138817`*^9, 
   3.5323026980138817`*^9}, {3.532636287104328*^9, 3.532636287104328*^9}, {
   3.5330803686035347`*^9, 3.533080368863535*^9}, {3.533248909289961*^9, 
   3.5332489110299635`*^9}, {3.534962469565694*^9, 3.5349624910757236`*^9}, {
   3.5349625224057674`*^9, 3.534962522655768*^9}, {3.534965265519608*^9, 
   3.534965265519608*^9}, {3.5349668288168063`*^9, 3.5349668292668066`*^9}, {
   3.5349747358481646`*^9, 3.5349747358481646`*^9}, {3.534975457949176*^9, 
   3.534975457949176*^9}, {3.5368633963532934`*^9, 3.536863396571694*^9}, {
   3.541644322511304*^9, 3.5416443577513533`*^9}, {3.5416458269234104`*^9, 
   3.541645839633428*^9}, {3.5416460312236967`*^9, 3.541646031483697*^9}, 
   3.541646227783972*^9, {3.541647080335165*^9, 3.5416471077052035`*^9}, {
   3.541648997607849*^9, 3.5416489978278494`*^9}, 3.541649088407976*^9, {
   3.5417848711596775`*^9, 3.541784902429721*^9}, {3.541786145331461*^9, 
   3.5417861455914617`*^9}, {3.5417870037166634`*^9, 3.5417870039366636`*^9}, 
   3.5462979620242105`*^9, {3.546298264094633*^9, 3.546298265344635*^9}, {
   3.546299000235664*^9, 3.546299001505666*^9}, {3.546315119909564*^9, 
   3.5463151267659564`*^9}, {3.547227356319705*^9, 3.547227356319705*^9}, 
   3.547918559331044*^9, {3.547918610170952*^9, 3.5479186304411116`*^9}, 
   3.548537755517557*^9, 3.5485441629685287`*^9, 3.5485517520154133`*^9, {
   3.5488740727887154`*^9, 3.5488740727887154`*^9}, {3.548874206618903*^9, 
   3.548874206808903*^9}, {3.5488771226154857`*^9, 3.5488771332055006`*^9}, {
   3.548878208637006*^9, 3.548878208777006*^9}, {3.5488822719350433`*^9, 
   3.5488822719350433`*^9}, {3.5488837630471315`*^9, 
   3.5488837632171316`*^9}, {3.5488838279872217`*^9, 
   3.5488838281672225`*^9}, {3.548884313497902*^9, 3.548884313657902*^9}, {
   3.5488863124207*^9, 3.5488863214107127`*^9}, {3.548886881361497*^9, 
   3.5488868816714973`*^9}, 3.5488873559521613`*^9, {3.548887457282303*^9, 
   3.548887457282303*^9}, {3.548887801722785*^9, 3.548887807902794*^9}, {
   3.5488882205933714`*^9, 3.548888238023396*^9}, 3.5532233767690625`*^9, {
   3.555117564159145*^9, 3.5551175646191454`*^9}, 3.5551184923504443`*^9, 
   3.574208884039203*^9, {3.574209004899372*^9, 3.574209006519375*^9}, {
   3.6730432069185486`*^9, 3.6730432198642893`*^9}, {3.673057984138757*^9, 
   3.6730579844407744`*^9}, 3.673060755573274*^9, {3.673291357416087*^9, 
   3.6732913576310997`*^9}, 3.6745802705917845`*^9, {3.6745807289460006`*^9, 
   3.6745807308221083`*^9}, {3.6745820114783573`*^9, 3.674582020544876*^9}, {
   3.674582493208911*^9, 3.67458249337392*^9}, {3.6971434395056314`*^9, 
   3.6971434496062093`*^9}, {3.697163641964302*^9, 3.69716364735861*^9}, {
   3.7280841732475424`*^9, 3.728084175457669*^9}},
 FontSize->12,
 FontWeight->"Bold"],

Cell["\<\
Derivation: Algebraic expansion\
\>", "Subsubsection",
 CellChangeTimes->{
  3.479686820458373*^9, {3.490657960122612*^9, 3.49065796095366*^9}, 
   3.495321989219356*^9, 3.4953835047314157`*^9, 3.4953841387223034`*^9, 
   3.495384868433325*^9, {3.728086382051878*^9, 3.7280863873451805`*^9}}],

Cell[TextData[{
 "Basis: ",
 Cell[BoxData[
  RowBox[{
   FractionBox["1", 
    RowBox[{"a", "+", 
     RowBox[{"b", " ", "z"}]}]], "\[Equal]", 
   RowBox[{
    FractionBox["1", "a"], "-", 
    FractionBox[
     RowBox[{"b", " ", "z"}], 
     RowBox[{"a", " ", 
      RowBox[{"(", 
       RowBox[{"a", "+", 
        RowBox[{"b", " ", "z"}]}], ")"}]}]]}]}]],
  CellChangeTimes->{{3.6745820787182035`*^9, 3.674582127358985*^9}}]
}], "Subsubsection",
 CellChangeTimes->{
  3.479686820458373*^9, {3.490657960122612*^9, 3.49065796095366*^9}, 
   3.495321989219356*^9, 3.4953835047314157`*^9, 3.4953841387223034`*^9, 
   3.495384868433325*^9, {3.550199696122528*^9, 3.5501997144425535`*^9}, {
   3.5501998242127075`*^9, 3.5501998242127075`*^9}, {3.5551175662191477`*^9, 
   3.5551175683491507`*^9}, {3.55511852465049*^9, 3.555118535740505*^9}, {
   3.574209037409418*^9, 3.5742090674294596`*^9}, {3.673043308218343*^9, 
   3.673043308218343*^9}, {3.6730790759472423`*^9, 3.6730790759472423`*^9}, {
   3.674582034184656*^9, 3.6745820508586097`*^9}, 3.674582135469449*^9, {
   3.7280863950466213`*^9, 3.7280863950466213`*^9}}],

Cell[TextData[{
 "Rule: If  ",
 Cell[BoxData[
  RowBox[{"m", "\[Element]", 
   SuperscriptBox["\[DoubleStruckCapitalZ]", "+"]}]]],
 ", then"
}], "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.495322033659418*^9, 3.495322068589467*^9}, {
   3.495332560963484*^9, 3.4953325798335104`*^9}, {3.4953342240158124`*^9, 
   3.4953342584258604`*^9}, 3.495335158717121*^9, {3.495341957376853*^9, 
   3.495341983631699*^9}, {3.4953421021607075`*^9, 3.495342121239541*^9}, {
   3.4953424507176905`*^9, 3.4953424521876926`*^9}, {3.4953424887677436`*^9, 
   3.4953424992877584`*^9}, {3.4953425925878887`*^9, 
   3.4953425939178905`*^9}, {3.495342746828105*^9, 3.495342777558148*^9}, {
   3.495383656421628*^9, 3.495383657121629*^9}, {3.4953841859423695`*^9, 
   3.495384187672372*^9}, 3.4953845466828747`*^9, {3.5488882363533936`*^9, 
   3.5488882363533936`*^9}, {3.673043227527727*^9, 3.673043227527727*^9}, {
   3.7280841880953913`*^9, 3.7280841880953913`*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"\[Integral]", 
   RowBox[{
    FractionBox[
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", " ", "x"}]}], ")"}], "m"], 
     RowBox[{"a", "+", 
      RowBox[{"b", " ", 
       SuperscriptBox[
        RowBox[{"(", 
         SuperscriptBox["F", 
          RowBox[{"g", 
           RowBox[{"(", 
            RowBox[{"e", "+", 
             RowBox[{"f", " ", "x"}]}], ")"}]}]], ")"}], "n"]}]}]], 
    RowBox[{"\[DifferentialD]", 
     RowBox[{"x", "  ", "\[LongRightArrow]", "  ", 
      FractionBox[
       SuperscriptBox[
        RowBox[{"(", 
         RowBox[{"c", "+", 
          RowBox[{"d", " ", "x"}]}], ")"}], 
        RowBox[{"m", "+", "1"}]], 
       RowBox[{"a", " ", "d", " ", 
        RowBox[{"(", 
         RowBox[{"m", "+", "1"}], ")"}]}]]}]}]}]}], "-", 
  RowBox[{
   FractionBox["b", "a"], 
   RowBox[{"\[Integral]", 
    RowBox[{
     FractionBox[
      RowBox[{
       SuperscriptBox[
        RowBox[{"(", 
         RowBox[{"c", "+", 
          RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
       SuperscriptBox[
        RowBox[{"(", 
         SuperscriptBox["F", 
          RowBox[{"g", 
           RowBox[{"(", 
            RowBox[{"e", "+", 
             RowBox[{"f", " ", "x"}]}], ")"}]}]], ")"}], "n"]}], 
      RowBox[{"a", "+", 
       RowBox[{"b", " ", 
        SuperscriptBox[
         RowBox[{"(", 
          SuperscriptBox["F", 
           RowBox[{"g", 
            RowBox[{"(", 
             RowBox[{"e", "+", 
              RowBox[{"f", " ", "x"}]}], ")"}]}]], ")"}], "n"]}]}]], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579723816*^9, 3.4796580261689425`*^9}, 
   3.479661191290163*^9, {3.479686720164157*^9, 3.479686720204214*^9}, {
   3.479686831554328*^9, 3.479686834819022*^9}, {3.4940970712485504`*^9, 
   3.4940971130286083`*^9}, 3.4940971455886545`*^9, 3.4953221152495327`*^9, 
   3.495322146199576*^9, {3.495322261579737*^9, 3.4953222704097495`*^9}, 
   3.4953325041434045`*^9, 3.4953325401034546`*^9, {3.4953327138936977`*^9, 
   3.4953327232437115`*^9}, {3.4953341614457245`*^9, 
   3.4953341867757597`*^9}, {3.4953351287770786`*^9, 3.4953351438071003`*^9}, 
   3.4953417122068224`*^9, {3.4953420082485423`*^9, 3.4953420357669907`*^9}, 
   3.49534216649522*^9, {3.4953422416249523`*^9, 3.495342254650975*^9}, {
   3.4953424044876256`*^9, 3.4953424293576603`*^9}, 3.4953425414678173`*^9, {
   3.4953426292179403`*^9, 3.4953426498179693`*^9}, 3.4953427170580635`*^9, {
   3.495342827258218*^9, 3.4953428329682255`*^9}, 3.495383554791486*^9, 
   3.495383593001539*^9, {3.4953842137524085`*^9, 3.4953842254724245`*^9}, {
   3.4953845765129166`*^9, 3.4953845937429404`*^9}, {3.4953849102933836`*^9, 
   3.495384926403406*^9}, {3.5551175723291564`*^9, 3.55511757470916*^9}, {
   3.555118494910448*^9, 3.555118504140461*^9}, {3.574208884629204*^9, 
   3.5742088857392054`*^9}, 3.5742089527792993`*^9, {3.5742090128193836`*^9, 
   3.574209030559408*^9}, 3.574211090492292*^9, {3.6730426771262465`*^9, 
   3.6730427210317574`*^9}, {3.673042810906898*^9, 3.6730428463569255`*^9}, {
   3.673043004576975*^9, 3.6730430779281707`*^9}, {3.6730791555847826`*^9, 
   3.6730791724952126`*^9}, {3.6745806668244476`*^9, 
   3.6745806717617297`*^9}, {3.674582152038397*^9, 3.674582205090431*^9}, {
   3.728086421145114*^9, 3.7280865068330154`*^9}, {3.7280866081308093`*^9, 
   3.7280866167233005`*^9}},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{"c_.", "+", 
        RowBox[{"d_.", "*", "x_"}]}], ")"}], "^", "m_."}], "/", 
     RowBox[{"(", 
      RowBox[{"a_", "+", 
       RowBox[{"b_.", "*", 
        RowBox[{
         RowBox[{"(", 
          RowBox[{"F_", "^", 
           RowBox[{"(", 
            RowBox[{"g_.", "*", 
             RowBox[{"(", 
              RowBox[{"e_.", "+", 
               RowBox[{"f_.", "*", "x_"}]}], ")"}]}], ")"}]}], ")"}], "^", 
         "n_."}]}]}], ")"}]}], ",", "x_Symbol"}], "]"}], " ", ":=", "\n", 
  "  ", 
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", "*", "x"}]}], ")"}], "^", 
      RowBox[{"(", 
       RowBox[{"m", "+", "1"}], ")"}]}], "/", 
     RowBox[{"(", 
      RowBox[{"a", "*", "d", "*", 
       RowBox[{"(", 
        RowBox[{"m", "+", "1"}], ")"}]}], ")"}]}], " ", "-", " ", 
    RowBox[{
     RowBox[{"b", "/", "a"}], "*", 
     RowBox[{"Int", "[", 
      RowBox[{
       RowBox[{
        RowBox[{
         RowBox[{"(", 
          RowBox[{"c", "+", 
           RowBox[{"d", "*", "x"}]}], ")"}], "^", "m"}], "*", 
        RowBox[{
         RowBox[{
          RowBox[{"(", 
           RowBox[{"F", "^", 
            RowBox[{"(", 
             RowBox[{"g", "*", 
              RowBox[{"(", 
               RowBox[{"e", "+", 
                RowBox[{"f", "*", "x"}]}], ")"}]}], ")"}]}], ")"}], "^", 
          "n"}], "/", 
         RowBox[{"(", 
          RowBox[{"a", "+", 
           RowBox[{"b", "*", 
            RowBox[{
             RowBox[{"(", 
              RowBox[{"F", "^", 
               RowBox[{"(", 
                RowBox[{"g", "*", 
                 RowBox[{"(", 
                  RowBox[{"e", "+", 
                   RowBox[{"f", "*", "x"}]}], ")"}]}], ")"}]}], ")"}], "^", 
             "n"}]}]}], ")"}]}]}], ",", "x"}], "]"}]}]}], " ", "/;", "\n", 
   RowBox[{
    RowBox[{"FreeQ", "[", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{
       "F", ",", "a", ",", "b", ",", "c", ",", "d", ",", "e", ",", "f", ",", 
        "g", ",", "n"}], "}"}], ",", "x"}], "]"}], " ", "&&", " ", 
    RowBox[{"IGtQ", "[", 
     RowBox[{"m", ",", "0"}], "]"}]}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, 
   3.4953222454097147`*^9, 3.4953324633733473`*^9, 3.4953326632436275`*^9, 
   3.4953341350856876`*^9, 3.4953350962570333`*^9, 3.495341936441616*^9, 
   3.495342089821086*^9, 3.495342306911067*^9, 3.4953424785377293`*^9, 
   3.4953425860178795`*^9, 3.495342687468022*^9, 3.4953835379514627`*^9, 
   3.4953841798023605`*^9, 3.4953845417128677`*^9, {3.49538487892334*^9, 
   3.4953848811733427`*^9}, 3.496087278649127*^9, 3.5488189303726234`*^9, 
   3.548819078896832*^9, {3.5488753442204957`*^9, 3.5488753452904973`*^9}, 
   3.5502002502633038`*^9, 3.550283333176357*^9, {3.555117575719161*^9, 
   3.5551175795091667`*^9}, {3.5551185923605843`*^9, 
   3.5551186395806503`*^9}, {3.5742089042092314`*^9, 
   3.5742089280792646`*^9}, {3.574209122579537*^9, 3.574209172579607*^9}, 
   3.673043333732802*^9, 3.673043404368842*^9, {3.673047771425624*^9, 
   3.6730477918977947`*^9}, 3.6730482952695856`*^9, {3.6730792086304884`*^9, 
   3.6730792349643345`*^9}, {3.6730793008160505`*^9, 
   3.6730793036084557`*^9}, {3.6745822501710095`*^9, 3.674582257739443*^9}, 
   3.6745823099244275`*^9, {3.6971434319251976`*^9, 3.6971434361314383`*^9}, {
   3.6971636548050365`*^9, 3.697163657085167*^9}, 3.7151093197022696`*^9, 
   3.7280841797709155`*^9, 3.7280866702293606`*^9},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9}]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{Cell[TextData[StyleBox["x:",
   FontFamily->"Arial",
   FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
  RowBox[{"\[Integral]", 
   RowBox[{
    FractionBox[
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", " ", "x"}]}], ")"}], "m"], 
     RowBox[{"a", "+", 
      RowBox[{"b", " ", 
       SuperscriptBox[
        RowBox[{"(", 
         SuperscriptBox["F", 
          RowBox[{"g", 
           RowBox[{"(", 
            RowBox[{"e", "+", 
             RowBox[{"f", " ", "x"}]}], ")"}]}]], ")"}], "n"]}]}]], 
    RowBox[{"\[DifferentialD]", "x"}], "  ", 
    StyleBox["when",
     FontFamily->"Arial",
     FontWeight->"Plain"], 
    StyleBox["  ",
     FontFamily->"Arial",
     FontWeight->"Plain"], Cell[TextData[Cell[BoxData[
     RowBox[{"m", "\[Element]", 
      SuperscriptBox["\[DoubleStruckCapitalZ]", "+"]}]]]], 
     "None"]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193422443984776`*^9, 3.5193422447260776`*^9}, 
   3.5193424488212357`*^9, {3.5193434708254333`*^9, 3.519343473103037*^9}, {
   3.5193435097631016`*^9, 3.51934353169674*^9}, {3.519345739319023*^9, 
   3.519345739599823*^9}, 3.5193462601259375`*^9, {3.5193475061060295`*^9, 
   3.5193475233908596`*^9}, {3.519351391880224*^9, 3.5193513931402254`*^9}, {
   3.519403773021052*^9, 3.5194037798564425`*^9}, 3.5194106793250647`*^9, {
   3.519537578341393*^9, 3.5195375792305946`*^9}, {3.5195376277778797`*^9, 
   3.51953767927357*^9}, {3.5197586260636144`*^9, 3.5197586260636144`*^9}, {
   3.51978053295651*^9, 3.519780542862528*^9}, {3.5197806526087203`*^9, 
   3.5197806657595434`*^9}, {3.5197807490168896`*^9, 3.5197807642581167`*^9}, 
   3.5197809004619555`*^9, {3.519782653514635*^9, 3.5197826623286505`*^9}, {
   3.5200207810164795`*^9, 3.520020785046485*^9}, {3.5213126833374643`*^9, 
   3.5213126833374643`*^9}, {3.521328235411002*^9, 3.521328235411002*^9}, {
   3.523315567893022*^9, 3.523315567893022*^9}, {3.5233157569332867`*^9, 
   3.5233157613032923`*^9}, {3.5323023198133526`*^9, 3.532302355723403*^9}, {
   3.5323026417638035`*^9, 3.5323026417638035`*^9}, {3.5323026980138817`*^9, 
   3.5323026980138817`*^9}, {3.532636287104328*^9, 3.532636287104328*^9}, {
   3.5330803686035347`*^9, 3.533080368863535*^9}, {3.533248909289961*^9, 
   3.5332489110299635`*^9}, {3.534962469565694*^9, 3.5349624910757236`*^9}, {
   3.5349625224057674`*^9, 3.534962522655768*^9}, {3.534965265519608*^9, 
   3.534965265519608*^9}, {3.5349668288168063`*^9, 3.5349668292668066`*^9}, {
   3.5349747358481646`*^9, 3.5349747358481646`*^9}, {3.534975457949176*^9, 
   3.534975457949176*^9}, {3.5368633963532934`*^9, 3.536863396571694*^9}, {
   3.541644322511304*^9, 3.5416443577513533`*^9}, {3.5416458269234104`*^9, 
   3.541645839633428*^9}, {3.5416460312236967`*^9, 3.541646031483697*^9}, 
   3.541646227783972*^9, {3.541647080335165*^9, 3.5416471077052035`*^9}, {
   3.541648997607849*^9, 3.5416489978278494`*^9}, 3.541649088407976*^9, {
   3.5417848711596775`*^9, 3.541784902429721*^9}, {3.541786145331461*^9, 
   3.5417861455914617`*^9}, {3.5417870037166634`*^9, 3.5417870039366636`*^9}, 
   3.5462979620242105`*^9, {3.546298264094633*^9, 3.546298265344635*^9}, {
   3.546299000235664*^9, 3.546299001505666*^9}, {3.546315119909564*^9, 
   3.5463151267659564`*^9}, {3.547227356319705*^9, 3.547227356319705*^9}, 
   3.547918559331044*^9, {3.547918610170952*^9, 3.5479186304411116`*^9}, 
   3.548537755517557*^9, 3.5485441629685287`*^9, 3.5485517520154133`*^9, {
   3.5488740727887154`*^9, 3.5488740727887154`*^9}, {3.548874206618903*^9, 
   3.548874206808903*^9}, {3.5488771226154857`*^9, 3.5488771332055006`*^9}, {
   3.548878208637006*^9, 3.548878208777006*^9}, {3.5488822719350433`*^9, 
   3.5488822719350433`*^9}, {3.5488837630471315`*^9, 
   3.5488837632171316`*^9}, {3.5488838279872217`*^9, 
   3.5488838281672225`*^9}, {3.548884313497902*^9, 3.548884313657902*^9}, {
   3.5488863124207*^9, 3.5488863214107127`*^9}, {3.548886881361497*^9, 
   3.5488868816714973`*^9}, 3.5488873559521613`*^9, {3.548887457282303*^9, 
   3.548887457282303*^9}, {3.548887801722785*^9, 3.548887807902794*^9}, {
   3.5488882205933714`*^9, 3.548888238023396*^9}, 3.5532233767690625`*^9, {
   3.555117564159145*^9, 3.5551175646191454`*^9}, 3.5551184923504443`*^9, 
   3.574208884039203*^9, {3.574209004899372*^9, 3.574209006519375*^9}, {
   3.6730432069185486`*^9, 3.6730432198642893`*^9}, {3.673057984138757*^9, 
   3.6730579844407744`*^9}, 3.673060755573274*^9, {3.673291357416087*^9, 
   3.6732913576310997`*^9}, 3.6745802705917845`*^9, {3.6745807289460006`*^9, 
   3.6745807308221083`*^9}, {3.6745820114783573`*^9, 3.674582020544876*^9}, {
   3.674582493208911*^9, 3.67458249337392*^9}, {3.6971434395056314`*^9, 
   3.6971434496062093`*^9}, {3.697163641964302*^9, 3.69716364735861*^9}, {
   3.7280841732475424`*^9, 3.728084175457669*^9}, {3.7280862960909615`*^9, 
   3.7280862963309755`*^9}},
 FontSize->12,
 FontWeight->"Bold"],

Cell["Derivation: Integration by parts", "Subsubsection",
 CellChangeTimes->{
  3.479686820458373*^9, {3.490657960122612*^9, 3.49065796095366*^9}, 
   3.495321989219356*^9, 3.4953835047314157`*^9, 3.4953841387223034`*^9, 
   3.495384868433325*^9}],

Cell[TextData[{
 "Basis: ",
 Cell[BoxData[
  RowBox[{
   FractionBox["1", 
    RowBox[{"a", "+", 
     RowBox[{"b", " ", 
      SuperscriptBox[
       RowBox[{"(", 
        SuperscriptBox["F", 
         RowBox[{"g", 
          RowBox[{"(", 
           RowBox[{"e", "+", 
            RowBox[{"f", " ", "x"}]}], ")"}]}]], ")"}], "n"]}]}]], "\[Equal]", 
   RowBox[{"-", 
    RowBox[{
     SubscriptBox["\[PartialD]", "x"], 
     FractionBox[
      RowBox[{"Log", "[", 
       RowBox[{"1", "+", 
        FractionBox["a", 
         RowBox[{"b", " ", 
          SuperscriptBox[
           RowBox[{"(", 
            SuperscriptBox["F", 
             RowBox[{"g", 
              RowBox[{"(", 
               RowBox[{"e", "+", 
                RowBox[{"f", " ", "x"}]}], ")"}]}]], ")"}], "n"]}]]}], "]"}], 
      RowBox[{"a", " ", "f", " ", "g", " ", "n", " ", 
       RowBox[{"Log", "[", "F", "]"}]}]]}]}]}]],
  CellChangeTimes->{{3.6745820787182035`*^9, 3.674582127358985*^9}}]
}], "Subsubsection",
 CellChangeTimes->{
  3.479686820458373*^9, {3.490657960122612*^9, 3.49065796095366*^9}, 
   3.495321989219356*^9, 3.4953835047314157`*^9, 3.4953841387223034`*^9, 
   3.495384868433325*^9, {3.550199696122528*^9, 3.5501997144425535`*^9}, {
   3.5501998242127075`*^9, 3.5501998242127075`*^9}, {3.5551175662191477`*^9, 
   3.5551175683491507`*^9}, {3.55511852465049*^9, 3.555118535740505*^9}, {
   3.574209037409418*^9, 3.5742090674294596`*^9}, {3.673043308218343*^9, 
   3.673043308218343*^9}, {3.6730790759472423`*^9, 3.6730790759472423`*^9}, {
   3.674582034184656*^9, 3.6745820508586097`*^9}, 3.674582135469449*^9}],

Cell[TextData[{
 "Rule: If  ",
 Cell[BoxData[
  RowBox[{"m", "\[Element]", 
   SuperscriptBox["\[DoubleStruckCapitalZ]", "+"]}]]],
 ", then"
}], "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.495322033659418*^9, 3.495322068589467*^9}, {
   3.495332560963484*^9, 3.4953325798335104`*^9}, {3.4953342240158124`*^9, 
   3.4953342584258604`*^9}, 3.495335158717121*^9, {3.495341957376853*^9, 
   3.495341983631699*^9}, {3.4953421021607075`*^9, 3.495342121239541*^9}, {
   3.4953424507176905`*^9, 3.4953424521876926`*^9}, {3.4953424887677436`*^9, 
   3.4953424992877584`*^9}, {3.4953425925878887`*^9, 
   3.4953425939178905`*^9}, {3.495342746828105*^9, 3.495342777558148*^9}, {
   3.495383656421628*^9, 3.495383657121629*^9}, {3.4953841859423695`*^9, 
   3.495384187672372*^9}, 3.4953845466828747`*^9, {3.5488882363533936`*^9, 
   3.5488882363533936`*^9}, {3.673043227527727*^9, 3.673043227527727*^9}, {
   3.7280841880953913`*^9, 3.7280841880953913`*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"\[Integral]", 
   RowBox[{
    FractionBox[
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", " ", "x"}]}], ")"}], "m"], 
     RowBox[{"a", "+", 
      RowBox[{"b", " ", 
       SuperscriptBox[
        RowBox[{"(", 
         SuperscriptBox["F", 
          RowBox[{"g", 
           RowBox[{"(", 
            RowBox[{"e", "+", 
             RowBox[{"f", " ", "x"}]}], ")"}]}]], ")"}], "n"]}]}]], 
    RowBox[{"\[DifferentialD]", 
     RowBox[{"x", "  ", "\[LongRightArrow]", "  ", 
      RowBox[{"-", 
       FractionBox[
        SuperscriptBox[
         RowBox[{"(", 
          RowBox[{"c", "+", 
           RowBox[{"d", " ", "x"}]}], ")"}], "m"], 
        RowBox[{"a", " ", "f", " ", "g", " ", "n", " ", 
         RowBox[{"Log", "[", "F", "]"}]}]]}]}]}], " ", 
    RowBox[{"Log", "[", 
     RowBox[{"1", "+", 
      FractionBox["a", 
       RowBox[{"b", " ", 
        SuperscriptBox[
         RowBox[{"(", 
          SuperscriptBox["F", 
           RowBox[{"g", 
            RowBox[{"(", 
             RowBox[{"e", "+", 
              RowBox[{"f", " ", "x"}]}], ")"}]}]], ")"}], "n"]}]]}], 
     "]"}]}]}], "+", 
  RowBox[{
   FractionBox[
    RowBox[{"d", " ", "m"}], 
    RowBox[{"a", " ", "f", " ", "g", " ", "n", " ", 
     RowBox[{"Log", "[", "F", "]"}]}]], 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", " ", "x"}]}], ")"}], 
      RowBox[{"m", "-", "1"}]], " ", 
     RowBox[{"Log", "[", 
      RowBox[{"1", "+", 
       FractionBox["a", 
        RowBox[{"b", " ", 
         SuperscriptBox[
          RowBox[{"(", 
           SuperscriptBox["F", 
            RowBox[{"g", 
             RowBox[{"(", 
              RowBox[{"e", "+", 
               RowBox[{"f", " ", "x"}]}], ")"}]}]], ")"}], "n"]}]]}], "]"}], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579723816*^9, 3.4796580261689425`*^9}, 
   3.479661191290163*^9, {3.479686720164157*^9, 3.479686720204214*^9}, {
   3.479686831554328*^9, 3.479686834819022*^9}, {3.4940970712485504`*^9, 
   3.4940971130286083`*^9}, 3.4940971455886545`*^9, 3.4953221152495327`*^9, 
   3.495322146199576*^9, {3.495322261579737*^9, 3.4953222704097495`*^9}, 
   3.4953325041434045`*^9, 3.4953325401034546`*^9, {3.4953327138936977`*^9, 
   3.4953327232437115`*^9}, {3.4953341614457245`*^9, 
   3.4953341867757597`*^9}, {3.4953351287770786`*^9, 3.4953351438071003`*^9}, 
   3.4953417122068224`*^9, {3.4953420082485423`*^9, 3.4953420357669907`*^9}, 
   3.49534216649522*^9, {3.4953422416249523`*^9, 3.495342254650975*^9}, {
   3.4953424044876256`*^9, 3.4953424293576603`*^9}, 3.4953425414678173`*^9, {
   3.4953426292179403`*^9, 3.4953426498179693`*^9}, 3.4953427170580635`*^9, {
   3.495342827258218*^9, 3.4953428329682255`*^9}, 3.495383554791486*^9, 
   3.495383593001539*^9, {3.4953842137524085`*^9, 3.4953842254724245`*^9}, {
   3.4953845765129166`*^9, 3.4953845937429404`*^9}, {3.4953849102933836`*^9, 
   3.495384926403406*^9}, {3.5551175723291564`*^9, 3.55511757470916*^9}, {
   3.555118494910448*^9, 3.555118504140461*^9}, {3.574208884629204*^9, 
   3.5742088857392054`*^9}, 3.5742089527792993`*^9, {3.5742090128193836`*^9, 
   3.574209030559408*^9}, 3.574211090492292*^9, {3.6730426771262465`*^9, 
   3.6730427210317574`*^9}, {3.673042810906898*^9, 3.6730428463569255`*^9}, {
   3.673043004576975*^9, 3.6730430779281707`*^9}, {3.6730791555847826`*^9, 
   3.6730791724952126`*^9}, {3.6745806668244476`*^9, 
   3.6745806717617297`*^9}, {3.674582152038397*^9, 3.674582205090431*^9}},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}}],

Cell[BoxData[
 RowBox[{"(*", " ", 
  RowBox[{
   RowBox[{"Int", "[", 
    RowBox[{
     RowBox[{
      RowBox[{
       RowBox[{"(", 
        RowBox[{"c_.", "+", 
         RowBox[{"d_.", "*", "x_"}]}], ")"}], "^", "m_."}], "/", 
      RowBox[{"(", 
       RowBox[{"a_", "+", 
        RowBox[{"b_.", "*", 
         RowBox[{
          RowBox[{"(", 
           RowBox[{"F_", "^", 
            RowBox[{"(", 
             RowBox[{"g_.", "*", 
              RowBox[{"(", 
               RowBox[{"e_.", "+", 
                RowBox[{"f_.", "*", "x_"}]}], ")"}]}], ")"}]}], ")"}], "^", 
          "n_."}]}]}], ")"}]}], ",", "x_Symbol"}], "]"}], " ", ":=", "\n", 
   "  ", 
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{
       RowBox[{"-", 
        RowBox[{
         RowBox[{"(", 
          RowBox[{"c", "+", 
           RowBox[{"d", "*", "x"}]}], ")"}], "^", "m"}]}], "/", 
       RowBox[{"(", 
        RowBox[{"a", "*", "f", "*", "g", "*", "n", "*", 
         RowBox[{"Log", "[", "F", "]"}]}], ")"}]}], "*", 
      RowBox[{"Log", "[", 
       RowBox[{"1", "+", 
        RowBox[{"a", "/", 
         RowBox[{"(", 
          RowBox[{"b", "*", 
           RowBox[{
            RowBox[{"(", 
             RowBox[{"F", "^", 
              RowBox[{"(", 
               RowBox[{"g", "*", 
                RowBox[{"(", 
                 RowBox[{"e", "+", 
                  RowBox[{"f", "*", "x"}]}], ")"}]}], ")"}]}], ")"}], "^", 
            "n"}]}], ")"}]}]}], "]"}]}], " ", "+", " ", "\n", "  ", 
     RowBox[{"d", "*", 
      RowBox[{"m", "/", 
       RowBox[{"(", 
        RowBox[{"a", "*", "f", "*", "g", "*", "n", "*", 
         RowBox[{"Log", "[", "F", "]"}]}], ")"}]}], "*", 
      RowBox[{"Int", "[", 
       RowBox[{
        RowBox[{
         RowBox[{
          RowBox[{"(", 
           RowBox[{"c", "+", 
            RowBox[{"d", "*", "x"}]}], ")"}], "^", 
          RowBox[{"(", 
           RowBox[{"m", "-", "1"}], ")"}]}], "*", 
         RowBox[{"Log", "[", 
          RowBox[{"1", "+", 
           RowBox[{"a", "/", 
            RowBox[{"(", 
             RowBox[{"b", "*", 
              RowBox[{
               RowBox[{"(", 
                RowBox[{"F", "^", 
                 RowBox[{"(", 
                  RowBox[{"g", "*", 
                   RowBox[{"(", 
                    RowBox[{"e", "+", 
                    RowBox[{"f", "*", "x"}]}], ")"}]}], ")"}]}], ")"}], "^", 
               "n"}]}], ")"}]}]}], "]"}]}], ",", "x"}], "]"}]}]}], " ", "/;", 
    "\n", 
    RowBox[{
     RowBox[{"FreeQ", "[", 
      RowBox[{
       RowBox[{"{", 
        RowBox[{
        "F", ",", "a", ",", "b", ",", "c", ",", "d", ",", "e", ",", "f", ",", 
         "g", ",", "n"}], "}"}], ",", "x"}], "]"}], " ", "&&", " ", 
     RowBox[{"IGtQ", "[", 
      RowBox[{"m", ",", "0"}], "]"}]}]}]}], " ", "*)"}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, 
   3.4953222454097147`*^9, 3.4953324633733473`*^9, 3.4953326632436275`*^9, 
   3.4953341350856876`*^9, 3.4953350962570333`*^9, 3.495341936441616*^9, 
   3.495342089821086*^9, 3.495342306911067*^9, 3.4953424785377293`*^9, 
   3.4953425860178795`*^9, 3.495342687468022*^9, 3.4953835379514627`*^9, 
   3.4953841798023605`*^9, 3.4953845417128677`*^9, {3.49538487892334*^9, 
   3.4953848811733427`*^9}, 3.496087278649127*^9, 3.5488189303726234`*^9, 
   3.548819078896832*^9, {3.5488753442204957`*^9, 3.5488753452904973`*^9}, 
   3.5502002502633038`*^9, 3.550283333176357*^9, {3.555117575719161*^9, 
   3.5551175795091667`*^9}, {3.5551185923605843`*^9, 
   3.5551186395806503`*^9}, {3.5742089042092314`*^9, 
   3.5742089280792646`*^9}, {3.574209122579537*^9, 3.574209172579607*^9}, 
   3.673043333732802*^9, 3.673043404368842*^9, {3.673047771425624*^9, 
   3.6730477918977947`*^9}, 3.6730482952695856`*^9, {3.6730792086304884`*^9, 
   3.6730792349643345`*^9}, {3.6730793008160505`*^9, 
   3.6730793036084557`*^9}, {3.6745822501710095`*^9, 3.674582257739443*^9}, 
   3.6745823099244275`*^9, {3.6971434319251976`*^9, 3.6971434361314383`*^9}, {
   3.6971636548050365`*^9, 3.697163657085167*^9}, 3.7151093197022696`*^9, 
   3.7280841797709155`*^9, {3.7280863013012595`*^9, 3.728086305242485*^9}},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9}]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{Cell[TextData[StyleBox["4:",
   FontFamily->"Arial",
   FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
  RowBox[{"\[Integral]", 
   RowBox[{
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"c", "+", 
       RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"a", "+", 
       RowBox[{"b", " ", 
        SuperscriptBox[
         RowBox[{"(", 
          SuperscriptBox["F", 
           RowBox[{"g", 
            RowBox[{"(", 
             RowBox[{"e", "+", 
              RowBox[{"f", " ", "x"}]}], ")"}]}]], ")"}], "n"]}]}], ")"}], 
     "p"], 
    RowBox[{"\[DifferentialD]", "x"}], "  ", 
    StyleBox["when",
     FontFamily->"Arial",
     FontWeight->"Plain"], 
    StyleBox["  ",
     FontFamily->"Arial",
     FontWeight->"Plain"], Cell[TextData[Cell[BoxData[
     RowBox[{
      RowBox[{"p", "\[Element]", 
       SuperscriptBox["\[DoubleStruckCapitalZ]", "-"]}], "\[And]", " ", 
      RowBox[{"m", "\[Element]", 
       SuperscriptBox["\[DoubleStruckCapitalZ]", "+"]}]}]]]], 
     "None"]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193422443984776`*^9, 3.5193422447260776`*^9}, 
   3.5193424488212357`*^9, {3.5193434708254333`*^9, 3.519343473103037*^9}, {
   3.5193435097631016`*^9, 3.51934353169674*^9}, {3.519345739319023*^9, 
   3.519345739599823*^9}, 3.5193462601259375`*^9, {3.5193475061060295`*^9, 
   3.5193475233908596`*^9}, {3.519351391880224*^9, 3.5193513931402254`*^9}, {
   3.519403773021052*^9, 3.5194037798564425`*^9}, 3.5194106793250647`*^9, {
   3.519537578341393*^9, 3.5195375792305946`*^9}, {3.5195376277778797`*^9, 
   3.51953767927357*^9}, {3.5197586260636144`*^9, 3.5197586260636144`*^9}, {
   3.51978053295651*^9, 3.519780542862528*^9}, {3.5197806526087203`*^9, 
   3.5197806657595434`*^9}, {3.5197807490168896`*^9, 3.5197807642581167`*^9}, 
   3.5197809004619555`*^9, {3.519782653514635*^9, 3.5197826623286505`*^9}, {
   3.5200207810164795`*^9, 3.520020785046485*^9}, {3.5213126833374643`*^9, 
   3.5213126833374643`*^9}, {3.521328235411002*^9, 3.521328235411002*^9}, {
   3.523315567893022*^9, 3.523315567893022*^9}, {3.5233157569332867`*^9, 
   3.5233157613032923`*^9}, {3.5323023198133526`*^9, 3.532302355723403*^9}, {
   3.5323026417638035`*^9, 3.5323026417638035`*^9}, {3.5323026980138817`*^9, 
   3.5323026980138817`*^9}, {3.532636287104328*^9, 3.532636287104328*^9}, {
   3.5330803686035347`*^9, 3.533080368863535*^9}, {3.533248909289961*^9, 
   3.5332489110299635`*^9}, {3.534962469565694*^9, 3.5349624910757236`*^9}, {
   3.5349625224057674`*^9, 3.534962522655768*^9}, {3.534965265519608*^9, 
   3.534965265519608*^9}, {3.5349668288168063`*^9, 3.5349668292668066`*^9}, {
   3.5349747358481646`*^9, 3.5349747358481646`*^9}, {3.534975457949176*^9, 
   3.534975457949176*^9}, {3.5368633963532934`*^9, 3.536863396571694*^9}, {
   3.541644322511304*^9, 3.5416443577513533`*^9}, {3.5416458269234104`*^9, 
   3.541645839633428*^9}, {3.5416460312236967`*^9, 3.541646031483697*^9}, 
   3.541646227783972*^9, {3.541647080335165*^9, 3.5416471077052035`*^9}, {
   3.541648997607849*^9, 3.5416489978278494`*^9}, 3.541649088407976*^9, {
   3.5417848711596775`*^9, 3.541784902429721*^9}, {3.541786145331461*^9, 
   3.5417861455914617`*^9}, {3.5417870037166634`*^9, 3.5417870039366636`*^9}, 
   3.5462979620242105`*^9, {3.546298264094633*^9, 3.546298265344635*^9}, {
   3.546299000235664*^9, 3.546299001505666*^9}, {3.546315119909564*^9, 
   3.5463151267659564`*^9}, {3.547227356319705*^9, 3.547227356319705*^9}, 
   3.547918559331044*^9, {3.547918610170952*^9, 3.5479186304411116`*^9}, 
   3.548537755517557*^9, 3.5485441629685287`*^9, 3.5485517520154133`*^9, {
   3.5488740727887154`*^9, 3.5488740727887154`*^9}, {3.548874206618903*^9, 
   3.548874206808903*^9}, {3.5488771226154857`*^9, 3.5488771332055006`*^9}, {
   3.548878208637006*^9, 3.548878208777006*^9}, {3.5488822719350433`*^9, 
   3.5488822719350433`*^9}, {3.5488837630471315`*^9, 
   3.5488837632171316`*^9}, {3.5488838279872217`*^9, 
   3.5488838281672225`*^9}, {3.548884313497902*^9, 3.548884313657902*^9}, {
   3.5488863124207*^9, 3.5488863214107127`*^9}, {3.548886881361497*^9, 
   3.5488868816714973`*^9}, 3.5488873559521613`*^9, {3.548887457282303*^9, 
   3.548887457282303*^9}, 3.5494981973468494`*^9, {3.5494983516935205`*^9, 
   3.549498353986725*^9}, {3.5494983855299797`*^9, 3.5494983862787814`*^9}, {
   3.549498568892702*^9, 3.54949857343231*^9}, 3.553223130431923*^9, 
   3.5551170302883973`*^9, {3.5589039463428574`*^9, 3.5589039479428596`*^9}, 
   3.5742077541576214`*^9, {3.5743644486996126`*^9, 3.574364448959613*^9}, {
   3.6730611092325025`*^9, 3.6730611160908947`*^9}, 3.6730611651697016`*^9, 
   3.674580267124586*^9, 3.6745804858200946`*^9, {3.67458184315973*^9, 
   3.6745818580135794`*^9}, {3.6745824917508273`*^9, 3.6745824919778404`*^9}, 
   3.674582610987647*^9, {3.6971421100115886`*^9, 3.6971421151298814`*^9}, {
   3.6971434437938766`*^9, 3.6971434440598917`*^9}, {3.6971578614116735`*^9, 
   3.6971578822238636`*^9}, {3.697163660344353*^9, 3.697163660573366*^9}},
 FontSize->12,
 FontWeight->"Bold"],

Cell["\<\
Derivation: Algebraic expansion\
\>", "Subsubsection",
 CellChangeTimes->{
  3.479686820458373*^9, {3.490657960122612*^9, 3.49065796095366*^9}, 
   3.495321989219356*^9, 3.4953835047314157`*^9, 3.4953842443724513`*^9, {
   3.6971421582683487`*^9, 3.697142163416643*^9}}],

Cell[TextData[{
 "Basis: ",
 Cell[BoxData[
  RowBox[{
   SuperscriptBox[
    RowBox[{"(", 
     RowBox[{"a", "+", 
      RowBox[{"b", " ", "z"}]}], ")"}], "p"], "\[Equal]", 
   RowBox[{
    FractionBox[
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"a", "+", 
        RowBox[{"b", " ", "z"}]}], ")"}], 
      RowBox[{"p", "+", "1"}]], "a"], "-", 
    FractionBox[
     RowBox[{"b", " ", "z", " ", 
      SuperscriptBox[
       RowBox[{"(", 
        RowBox[{"a", "+", 
         RowBox[{"b", " ", "z"}]}], ")"}], "p"]}], "a"]}]}]], "Input",
  CellChangeTimes->{{3.6971423198105884`*^9, 3.697142320477627*^9}, {
   3.6971423644061394`*^9, 3.6971423713605366`*^9}}]
}], "Subsubsection",
 CellChangeTimes->{
  3.479686820458373*^9, {3.490657960122612*^9, 3.49065796095366*^9}, 
   3.495321989219356*^9, 3.4953835047314157`*^9, 3.4953842443724513`*^9, {
   3.6971421582683487`*^9, 3.697142180576625*^9}, 3.697142407985632*^9}],

Cell[TextData[{
 "Rule: If  ",
 Cell[BoxData[
  RowBox[{
   RowBox[{"p", "\[Element]", 
    SuperscriptBox["\[DoubleStruckCapitalZ]", "-"]}], "\[And]", " ", 
   RowBox[{"m", "\[Element]", 
    SuperscriptBox["\[DoubleStruckCapitalZ]", "+"]}]}]]],
 ", then"
}], "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.495322033659418*^9, 3.495322068589467*^9}, {
   3.495332560963484*^9, 3.4953325798335104`*^9}, {3.4953342240158124`*^9, 
   3.4953342584258604`*^9}, 3.495335158717121*^9, {3.495341957376853*^9, 
   3.495341983631699*^9}, {3.4953421021607075`*^9, 3.495342121239541*^9}, {
   3.4953424507176905`*^9, 3.4953424521876926`*^9}, {3.4953424887677436`*^9, 
   3.4953424992877584`*^9}, {3.4953425925878887`*^9, 
   3.4953425939178905`*^9}, {3.495342746828105*^9, 3.495342777558148*^9}, {
   3.495383656421628*^9, 3.495383657121629*^9}, {3.4953837077317*^9, 
   3.495383711161705*^9}, {3.495383774431793*^9, 3.4953837774217978`*^9}, {
   3.4953838495818987`*^9, 3.4953838560319076`*^9}, {3.4953839845720873`*^9, 
   3.4953839850920887`*^9}, {3.5494983915203905`*^9, 
   3.5494983915203905`*^9}, {3.673061123204302*^9, 3.673061123204302*^9}, {
   3.6971422012788086`*^9, 3.6971422012788086`*^9}, {3.6971578877031765`*^9, 
   3.6971578877031765`*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"\[Integral]", 
   RowBox[{
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"c", "+", 
       RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"a", "+", 
       RowBox[{"b", " ", 
        SuperscriptBox[
         RowBox[{"(", 
          SuperscriptBox["F", 
           RowBox[{"g", 
            RowBox[{"(", 
             RowBox[{"e", "+", 
              RowBox[{"f", " ", "x"}]}], ")"}]}]], ")"}], "n"]}]}], ")"}], 
     "p"], 
    RowBox[{"\[DifferentialD]", 
     RowBox[{"x", "  ", "\[LongRightArrow]", "  ", "\[IndentingNewLine]", 
      FractionBox["1", "a"]}]}], " ", 
    RowBox[{"\[Integral]", 
     RowBox[{
      SuperscriptBox[
       RowBox[{"(", 
        RowBox[{"c", "+", 
         RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
      SuperscriptBox[
       RowBox[{"(", 
        RowBox[{"a", "+", 
         RowBox[{"b", " ", 
          SuperscriptBox[
           RowBox[{"(", 
            SuperscriptBox["F", 
             RowBox[{"g", 
              RowBox[{"(", 
               RowBox[{"e", "+", 
                RowBox[{"f", " ", "x"}]}], ")"}]}]], ")"}], "n"]}]}], ")"}], 
       RowBox[{"p", "+", "1"}]], 
      RowBox[{"\[DifferentialD]", "x"}]}]}]}]}], "-", 
  RowBox[{
   FractionBox["b", "a"], " ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
     SuperscriptBox[
      RowBox[{"(", 
       SuperscriptBox["F", 
        RowBox[{"g", 
         RowBox[{"(", 
          RowBox[{"e", "+", 
           RowBox[{"f", " ", "x"}]}], ")"}]}]], ")"}], "n"], " ", 
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"a", "+", 
        RowBox[{"b", " ", 
         SuperscriptBox[
          RowBox[{"(", 
           SuperscriptBox["F", 
            RowBox[{"g", 
             RowBox[{"(", 
              RowBox[{"e", "+", 
               RowBox[{"f", " ", "x"}]}], ")"}]}]], ")"}], "n"]}]}], ")"}], 
      "p"], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579723816*^9, 3.4796580261689425`*^9}, 
   3.479661191290163*^9, {3.479686720164157*^9, 3.479686720204214*^9}, {
   3.479686831554328*^9, 3.479686834819022*^9}, {3.4940970712485504`*^9, 
   3.4940971130286083`*^9}, 3.4940971455886545`*^9, 3.4953221152495327`*^9, 
   3.495322146199576*^9, {3.495322261579737*^9, 3.4953222704097495`*^9}, 
   3.4953325041434045`*^9, 3.4953325401034546`*^9, {3.4953327138936977`*^9, 
   3.4953327232437115`*^9}, {3.4953341614457245`*^9, 
   3.4953341867757597`*^9}, {3.4953351287770786`*^9, 3.4953351438071003`*^9}, 
   3.4953417122068224`*^9, {3.4953420082485423`*^9, 3.4953420357669907`*^9}, 
   3.49534216649522*^9, {3.4953422416249523`*^9, 3.495342254650975*^9}, {
   3.4953424044876256`*^9, 3.4953424293576603`*^9}, 3.4953425414678173`*^9, {
   3.4953426292179403`*^9, 3.4953426498179693`*^9}, 3.4953427170580635`*^9, {
   3.495342827258218*^9, 3.4953428329682255`*^9}, 3.495383554791486*^9, 
   3.495383593001539*^9, 3.495383741531747*^9, {3.4953838030218334`*^9, 
   3.495383812201846*^9}, 3.495383876631936*^9, {3.4953839464920344`*^9, 
   3.495383955552047*^9}, {3.549498408976821*^9, 3.5494984286484556`*^9}, {
   3.549498583541128*^9, 3.5494985849763308`*^9}, {3.555117033538402*^9, 
   3.555117036978407*^9}, {3.574207756977625*^9, 3.5742077593776283`*^9}, {
   3.673061126608496*^9, 3.6730611321118107`*^9}, {3.6730611697039614`*^9, 
   3.673061173998207*^9}, {3.6745804921814585`*^9, 3.6745804956606574`*^9}, {
   3.674581924027355*^9, 3.6745819381811647`*^9}, 3.674582617718032*^9, {
   3.6971424296108685`*^9, 3.6971424839449763`*^9}, 3.6971426514925594`*^9, 
   3.6971426866685715`*^9, {3.6971428839388547`*^9, 3.6971428893871665`*^9}},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{"c_.", "+", 
        RowBox[{"d_.", "*", "x_"}]}], ")"}], "^", "m_."}], "*", 
     RowBox[{
      RowBox[{"(", 
       RowBox[{"a_", "+", 
        RowBox[{"b_.", "*", 
         RowBox[{
          RowBox[{"(", 
           RowBox[{"F_", "^", 
            RowBox[{"(", 
             RowBox[{"g_.", "*", 
              RowBox[{"(", 
               RowBox[{"e_.", "+", 
                RowBox[{"f_.", "*", "x_"}]}], ")"}]}], ")"}]}], ")"}], "^", 
          "n_."}]}]}], ")"}], "^", "p_"}]}], ",", "x_Symbol"}], "]"}], " ", ":=",
   "\n", "  ", 
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{"1", "/", "a"}], "*", 
     RowBox[{"Int", "[", 
      RowBox[{
       RowBox[{
        RowBox[{
         RowBox[{"(", 
          RowBox[{"c", "+", 
           RowBox[{"d", "*", "x"}]}], ")"}], "^", "m"}], "*", 
        RowBox[{
         RowBox[{"(", 
          RowBox[{"a", "+", 
           RowBox[{"b", "*", 
            RowBox[{
             RowBox[{"(", 
              RowBox[{"F", "^", 
               RowBox[{"(", 
                RowBox[{"g", "*", 
                 RowBox[{"(", 
                  RowBox[{"e", "+", 
                   RowBox[{"f", "*", "x"}]}], ")"}]}], ")"}]}], ")"}], "^", 
             "n"}]}]}], ")"}], "^", 
         RowBox[{"(", 
          RowBox[{"p", "+", "1"}], ")"}]}]}], ",", "x"}], "]"}]}], " ", "-", 
    " ", "\n", "  ", 
    RowBox[{
     RowBox[{"b", "/", "a"}], "*", 
     RowBox[{"Int", "[", 
      RowBox[{
       RowBox[{
        RowBox[{
         RowBox[{"(", 
          RowBox[{"c", "+", 
           RowBox[{"d", "*", "x"}]}], ")"}], "^", "m"}], "*", 
        RowBox[{
         RowBox[{"(", 
          RowBox[{"F", "^", 
           RowBox[{"(", 
            RowBox[{"g", "*", 
             RowBox[{"(", 
              RowBox[{"e", "+", 
               RowBox[{"f", "*", "x"}]}], ")"}]}], ")"}]}], ")"}], "^", "n"}],
         "*", 
        RowBox[{
         RowBox[{"(", 
          RowBox[{"a", "+", 
           RowBox[{"b", "*", 
            RowBox[{
             RowBox[{"(", 
              RowBox[{"F", "^", 
               RowBox[{"(", 
                RowBox[{"g", "*", 
                 RowBox[{"(", 
                  RowBox[{"e", "+", 
                   RowBox[{"f", "*", "x"}]}], ")"}]}], ")"}]}], ")"}], "^", 
             "n"}]}]}], ")"}], "^", "p"}]}], ",", "x"}], "]"}]}]}], " ", "/;",
    "\n", 
   RowBox[{
    RowBox[{"FreeQ", "[", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{
       "F", ",", "a", ",", "b", ",", "c", ",", "d", ",", "e", ",", "f", ",", 
        "g", ",", "n"}], "}"}], ",", "x"}], "]"}], " ", "&&", " ", 
    RowBox[{"ILtQ", "[", 
     RowBox[{"p", ",", "0"}], "]"}], " ", "&&", " ", 
    RowBox[{"IGtQ", "[", 
     RowBox[{"m", ",", "0"}], "]"}]}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, 
   3.4953222454097147`*^9, 3.4953324633733473`*^9, 3.4953326632436275`*^9, 
   3.4953341350856876`*^9, 3.4953350962570333`*^9, 3.495341936441616*^9, 
   3.495342089821086*^9, 3.495342306911067*^9, 3.4953424785377293`*^9, 
   3.4953425860178795`*^9, 3.495342687468022*^9, 3.4953835379514627`*^9, 
   3.495383725451725*^9, 3.495383788381813*^9, 3.4953838337518764`*^9, 
   3.495383924742004*^9, 3.5488189132125998`*^9, 3.548819042776781*^9, {
   3.548887694932636*^9, 3.5488876962026377`*^9}, {3.5494983742511606`*^9, 
   3.549498381957574*^9}, {3.5494984415496783`*^9, 3.5494984587877088`*^9}, {
   3.5494985906079407`*^9, 3.5494986038367634`*^9}, {3.555117084148473*^9, 
   3.5551170938084865`*^9}, {3.5742077842576637`*^9, 
   3.5742077979476824`*^9}, {3.6730611362840495`*^9, 3.673061146553637*^9}, {
   3.673061190537153*^9, 3.6730612108183126`*^9}, {3.674580503211089*^9, 
   3.6745805368530135`*^9}, {3.6745819488317738`*^9, 3.674581981350634*^9}, {
   3.697142709399872*^9, 3.697142717657344*^9}, 3.69714277527664*^9, {
   3.6971428969275975`*^9, 3.6971428990077167`*^9}, {3.6971578961226587`*^9, 
   3.697157929678578*^9}, {3.7151094442493925`*^9, 3.715109476246223*^9}},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9}]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{Cell[TextData[StyleBox["5:",
   FontFamily->"Arial",
   FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
  RowBox[{"\[Integral]", 
   RowBox[{
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"c", "+", 
       RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"a", "+", 
       RowBox[{"b", " ", 
        SuperscriptBox[
         RowBox[{"(", 
          SuperscriptBox["F", 
           RowBox[{"g", 
            RowBox[{"(", 
             RowBox[{"e", "+", 
              RowBox[{"f", " ", "x"}]}], ")"}]}]], ")"}], "n"]}]}], ")"}], 
     "p"], 
    RowBox[{"\[DifferentialD]", "x"}], "  ", 
    StyleBox["when",
     FontFamily->"Arial",
     FontWeight->"Plain"], 
    StyleBox["  ",
     FontFamily->"Arial",
     FontWeight->"Plain"], Cell[TextData[Cell[BoxData[
     RowBox[{
      RowBox[{"m", "\[Element]", 
       SuperscriptBox["\[DoubleStruckCapitalZ]", "+"]}], "\[And]", " ", 
      RowBox[{"p", "<", 
       RowBox[{"-", "1"}]}]}]]]], "None"]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193422443984776`*^9, 3.5193422447260776`*^9}, 
   3.5193424488212357`*^9, {3.5193434708254333`*^9, 3.519343473103037*^9}, {
   3.5193435097631016`*^9, 3.51934353169674*^9}, {3.519345739319023*^9, 
   3.519345739599823*^9}, 3.5193462601259375`*^9, {3.5193475061060295`*^9, 
   3.5193475233908596`*^9}, {3.519351391880224*^9, 3.5193513931402254`*^9}, {
   3.519403773021052*^9, 3.5194037798564425`*^9}, 3.5194106793250647`*^9, {
   3.519537578341393*^9, 3.5195375792305946`*^9}, {3.5195376277778797`*^9, 
   3.51953767927357*^9}, {3.5197586260636144`*^9, 3.5197586260636144`*^9}, {
   3.51978053295651*^9, 3.519780542862528*^9}, {3.5197806526087203`*^9, 
   3.5197806657595434`*^9}, {3.5197807490168896`*^9, 3.5197807642581167`*^9}, 
   3.5197809004619555`*^9, {3.519782653514635*^9, 3.5197826623286505`*^9}, {
   3.5200207810164795`*^9, 3.520020785046485*^9}, {3.5213126833374643`*^9, 
   3.5213126833374643`*^9}, {3.521328235411002*^9, 3.521328235411002*^9}, {
   3.523315567893022*^9, 3.523315567893022*^9}, {3.5233157569332867`*^9, 
   3.5233157613032923`*^9}, {3.5323023198133526`*^9, 3.532302355723403*^9}, {
   3.5323026417638035`*^9, 3.5323026417638035`*^9}, {3.5323026980138817`*^9, 
   3.5323026980138817`*^9}, {3.532636287104328*^9, 3.532636287104328*^9}, {
   3.5330803686035347`*^9, 3.533080368863535*^9}, {3.533248909289961*^9, 
   3.5332489110299635`*^9}, {3.534962469565694*^9, 3.5349624910757236`*^9}, {
   3.5349625224057674`*^9, 3.534962522655768*^9}, {3.534965265519608*^9, 
   3.534965265519608*^9}, {3.5349668288168063`*^9, 3.5349668292668066`*^9}, {
   3.5349747358481646`*^9, 3.5349747358481646`*^9}, {3.534975457949176*^9, 
   3.534975457949176*^9}, {3.5368633963532934`*^9, 3.536863396571694*^9}, {
   3.541644322511304*^9, 3.5416443577513533`*^9}, {3.5416458269234104`*^9, 
   3.541645839633428*^9}, {3.5416460312236967`*^9, 3.541646031483697*^9}, 
   3.541646227783972*^9, {3.541647080335165*^9, 3.5416471077052035`*^9}, {
   3.541648997607849*^9, 3.5416489978278494`*^9}, 3.541649088407976*^9, {
   3.5417848711596775`*^9, 3.541784902429721*^9}, {3.541786145331461*^9, 
   3.5417861455914617`*^9}, {3.5417870037166634`*^9, 3.5417870039366636`*^9}, 
   3.5462979620242105`*^9, {3.546298264094633*^9, 3.546298265344635*^9}, {
   3.546299000235664*^9, 3.546299001505666*^9}, {3.546315119909564*^9, 
   3.5463151267659564`*^9}, {3.547227356319705*^9, 3.547227356319705*^9}, 
   3.547918559331044*^9, {3.547918610170952*^9, 3.5479186304411116`*^9}, 
   3.548537755517557*^9, 3.5485441629685287`*^9, 3.5485517520154133`*^9, {
   3.5488740727887154`*^9, 3.5488740727887154`*^9}, {3.548874206618903*^9, 
   3.548874206808903*^9}, {3.5488771226154857`*^9, 3.5488771332055006`*^9}, {
   3.548878208637006*^9, 3.548878208777006*^9}, {3.5488822719350433`*^9, 
   3.5488822719350433`*^9}, {3.5488837630471315`*^9, 
   3.5488837632171316`*^9}, {3.5488838279872217`*^9, 
   3.5488838281672225`*^9}, {3.548884313497902*^9, 3.548884313657902*^9}, {
   3.5488863124207*^9, 3.5488863214107127`*^9}, {3.548886881361497*^9, 
   3.5488868816714973`*^9}, 3.5488873559521613`*^9, {3.548887457282303*^9, 
   3.548887457282303*^9}, {3.548887801722785*^9, 3.5488878343728313`*^9}, {
   3.548887867962878*^9, 3.548887868122878*^9}, {3.549498615599184*^9, 
   3.5494986157863846`*^9}, 3.5532231320400147`*^9, {3.5551168810381885`*^9, 
   3.5551169040482206`*^9}, {3.558903916752816*^9, 3.5589039169328165`*^9}, 
   3.574207849207754*^9, 3.5742080078479767`*^9, {3.6730606874963803`*^9, 
   3.6730606975319543`*^9}, 3.6730609094650764`*^9, 3.6745802715628395`*^9, {
   3.6745807348613386`*^9, 3.6745807375444927`*^9}, 3.674582341387227*^9, {
   3.674582494560988*^9, 3.6745824947299976`*^9}, 3.6745826386022267`*^9, {
   3.6971636686738296`*^9, 3.6971636688668404`*^9}, {3.728085310656598*^9, 
   3.728085310656598*^9}, {3.7280853866859465`*^9, 3.7280853866859465`*^9}},
 FontSize->12,
 FontWeight->"Bold"],

Cell["Derivation: Integration by parts", "Subsubsection",
 CellChangeTimes->{
  3.479686820458373*^9, {3.490657960122612*^9, 3.49065796095366*^9}, 
   3.495321989219356*^9, 3.4953835047314157`*^9, 3.4953842443724513`*^9}],

Cell[TextData[{
 "Rule: If  ",
 Cell[BoxData[
  RowBox[{
   RowBox[{"m", "\[Element]", 
    SuperscriptBox["\[DoubleStruckCapitalZ]", "+"]}], "\[And]", " ", 
   RowBox[{"p", "<", 
    RowBox[{"-", "1"}]}]}]]],
 ", let ",
 Cell[BoxData[
  RowBox[{"u", "=", 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"a", "+", 
        RowBox[{"b", " ", 
         SuperscriptBox[
          RowBox[{"(", 
           SuperscriptBox["F", 
            RowBox[{"g", 
             RowBox[{"(", 
              RowBox[{"e", "+", 
               RowBox[{"f", " ", "x"}]}], ")"}]}]], ")"}], "n"]}]}], ")"}], 
      "p"], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}]],
  CellDingbat->None,
  CellChangeTimes->{
   3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
    3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
    3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
    3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
    3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
    3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
    3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
    3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
    3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
    3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
    3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
    3.5193325694253187`*^9, {3.5193422443984776`*^9, 3.5193422447260776`*^9}, 
    3.5193424488212357`*^9, {3.5193434708254333`*^9, 3.519343473103037*^9}, {
    3.5193435097631016`*^9, 3.51934353169674*^9}, {3.519345739319023*^9, 
    3.519345739599823*^9}, 3.5193462601259375`*^9, {3.5193475061060295`*^9, 
    3.5193475233908596`*^9}, {3.519351391880224*^9, 3.5193513931402254`*^9}, {
    3.519403773021052*^9, 3.5194037798564425`*^9}, 3.5194106793250647`*^9, {
    3.519537578341393*^9, 3.5195375792305946`*^9}, {3.5195376277778797`*^9, 
    3.51953767927357*^9}, {3.5197586260636144`*^9, 3.5197586260636144`*^9}, {
    3.51978053295651*^9, 3.519780542862528*^9}, {3.5197806526087203`*^9, 
    3.5197806657595434`*^9}, {3.5197807490168896`*^9, 3.5197807642581167`*^9},
     3.5197809004619555`*^9, {3.519782653514635*^9, 3.5197826623286505`*^9}, {
    3.5200207810164795`*^9, 3.520020785046485*^9}, {3.5213126833374643`*^9, 
    3.5213126833374643`*^9}, {3.521328235411002*^9, 3.521328235411002*^9}, {
    3.523315567893022*^9, 3.523315567893022*^9}, {3.5233157569332867`*^9, 
    3.5233157613032923`*^9}, {3.5323023198133526`*^9, 3.532302355723403*^9}, {
    3.5323026417638035`*^9, 3.5323026417638035`*^9}, {3.5323026980138817`*^9, 
    3.5323026980138817`*^9}, {3.532636287104328*^9, 3.532636287104328*^9}, {
    3.5330803686035347`*^9, 3.533080368863535*^9}, {3.533248909289961*^9, 
    3.5332489110299635`*^9}, {3.534962469565694*^9, 3.5349624910757236`*^9}, {
    3.5349625224057674`*^9, 3.534962522655768*^9}, {3.534965265519608*^9, 
    3.534965265519608*^9}, {3.5349668288168063`*^9, 3.5349668292668066`*^9}, {
    3.5349747358481646`*^9, 3.5349747358481646`*^9}, {3.534975457949176*^9, 
    3.534975457949176*^9}, {3.5368633963532934`*^9, 3.536863396571694*^9}, {
    3.541644322511304*^9, 3.5416443577513533`*^9}, {3.5416458269234104`*^9, 
    3.541645839633428*^9}, {3.5416460312236967`*^9, 3.541646031483697*^9}, 
    3.541646227783972*^9, {3.541647080335165*^9, 3.5416471077052035`*^9}, {
    3.541648997607849*^9, 3.5416489978278494`*^9}, 3.541649088407976*^9, {
    3.5417848711596775`*^9, 3.541784902429721*^9}, {3.541786145331461*^9, 
    3.5417861455914617`*^9}, {3.5417870037166634`*^9, 3.5417870039366636`*^9},
     3.5462979620242105`*^9, {3.546298264094633*^9, 3.546298265344635*^9}, {
    3.546299000235664*^9, 3.546299001505666*^9}, {3.546315119909564*^9, 
    3.5463151267659564`*^9}, {3.547227356319705*^9, 3.547227356319705*^9}, 
    3.547918559331044*^9, {3.547918610170952*^9, 3.5479186304411116`*^9}, 
    3.548537755517557*^9, 3.5485441629685287`*^9, 3.5485517520154133`*^9, {
    3.5488740727887154`*^9, 3.5488740727887154`*^9}, {3.548874206618903*^9, 
    3.548874206808903*^9}, {3.5488771226154857`*^9, 3.5488771332055006`*^9}, {
    3.548878208637006*^9, 3.548878208777006*^9}, {3.5488822719350433`*^9, 
    3.5488822719350433`*^9}, {3.5488837630471315`*^9, 
    3.5488837632171316`*^9}, {3.5488838279872217`*^9, 
    3.5488838281672225`*^9}, {3.548884313497902*^9, 3.548884313657902*^9}, {
    3.5488863124207*^9, 3.5488863214107127`*^9}, {3.548886881361497*^9, 
    3.5488868816714973`*^9}, 3.5488873559521613`*^9, {3.548887457282303*^9, 
    3.548887457282303*^9}, {3.548887801722785*^9, 3.5488878343728313`*^9}, {
    3.548887867962878*^9, 3.548887868122878*^9}, {3.549498615599184*^9, 
    3.5494986157863846`*^9}, 3.5532231320400147`*^9, {3.5551168810381885`*^9, 
    3.5551169040482206`*^9}, {3.558903916752816*^9, 3.5589039169328165`*^9}, 
    3.574207849207754*^9, 3.5742080078479767`*^9, {3.6730606874963803`*^9, 
    3.6730606975319543`*^9}, 3.6730609094650764`*^9},
  FontSize->12,
  FontWeight->"Bold"],
 ", then"
}], "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.495322033659418*^9, 3.495322068589467*^9}, {
   3.495332560963484*^9, 3.4953325798335104`*^9}, {3.4953342240158124`*^9, 
   3.4953342584258604`*^9}, 3.495335158717121*^9, {3.495341957376853*^9, 
   3.495341983631699*^9}, {3.4953421021607075`*^9, 3.495342121239541*^9}, {
   3.4953424507176905`*^9, 3.4953424521876926`*^9}, {3.4953424887677436`*^9, 
   3.4953424992877584`*^9}, {3.4953425925878887`*^9, 
   3.4953425939178905`*^9}, {3.495342746828105*^9, 3.495342777558148*^9}, {
   3.495383656421628*^9, 3.495383657121629*^9}, {3.4953837077317*^9, 
   3.495383711161705*^9}, {3.495384263492478*^9, 3.495384279642501*^9}, {
   3.6730607044283485`*^9, 3.6730607044283485`*^9}, {3.6730609579688506`*^9, 
   3.6730609803261294`*^9}, 3.673061018889335*^9, {3.673078922823572*^9, 
   3.673078932667189*^9}, {3.6745823640955257`*^9, 3.6745823640955257`*^9}, {
   3.728085312813721*^9, 3.728085312813721*^9}, {3.728085383130743*^9, 
   3.728085383130743*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"\[Integral]", 
   RowBox[{
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"c", "+", 
       RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"a", "+", 
       RowBox[{"b", " ", 
        SuperscriptBox[
         RowBox[{"(", 
          SuperscriptBox["F", 
           RowBox[{"g", 
            RowBox[{"(", 
             RowBox[{"e", "+", 
              RowBox[{"f", " ", "x"}]}], ")"}]}]], ")"}], "n"]}]}], ")"}], 
     "p"], 
    RowBox[{"\[DifferentialD]", 
     RowBox[{"x", "  ", "\[LongRightArrow]", "  ", "u"}]}], " ", 
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"c", "+", 
       RowBox[{"d", " ", "x"}]}], ")"}], "m"]}]}], "-", 
  RowBox[{"d", " ", "m", 
   RowBox[{"\[Integral]", 
    RowBox[{"u", " ", 
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", " ", "x"}]}], ")"}], 
      RowBox[{"m", "-", "1"}]], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579723816*^9, 3.4796580261689425`*^9}, 
   3.479661191290163*^9, {3.479686720164157*^9, 3.479686720204214*^9}, {
   3.479686831554328*^9, 3.479686834819022*^9}, {3.4940970712485504`*^9, 
   3.4940971130286083`*^9}, 3.4940971455886545`*^9, 3.4953221152495327`*^9, 
   3.495322146199576*^9, {3.495322261579737*^9, 3.4953222704097495`*^9}, 
   3.4953325041434045`*^9, 3.4953325401034546`*^9, {3.4953327138936977`*^9, 
   3.4953327232437115`*^9}, {3.4953341614457245`*^9, 
   3.4953341867757597`*^9}, {3.4953351287770786`*^9, 3.4953351438071003`*^9}, 
   3.4953417122068224`*^9, {3.4953420082485423`*^9, 3.4953420357669907`*^9}, 
   3.49534216649522*^9, {3.4953422416249523`*^9, 3.495342254650975*^9}, {
   3.4953424044876256`*^9, 3.4953424293576603`*^9}, 3.4953425414678173`*^9, {
   3.4953426292179403`*^9, 3.4953426498179693`*^9}, 3.4953427170580635`*^9, {
   3.495342827258218*^9, 3.4953428329682255`*^9}, 3.495383554791486*^9, 
   3.495383593001539*^9, 3.495383741531747*^9, {3.495384375472635*^9, 
   3.4953843868926506`*^9}, {3.555116638727849*^9, 3.5551166646478853`*^9}, {
   3.5551169070082245`*^9, 3.555116913248234*^9}, {3.5742080135179844`*^9, 
   3.574208019897993*^9}, {3.574208093228096*^9, 3.5742080986881037`*^9}, {
   3.673060709472637*^9, 3.6730607144809237`*^9}, {3.673060911693204*^9, 
   3.6730609202396927`*^9}, {3.673061029571946*^9, 3.673061049643094*^9}, {
   3.6745823517158175`*^9, 3.6745823815525246`*^9}, {3.6745826431964893`*^9, 
   3.6745826498428698`*^9}},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{"c_.", "+", 
        RowBox[{"d_.", "*", "x_"}]}], ")"}], "^", "m_."}], "*", 
     RowBox[{
      RowBox[{"(", 
       RowBox[{"a_", "+", 
        RowBox[{"b_.", "*", 
         RowBox[{
          RowBox[{"(", 
           RowBox[{"F_", "^", 
            RowBox[{"(", 
             RowBox[{"g_.", "*", 
              RowBox[{"(", 
               RowBox[{"e_.", "+", 
                RowBox[{"f_.", "*", "x_"}]}], ")"}]}], ")"}]}], ")"}], "^", 
          "n_."}]}]}], ")"}], "^", "p_"}]}], ",", "x_Symbol"}], "]"}], " ", ":=",
   "\n", "  ", 
  RowBox[{
   RowBox[{"With", "[", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{"u", "=", 
       RowBox[{"IntHide", "[", 
        RowBox[{
         RowBox[{
          RowBox[{"(", 
           RowBox[{"a", "+", 
            RowBox[{"b", "*", 
             RowBox[{
              RowBox[{"(", 
               RowBox[{"F", "^", 
                RowBox[{"(", 
                 RowBox[{"g", "*", 
                  RowBox[{"(", 
                   RowBox[{"e", "+", 
                    RowBox[{"f", "*", "x"}]}], ")"}]}], ")"}]}], ")"}], "^", 
              "n"}]}]}], ")"}], "^", "p"}], ",", "x"}], "]"}]}], "}"}], ",", 
     "\n", "  ", 
     RowBox[{
      RowBox[{"Dist", "[", 
       RowBox[{
        RowBox[{
         RowBox[{"(", 
          RowBox[{"c", "+", 
           RowBox[{"d", "*", "x"}]}], ")"}], "^", "m"}], ",", "u", ",", "x"}],
        "]"}], " ", "-", " ", 
      RowBox[{"d", "*", "m", "*", 
       RowBox[{"Int", "[", 
        RowBox[{
         RowBox[{
          RowBox[{
           RowBox[{"(", 
            RowBox[{"c", "+", 
             RowBox[{"d", "*", "x"}]}], ")"}], "^", 
           RowBox[{"(", 
            RowBox[{"m", "-", "1"}], ")"}]}], "*", "u"}], ",", "x"}], 
        "]"}]}]}]}], "]"}], " ", "/;", "\n", 
   RowBox[{
    RowBox[{"FreeQ", "[", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{
       "F", ",", "a", ",", "b", ",", "c", ",", "d", ",", "e", ",", "f", ",", 
        "g", ",", "n"}], "}"}], ",", "x"}], "]"}], " ", "&&", " ", 
    RowBox[{"IGtQ", "[", 
     RowBox[{"m", ",", "0"}], "]"}], " ", "&&", " ", 
    RowBox[{"LtQ", "[", 
     RowBox[{"p", ",", 
      RowBox[{"-", "1"}]}], "]"}]}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, 
   3.4953222454097147`*^9, 3.4953324633733473`*^9, 3.4953326632436275`*^9, 
   3.4953341350856876`*^9, 3.4953350962570333`*^9, 3.495341936441616*^9, 
   3.495342089821086*^9, 3.495342306911067*^9, 3.4953424785377293`*^9, 
   3.4953425860178795`*^9, 3.495342687468022*^9, 3.4953835379514627`*^9, 
   3.495383725451725*^9, 3.495384254702466*^9, {3.548818805580448*^9, 
   3.548818807250451*^9}, 3.5488189216926117`*^9, 3.548819063986811*^9, {
   3.550161631836937*^9, 3.550161643166953*^9}, 3.550283324221941*^9, 
   3.55511668902792*^9, {3.5551168376881275`*^9, 3.5551168495181446`*^9}, {
   3.5551169473682814`*^9, 3.555116985178334*^9}, {3.574208033888013*^9, 
   3.5742080714080653`*^9}, {3.6218140458407564`*^9, 3.621814058166462*^9}, {
   3.6414120172506447`*^9, 3.641412018202699*^9}, 3.6414121203865433`*^9, {
   3.6730607196592197`*^9, 3.6730607304588375`*^9}, {3.6730609938959055`*^9, 
   3.6730610063336167`*^9}, {3.674582417855601*^9, 3.6745824576728783`*^9}, {
   3.715109357947457*^9, 3.715109379568693*^9}, 3.7280853004530144`*^9},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9}]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{Cell[TextData[StyleBox["6.",
   FontFamily->"Arial"]], "None"], "  ", 
  RowBox[{"\[Integral]", 
   RowBox[{
    SuperscriptBox["u", "m"], " ", 
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"a", "+", 
       RowBox[{"b", " ", 
        SuperscriptBox[
         RowBox[{"(", 
          SuperscriptBox["F", 
           RowBox[{"g", " ", "v"}]], ")"}], "n"]}]}], ")"}], "p"], 
    RowBox[{"\[DifferentialD]", "x"}], "  ", 
    StyleBox["when",
     FontFamily->"Arial",
     FontWeight->"Plain"], 
    StyleBox["  ",
     FontFamily->"Arial",
     FontWeight->"Plain"], Cell[TextData[Cell[BoxData[
     RowBox[{
      RowBox[{"v", "\[Equal]", 
       RowBox[{"e", "+", 
        RowBox[{"f", " ", "x"}]}]}], " ", "\[And]", " ", 
      RowBox[{"u", "\[Equal]", 
       SuperscriptBox[
        RowBox[{"(", 
         RowBox[{"c", "+", 
          RowBox[{"d", " ", "x"}]}], ")"}], "q"]}]}]]]], 
     "None"]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.4964645213514385`*^9, 
   3.5192470594155855`*^9, {3.519247288125906*^9, 3.5192473207459517`*^9}, {
   3.519250976620308*^9, 3.5192509795375133`*^9}, 3.5192520188581386`*^9, {
   3.5193215829334736`*^9, 3.519321595725496*^9}, {3.5194112531544743`*^9, 
   3.5194112537784753`*^9}, {3.519411537636574*^9, 3.519411568711828*^9}, {
   3.519793882583559*^9, 3.519793883878361*^9}, {3.5214731238817034`*^9, 
   3.521473124371704*^9}, {3.523316505044334*^9, 3.523316505044334*^9}, {
   3.5233184226511183`*^9, 3.523318422861119*^9}, 3.5328367822219863`*^9, {
   3.5328742732655816`*^9, 3.5328742968684235`*^9}, {3.532874332701686*^9, 
   3.532874341843302*^9}, {3.532874546156861*^9, 3.532874552443672*^9}, {
   3.5328756408887835`*^9, 3.5328756426671867`*^9}, {3.5328756840852594`*^9, 
   3.532875690340871*^9}, {3.5328775995694237`*^9, 3.532877607619038*^9}, {
   3.5330804511936502`*^9, 3.5330804511936502`*^9}, {3.5330827213068285`*^9, 
   3.5330827218568287`*^9}, {3.5368634501889877`*^9, 
   3.5368634503449883`*^9}, {3.537654628221073*^9, 3.537654628221073*^9}, {
   3.5376556650325246`*^9, 3.5376556690825305`*^9}, {3.5376558319727583`*^9, 
   3.537655832122759*^9}, {3.5403190180369616`*^9, 3.540319018277975*^9}, {
   3.540320078404611*^9, 3.5403200969116697`*^9}, {3.54497301407633*^9, 
   3.5449730147315316`*^9}, {3.5449730455259857`*^9, 3.54497310485289*^9}, 
   3.5449734027198133`*^9, {3.544973705703545*^9, 3.5449737123023567`*^9}, {
   3.5449743847103376`*^9, 3.5449743847103376`*^9}, {3.5449897190542192`*^9, 
   3.5449897371142445`*^9}, 3.54501330464754*^9, 3.5450140205483975`*^9, {
   3.5450140745556927`*^9, 3.545014074867693*^9}, {3.545014174442668*^9, 
   3.5450141765018716`*^9}, 3.5450143633121996`*^9, {3.5450664467216406`*^9, 
   3.5450664468916407`*^9}, {3.5450708578278165`*^9, 3.545070858737818*^9}, {
   3.5454157832263145`*^9, 3.5454158086387587`*^9}, {3.5454238599089427`*^9, 
   3.5454238602689433`*^9}, 3.5454965482568398`*^9, {3.545497042817532*^9, 
   3.545497042817532*^9}, {3.545497076297579*^9, 3.5454970772875805`*^9}, {
   3.545503623646745*^9, 3.5455036255667477`*^9}, {3.546105323118441*^9, 
   3.5461053352552624`*^9}, 3.5462961408916607`*^9, {3.5462968426126432`*^9, 
   3.5462968519326563`*^9}, 3.546297534403612*^9, 3.5485441477885075`*^9, {
   3.548544198208578*^9, 3.548544198208578*^9}, 3.54887390449848*^9, {
   3.5488740223586454`*^9, 3.5488740437186747`*^9}, 3.548877131725498*^9, {
   3.5488782126170115`*^9, 3.548878212797012*^9}, {3.5488823275851216`*^9, 
   3.5488823275851216`*^9}, {3.548883756427122*^9, 3.5488837567271223`*^9}, {
   3.5488838217072134`*^9, 3.5488838219072137`*^9}, {3.5488843097178965`*^9, 
   3.5488843099378967`*^9}, {3.550105104734296*^9, 3.5501051093043027`*^9}, {
   3.55010517183439*^9, 3.550105192284419*^9}, 3.5501052253544655`*^9, {
   3.550108250838701*^9, 3.550108250838701*^9}, 3.5528423038268895`*^9, {
   3.5528426837174215`*^9, 3.5528426844774227`*^9}, 3.552842965457816*^9, 
   3.552843559138647*^9, {3.552843844929047*^9, 3.552843844929047*^9}, 
   3.5528471159536266`*^9, {3.5529260051623287`*^9, 3.5529260232427607`*^9}, 
   3.5532224065361104`*^9, {3.573076009630292*^9, 3.573076014570298*^9}, 
   3.5795486827948823`*^9, 3.579548954875263*^9, 3.579549298935745*^9, 
   3.5795493579558277`*^9, 3.5795585816463127`*^9, 3.579558846416683*^9, {
   3.5795664832515965`*^9, 3.5795665009316216`*^9}, {3.579567522703052*^9, 
   3.579567532633066*^9}, {3.5795675729431224`*^9, 3.5795675787231307`*^9}, {
   3.5795676334332075`*^9, 3.5795676351132097`*^9}, {3.6144488225490737`*^9, 
   3.614448826297288*^9}, {3.67381553572453*^9, 3.673815535939543*^9}, {
   3.673815655509382*^9, 3.673815655509382*^9}, {3.673815764735629*^9, 
   3.673815772665083*^9}, {3.6745848139966516`*^9, 3.6745848141596613`*^9}, {
   3.674585299864442*^9, 3.674585314834298*^9}, {3.6749278425116143`*^9, 
   3.674927844224658*^9}, {3.6749279880799503`*^9, 3.6749279901509533`*^9}, {
   3.6749283536625204`*^9, 3.674928363176536*^9}, {3.6749284225586247`*^9, 
   3.674928449466666*^9}, 3.67492849486474*^9, {3.674928545865817*^9, 
   3.6749285568998337`*^9}, {3.6749289693746076`*^9, 3.674928969574608*^9}, {
   3.6971636707509484`*^9, 3.697163670986962*^9}},
 FontSize->12,
 FontWeight->"Bold"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t", 
  RowBox[{Cell[TextData[StyleBox["1:",
    FontFamily->"Arial",
    FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox["u", "m"], " ", 
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"a", "+", 
        RowBox[{"b", " ", 
         SuperscriptBox[
          RowBox[{"(", 
           SuperscriptBox["F", 
            RowBox[{"g", " ", "v"}]], ")"}], "n"]}]}], ")"}], "p"], 
     RowBox[{"\[DifferentialD]", "x"}], "  ", 
     StyleBox["when",
      FontFamily->"Arial",
      FontWeight->"Plain"], 
     StyleBox["  ",
      FontFamily->"Arial",
      FontWeight->"Plain"], Cell[TextData[Cell[BoxData[
      RowBox[{
       RowBox[{"v", "\[Equal]", 
        RowBox[{"e", "+", 
         RowBox[{"f", " ", "x"}]}]}], " ", "\[And]", " ", 
       RowBox[{"u", "\[Equal]", 
        SuperscriptBox[
         RowBox[{"(", 
          RowBox[{"c", "+", 
           RowBox[{"d", " ", "x"}]}], ")"}], "q"]}], " ", "\[And]", " ", 
       RowBox[{"m", "\[Element]", "\[DoubleStruckCapitalZ]"}]}]]]], 
      "None"]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.4964645213514385`*^9, 
   3.5192470594155855`*^9, {3.519247288125906*^9, 3.5192473207459517`*^9}, {
   3.519247363296011*^9, 3.519247363796012*^9}, 3.519258420199427*^9, 
   3.5192584863895197`*^9, {3.519320760160028*^9, 3.519320767367241*^9}, 
   3.5193299099191504`*^9, {3.5193300315093203`*^9, 3.519330031709321*^9}, {
   3.5193323099968634`*^9, 3.5193324414114943`*^9}, {3.5193327317904043`*^9, 
   3.519332762148057*^9}, {3.519340455685931*^9, 3.519340463318947*^9}, 
   3.519340635545749*^9, {3.5193422478304834`*^9, 3.519342248704085*^9}, {
   3.519342492766513*^9, 3.519342493156514*^9}, {3.519345750301442*^9, 
   3.5193457505978427`*^9}, 3.519346263464343*^9, {3.519349307658399*^9, 
   3.519349310965605*^9}, {3.519352294781488*^9, 3.5193522950014887`*^9}, {
   3.5194134359455075`*^9, 3.519413446054325*^9}, {3.5194134908888044`*^9, 
   3.5194134927764077`*^9}, {3.5194141847624226`*^9, 
   3.5194142323581066`*^9}, {3.519414342899901*^9, 3.519414474720132*^9}, {
   3.5197939535480833`*^9, 3.5197939539380836`*^9}, {3.5197940297698174`*^9, 
   3.5197940299726176`*^9}, 3.519795153580191*^9, {3.5200103027397947`*^9, 
   3.5200103044997973`*^9}, {3.5232186878169007`*^9, 
   3.5232186881569014`*^9}, {3.5232193955778913`*^9, 
   3.5232193955778913`*^9}, {3.5233166487845354`*^9, 
   3.5233166487845354`*^9}, {3.5233167010346084`*^9, 
   3.5233167010346084`*^9}, {3.523318546315711*^9, 3.523318573368556*^9}, {
   3.5239220709492664`*^9, 3.5239220924617043`*^9}, {3.5239221279205666`*^9, 
   3.5239221365629816`*^9}, {3.523930270899587*^9, 3.523930270899587*^9}, {
   3.5240761703528385`*^9, 3.5240761703528385`*^9}, {3.52407843124921*^9, 
   3.5240784332304134`*^9}, {3.5240784932749186`*^9, 
   3.5240784932749186`*^9}, {3.524079230095413*^9, 3.524079246553442*^9}, {
   3.5240858499762287`*^9, 3.524085860178646*^9}, {3.524085945261196*^9, 
   3.524085945261196*^9}, {3.5242763399632483`*^9, 3.5242763411488504`*^9}, {
   3.5242771575294847`*^9, 3.5242771578102846`*^9}, {3.530500224433988*^9, 
   3.530500225113989*^9}, {3.5313467876910577`*^9, 3.5313467883119507`*^9}, {
   3.5332231066826243`*^9, 3.5332231066826243`*^9}, {3.5332248881251183`*^9, 
   3.533224890245121*^9}, {3.5361011385491977`*^9, 3.5361011394332485`*^9}, {
   3.5363501469745007`*^9, 3.5363501506345057`*^9}, {3.5363617411167684`*^9, 
   3.5363617534067855`*^9}, {3.536366615151597*^9, 3.5363666230616083`*^9}, {
   3.5363667754518213`*^9, 3.5363667821318307`*^9}, {3.536366906722005*^9, 
   3.536366922022027*^9}, {3.536596861742076*^9, 3.5365968701504908`*^9}, {
   3.5366042542749715`*^9, 3.536604254462172*^9}, {3.536625554425806*^9, 
   3.536625554775807*^9}, {3.5366276426139326`*^9, 3.536627642853933*^9}, {
   3.536629402336396*^9, 3.536629402746397*^9}, {3.5366940603006477`*^9, 
   3.536694061273703*^9}, {3.5366943323932104`*^9, 3.536694332661226*^9}, {
   3.536694390476533*^9, 3.536694402176202*^9}, {3.5368634751022315`*^9, 
   3.5368634756170325`*^9}, {3.536980474148706*^9, 3.5369804925387316`*^9}, {
   3.5369807910591497`*^9, 3.5369808136991816`*^9}, 3.538443999546241*^9, {
   3.538444040229568*^9, 3.5384440801737804`*^9}, {3.544471201864182*^9, 
   3.5444712020241823`*^9}, {3.5499343621399236`*^9, 
   3.5499343628887253`*^9}, {3.5693429012648573`*^9, 3.569342923814889*^9}, {
   3.5693432071452856`*^9, 3.5693432074652863`*^9}, {3.569344801387518*^9, 
   3.569344801387518*^9}, {3.5693533425947766`*^9, 3.5693533510147886`*^9}, {
   3.569353493264988*^9, 3.5693535115050135`*^9}, {3.5694290997903204`*^9, 
   3.569429100480322*^9}, {3.5718791895851946`*^9, 3.571879189775195*^9}, {
   3.572109875965929*^9, 3.572109876175929*^9}, {3.5722014327630053`*^9, 
   3.5722014341930075`*^9}, {3.5722014837230773`*^9, 
   3.5722014840130777`*^9}, {3.572205109828057*^9, 3.572205110068057*^9}, {
   3.5722827529728394`*^9, 3.57228275320284*^9}, {3.5728333915121145`*^9, 
   3.5728334104821405`*^9}, {3.5729228769791255`*^9, 3.572922877149126*^9}, {
   3.5729229234691906`*^9, 3.572922991489286*^9}, {3.5795507933078365`*^9, 
   3.5795508236878796`*^9}, {3.5795516549990435`*^9, 
   3.5795516630790544`*^9}, {3.5795525734903293`*^9, 
   3.5795525789003367`*^9}, {3.579558592186327*^9, 3.579558592186327*^9}, {
   3.579558853016692*^9, 3.579558853016692*^9}, {3.579560242101156*^9, 
   3.579560248491165*^9}, {3.579563052515091*^9, 3.579563061675103*^9}, {
   3.5795662562075777`*^9, 3.579566302467643*^9}, {3.5795665034316254`*^9, 
   3.5795665052116275`*^9}, {3.5795675921531496`*^9, 3.5795675921531496`*^9}, 
   3.614448848452555*^9, {3.673815666717023*^9, 3.673815666717023*^9}, 
   3.673815779681484*^9, {3.6745853002534647`*^9, 3.6745853166113997`*^9}, {
   3.674928423358626*^9, 3.6749284510366683`*^9}, {3.674928489172727*^9, 
   3.6749284926647367`*^9}, {3.674928570583855*^9, 3.674928590888894*^9}, 
   3.6749289793206253`*^9},
 FontSize->12,
 FontWeight->"Bold"],

Cell["Derivation: Algebraic normalization", "Subsubsection",
 CellChangeTimes->{
  3.4953915070313115`*^9, 3.495840214754669*^9, {3.4958483177760134`*^9, 
   3.495848318886015*^9}, {3.495925911922002*^9, 3.495925920122013*^9}, 
   3.4961847242848682`*^9, 3.496184957525194*^9, 3.496188449500083*^9, 
   3.4961908979261436`*^9, 3.4961924998683863`*^9, 3.4961928754389124`*^9, {
   3.51849075797458*^9, 3.5184907698562593`*^9}, {3.5747099413791857`*^9, 
   3.5747099436591883`*^9}}],

Cell[TextData[{
 "Rule: If  ",
 Cell[BoxData[
  RowBox[{
   RowBox[{"v", "\[Equal]", 
    RowBox[{"e", "+", 
     RowBox[{"f", " ", "x"}]}]}], " ", "\[And]", " ", 
   RowBox[{"u", "\[Equal]", 
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"c", "+", 
       RowBox[{"d", " ", "x"}]}], ")"}], "q"]}], " ", "\[And]", " ", 
   RowBox[{"m", "\[Element]", "\[DoubleStruckCapitalZ]"}]}]]],
 ", then"
}], "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.495391630941485*^9, 3.4953916682515373`*^9}, 
   3.495398892616804*^9, {3.495402251391507*^9, 3.4954022711315346`*^9}, {
   3.495488783827441*^9, 3.4954888500775337`*^9}, {3.4954889799777155`*^9, 
   3.495489009627757*^9}, {3.495489141407942*^9, 3.495489151717956*^9}, {
   3.495489235178073*^9, 3.495489239268079*^9}, {3.495489386068284*^9, 
   3.4954894212383337`*^9}, {3.495489464438394*^9, 3.4954894647383947`*^9}, {
   3.495489500148444*^9, 3.4954895034984484`*^9}, {3.4954897331687703`*^9, 
   3.495489780118836*^9}, {3.495493913565876*^9, 3.495493935715907*^9}, {
   3.495494084919718*^9, 3.4954941083997507`*^9}, 3.495494511560315*^9, {
   3.495494580840412*^9, 3.4954945834404163`*^9}, {3.495494615970462*^9, 
   3.495494740200636*^9}, {3.495494880710832*^9, 3.4954948880508423`*^9}, {
   3.4954949819649754`*^9, 3.495494999445*^9}, {3.495495046515066*^9, 
   3.495495072875103*^9}, {3.4954951239351745`*^9, 3.4954951241651745`*^9}, {
   3.4954953673051157`*^9, 3.4954953726551237`*^9}, {3.495495450474448*^9, 
   3.495495453173253*^9}, {3.495495542140209*^9, 3.495495583792282*^9}, {
   3.4954957431465626`*^9, 3.4954957746118174`*^9}, {3.495495914419263*^9, 
   3.495495917804469*^9}, {3.495495955322535*^9, 3.4954960459586945`*^9}, {
   3.495496189588146*^9, 3.4954961923181515`*^9}, {3.4954973384365644`*^9, 
   3.495497357577798*^9}, {3.5240858715198665`*^9, 3.52408591925595*^9}, {
   3.5331536712917595`*^9, 3.5331536971617956`*^9}, {3.533223063022563*^9, 
   3.5332230658125668`*^9}, {3.533224894045127*^9, 3.5332248954551287`*^9}, 
   3.5363501685345306`*^9, {3.538444353705845*^9, 3.5384443676658645`*^9}, {
   3.5499343655595293`*^9, 3.549934366379531*^9}, {3.5693429326149015`*^9, 
   3.5693429326149015`*^9}, {3.5693448172975397`*^9, 
   3.5693448172975397`*^9}, {3.5693533761948233`*^9, 
   3.5693533761948233`*^9}, {3.572923022359329*^9, 3.572923022359329*^9}, {
   3.579550848297914*^9, 3.579550848297914*^9}, {3.579551672029067*^9, 
   3.579551672029067*^9}, {3.5795525826803417`*^9, 3.5795525826803417`*^9}, {
   3.5795585969663343`*^9, 3.5795585969663343`*^9}, {3.5795588543766947`*^9, 
   3.5795588543766947`*^9}, {3.57956025200117*^9, 3.57956025200117*^9}, {
   3.5795630745851216`*^9, 3.5795630745851216`*^9}, {3.579566316937663*^9, 
   3.579566316937663*^9}, {3.579567604493167*^9, 3.579567604493167*^9}, {
   3.673815671708308*^9, 3.673815671708308*^9}, {3.6745853185845127`*^9, 
   3.6745853185845127`*^9}, {3.674928425152629*^9, 3.674928453037672*^9}, {
   3.6749286041569176`*^9, 3.6749286041569176`*^9}}],

Cell[BoxData[
 RowBox[{"\[Integral]", 
  RowBox[{
   SuperscriptBox["u", "m"], " ", 
   SuperscriptBox[
    RowBox[{"(", 
     RowBox[{"a", "+", 
      RowBox[{"b", " ", 
       SuperscriptBox[
        RowBox[{"(", 
         SuperscriptBox["F", 
          RowBox[{"g", " ", "v"}]], ")"}], "n"]}]}], ")"}], "p"], 
   RowBox[{"\[DifferentialD]", 
    RowBox[{"x", "  ", "\[LongRightArrow]", "  ", 
     RowBox[{"\[Integral]", 
      RowBox[{
       SuperscriptBox[
        RowBox[{"(", 
         RowBox[{"c", "+", 
          RowBox[{"d", " ", "x"}]}], ")"}], 
        RowBox[{"m", " ", "q"}]], " ", 
       SuperscriptBox[
        RowBox[{"(", 
         RowBox[{"a", "+", 
          RowBox[{"b", " ", 
           SuperscriptBox[
            RowBox[{"(", 
             SuperscriptBox["F", 
              RowBox[{"g", 
               RowBox[{"(", 
                RowBox[{"e", "+", 
                 RowBox[{"f", " ", "x"}]}], ")"}]}]], ")"}], "n"]}]}], ")"}], 
        "p"], 
       RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579723816*^9, 3.4796580261689425`*^9}, 
   3.479661191290163*^9, {3.479686720164157*^9, 3.479686720204214*^9}, {
   3.479686831554328*^9, 3.479686834819022*^9}, {3.4940970712485504`*^9, 
   3.4940971130286083`*^9}, 3.4940971455886545`*^9, 3.4953915848214207`*^9, 
   3.495391696001576*^9, {3.495401987881138*^9, 3.49540199674115*^9}, {
   3.495402119591322*^9, 3.4954021482513623`*^9}, {3.4954887467773895`*^9, 
   3.4954887749874287`*^9}, 3.495488944397666*^9, 3.495489263778113*^9, 
   3.4954893636582527`*^9, 3.4954898177188883`*^9, 3.495489875358969*^9, 
   3.495493833545764*^9, 3.495493864175807*^9, 3.4954940303560395`*^9, 
   3.4954941516698112`*^9, 3.4954942700599775`*^9, 3.4954948473707857`*^9, {
   3.495495264965372*^9, 3.4954952782653904`*^9}, {3.4954956850364604`*^9, 
   3.4954957088577023`*^9}, {3.495495865341577*^9, 3.4954958884920177`*^9}, {
   3.495496084225561*^9, 3.4954961029143944`*^9}, {3.495496211459385*^9, 
   3.4954962213810024`*^9}, 3.495496440218187*^9, {3.495497026732417*^9, 
   3.4954970596328745`*^9}, {3.495497256957621*^9, 3.495497302338101*^9}, 
   3.533151648818928*^9, 3.5331516797189713`*^9, 3.533151717349024*^9, {
   3.5331518776292486`*^9, 3.533151970249378*^9}, {3.5331520176994443`*^9, 
   3.5331520227894516`*^9}, {3.533152054329496*^9, 3.5331520626995077`*^9}, 
   3.533152270349798*^9, {3.533153065130911*^9, 3.5331530971109557`*^9}, 
   3.53315335008131*^9, {3.5331535194015465`*^9, 3.5331535356615696`*^9}, 
   3.5331536017616625`*^9, {3.533221927890974*^9, 3.533221930830978*^9}, {
   3.533222264011444*^9, 3.533222283731472*^9}, {3.5332223304615374`*^9, 
   3.5332223353515444`*^9}, 3.5363501604045196`*^9, {3.536361767566806*^9, 
   3.5363617697768087`*^9}, {3.5363666287016163`*^9, 
   3.5363666313816195`*^9}, {3.536366784271834*^9, 3.5363667868618374`*^9}, {
   3.536366923782029*^9, 3.5363669254320316`*^9}, {3.5365968786057057`*^9, 
   3.536596881944112*^9}, {3.5366940930205193`*^9, 3.53669410440917*^9}, {
   3.5366942507975435`*^9, 3.536694251403578*^9}, {3.536694493325415*^9, 
   3.536694495295528*^9}, 3.536980387788585*^9, {3.5384441491255584`*^9, 
   3.538444158185571*^9}, 3.5384445157160716`*^9, {3.5693429405449123`*^9, 
   3.569342962434943*^9}, 3.5693487483783445`*^9, 3.5693533713048167`*^9, {
   3.56935351683502*^9, 3.5693535184850225`*^9}, {3.572833444812189*^9, 
   3.5728334591122093`*^9}, {3.572922997329294*^9, 3.5729229979692955`*^9}, {
   3.5729231499495077`*^9, 3.5729231561595163`*^9}, 3.5795508621679335`*^9, {
   3.579551676949074*^9, 3.579551678979077*^9}, 3.5795585977163353`*^9, 
   3.5795588554866962`*^9, {3.579560255261174*^9, 3.579560255591175*^9}, {
   3.5795630788551273`*^9, 3.579563079755129*^9}, {3.579566324877674*^9, 
   3.5795663330776854`*^9}, 3.673815674865489*^9, {3.673815782494645*^9, 
   3.673815792968244*^9}, {3.6745848725370007`*^9, 3.6745848886889243`*^9}, {
   3.6749284269326315`*^9, 3.674928456325679*^9}, 3.67492850201575*^9, 
   3.6749285739378653`*^9, {3.674928627249955*^9, 3.6749286643180113`*^9}},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{"u_", "^", "m_."}], "*", 
     RowBox[{
      RowBox[{"(", 
       RowBox[{"a_.", "+", 
        RowBox[{"b_.", "*", 
         RowBox[{
          RowBox[{"(", 
           RowBox[{"F_", "^", 
            RowBox[{"(", 
             RowBox[{"g_.", "*", "v_"}], ")"}]}], ")"}], "^", "n_."}]}]}], 
       ")"}], "^", "p_."}]}], ",", "x_Symbol"}], "]"}], " ", ":=", "\n", "  ", 
  RowBox[{
   RowBox[{"Int", "[", 
    RowBox[{
     RowBox[{
      RowBox[{
       RowBox[{"NormalizePowerOfLinear", "[", 
        RowBox[{"u", ",", "x"}], "]"}], "^", "m"}], "*", 
      RowBox[{
       RowBox[{"(", 
        RowBox[{"a", "+", 
         RowBox[{"b", "*", 
          RowBox[{
           RowBox[{"(", 
            RowBox[{"F", "^", 
             RowBox[{"(", 
              RowBox[{"g", "*", 
               RowBox[{"ExpandToSum", "[", 
                RowBox[{"v", ",", "x"}], "]"}]}], ")"}]}], ")"}], "^", 
           "n"}]}]}], ")"}], "^", "p"}]}], ",", "x"}], "]"}], " ", "/;", "\n", 
   RowBox[{
    RowBox[{"FreeQ", "[", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{"F", ",", "a", ",", "b", ",", "g", ",", "n", ",", "p"}], "}"}],
       ",", "x"}], "]"}], " ", "&&", " ", 
    RowBox[{"LinearQ", "[", 
     RowBox[{"v", ",", "x"}], "]"}], " ", "&&", " ", 
    RowBox[{"PowerOfLinearQ", "[", 
     RowBox[{"u", ",", "x"}], "]"}], " ", "&&", " ", 
    RowBox[{"Not", "[", 
     RowBox[{
      RowBox[{"LinearMatchQ", "[", 
       RowBox[{"v", ",", "x"}], "]"}], " ", "&&", " ", 
      RowBox[{"PowerOfLinearMatchQ", "[", 
       RowBox[{"u", ",", "x"}], "]"}]}], "]"}], " ", "&&", " ", 
    RowBox[{"IntegerQ", "[", "m", "]"}]}]}]}]], "Code",
 CellChangeTimes->{{3.494097165958683*^9, 3.4940971894087152`*^9}, {
   3.4940972668388243`*^9, 3.4940972723188314`*^9}, 3.49539148440128*^9, 
   3.4954018196509027`*^9, {3.495401867050969*^9, 3.4954018672609687`*^9}, {
   3.4954022872215567`*^9, 3.4954022882115583`*^9}, 3.495488713187342*^9, 
   3.4954889210576334`*^9, 3.4954892270180616`*^9, 3.4954893399982195`*^9, {
   3.4954894833184204`*^9, 3.495489485938424*^9}, 3.4954897194087505`*^9, 
   3.495493755131653*^9, {3.4954938826158323`*^9, 3.495493884975836*^9}, 
   3.4954939962159915`*^9, {3.4954942766399865`*^9, 3.4954942945800114`*^9}, 
   3.4954948208307486`*^9, {3.495495087335123*^9, 3.495495087705124*^9}, 
   3.4954951805652537`*^9, 3.495495346275087*^9, 3.495495665489626*^9, 
   3.4954958533919563`*^9, {3.49549609483358*^9, 3.495496110792408*^9}, 
   3.4954961789333277`*^9, 3.495497173544275*^9, {3.495681286888199*^9, 
   3.4956812892482023`*^9}, {3.5240859566180162`*^9, 3.52408598201486*^9}, {
   3.524086057533393*^9, 3.524086058937395*^9}, {3.5242757161181526`*^9, 
   3.5242757216717625`*^9}, {3.5242763445964565`*^9, 
   3.5242763477320623`*^9}, {3.5331525961202545`*^9, 3.533152618700286*^9}, {
   3.5331527018704023`*^9, 3.5331528058905478`*^9}, {3.533153282981216*^9, 
   3.5331533342812877`*^9}, {3.533153565251611*^9, 3.533153599281659*^9}, {
   3.5332224159616575`*^9, 3.533222435081684*^9}, {3.5332224656717267`*^9, 
   3.533222494301767*^9}, {3.533223281652869*^9, 3.53322331799292*^9}, {
   3.5332234651831264`*^9, 3.5332234684731307`*^9}, {3.533223854473671*^9, 
   3.5332238609936805`*^9}, {3.5332239572838154`*^9, 3.5332239599938188`*^9}, 
   3.533224847425061*^9, {3.5332249144151554`*^9, 3.533224915905157*^9}, 
   3.533246982034763*^9, 3.5332605459462523`*^9, {3.536350185774555*^9, 
   3.536350207334585*^9}, {3.5363506584052167`*^9, 3.5363506914352627`*^9}, {
   3.5363508662655077`*^9, 3.5363509090255675`*^9}, {3.5363510097557087`*^9, 
   3.5363510569557743`*^9}, {3.5363512834660916`*^9, 3.536351309536128*^9}, {
   3.5363515550764723`*^9, 3.5363516176765594`*^9}, {3.5363516722566357`*^9, 
   3.5363516724366364`*^9}, {3.536356352083194*^9, 3.536356353383196*^9}, {
   3.5363617881968346`*^9, 3.536361833056897*^9}, {3.536366667801671*^9, 
   3.5363667247217503`*^9}, {3.5363667904918423`*^9, 3.536366799971856*^9}, {
   3.536366941382054*^9, 3.536366980882109*^9}, {3.5365968943773336`*^9, 
   3.536596936123007*^9}, {3.536596976558278*^9, 3.536597025589164*^9}, 
   3.536602461258152*^9, {3.5366940777176437`*^9, 3.5366940881722417`*^9}, 
   3.5366943020244737`*^9, {3.536694516327731*^9, 3.5366945346767807`*^9}, {
   3.5369801281282215`*^9, 3.5369801381582355`*^9}, 3.536980329338503*^9, {
   3.53698041236862*^9, 3.5369804166186256`*^9}, {3.5369807322990675`*^9, 
   3.536980746969088*^9}, 3.538444106556738*^9, {3.5384444438259706`*^9, 
   3.5384444863560305`*^9}, 3.544471282097495*^9, {3.549934369376335*^9, 
   3.549934370546336*^9}, {3.555274643817338*^9, 3.555274658697359*^9}, {
   3.5693429690149527`*^9, 3.569343004485002*^9}, {3.569343115935158*^9, 
   3.5693431274951744`*^9}, {3.5693438869262376`*^9, 
   3.5693438923862453`*^9}, {3.5693448264475527`*^9, 3.56934483149756*^9}, {
   3.5693449570477357`*^9, 3.5693449590377383`*^9}, {3.569345094427928*^9, 
   3.569345098257933*^9}, {3.569345161418022*^9, 3.5693451624280233`*^9}, {
   3.5693487017082796`*^9, 3.5693487044182835`*^9}, {3.5693492595090604`*^9, 
   3.5693493605292015`*^9}, {3.569349958830039*^9, 3.5693499638300467`*^9}, {
   3.5693533958348513`*^9, 3.5693534053048644`*^9}, {3.569353530255039*^9, 
   3.5693535588850794`*^9}, {3.5728334684022217`*^9, 3.572833498292264*^9}, {
   3.572922999249297*^9, 3.5729230018393*^9}, {3.5729230511593695`*^9, 
   3.5729231183194637`*^9}, {3.579550879927958*^9, 3.5795508902279725`*^9}, {
   3.5795509383780403`*^9, 3.5795509459380503`*^9}, {3.579551686859088*^9, 
   3.5795516944490986`*^9}, {3.5795602638411865`*^9, 3.579560288641221*^9}, {
   3.57956309471515*^9, 3.5795631845052757`*^9}, {3.579563216945321*^9, 
   3.579563227765336*^9}, {3.5795661043473654`*^9, 3.57956613599741*^9}, {
   3.5795661679974546`*^9, 3.579566168317455*^9}, {3.57956635769772*^9, 
   3.579566373517742*^9}, {3.579567662623248*^9, 3.5795676764832673`*^9}, {
   3.579569732548646*^9, 3.579569743578662*^9}, 3.5795710551004972`*^9, {
   3.614449366954212*^9, 3.6144493703714075`*^9}, {3.6144511524543366`*^9, 
   3.614451152636347*^9}, {3.6738158230939674`*^9, 3.673815866993478*^9}, {
   3.674584792335413*^9, 3.6745847965176525`*^9}, {3.6749280818331017`*^9, 
   3.674928085395108*^9}, {3.67492870227907*^9, 3.674928774648302*^9}, {
   3.6749288248243866`*^9, 3.6749288250243874`*^9}, {3.6749288961994953`*^9, 
   3.6749289295875473`*^9}},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{
  3.4796643211106243`*^9, {3.536351324436149*^9, 3.536351326496152*^9}, {
   3.5363515481064625`*^9, 3.536351548866463*^9}}]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t", 
  RowBox[{Cell[TextData[StyleBox["2:",
    FontFamily->"Arial",
    FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox["u", "m"], " ", 
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"a", "+", 
        RowBox[{"b", " ", 
         SuperscriptBox[
          RowBox[{"(", 
           SuperscriptBox["F", 
            RowBox[{"g", " ", "v"}]], ")"}], "n"]}]}], ")"}], "p"], 
     RowBox[{"\[DifferentialD]", "x"}], "  ", 
     StyleBox["when",
      FontFamily->"Arial",
      FontWeight->"Plain"], 
     StyleBox["  ",
      FontFamily->"Arial",
      FontWeight->"Plain"], Cell[TextData[Cell[BoxData[
      RowBox[{
       RowBox[{"v", "\[Equal]", 
        RowBox[{"e", "+", 
         RowBox[{"f", " ", "x"}]}]}], " ", "\[And]", " ", 
       RowBox[{"u", "\[Equal]", 
        SuperscriptBox[
         RowBox[{"(", 
          RowBox[{"c", "+", 
           RowBox[{"d", " ", "x"}]}], ")"}], "q"]}], " ", "\[And]", " ", 
       RowBox[{"m", "\[NotElement]", "\[DoubleStruckCapitalZ]"}]}]]]], 
      "None"]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.4964645213514385`*^9, 
   3.5192470594155855`*^9, {3.519247288125906*^9, 3.5192473207459517`*^9}, {
   3.519247363296011*^9, 3.519247363796012*^9}, 3.519258420199427*^9, 
   3.5192584863895197`*^9, {3.519320760160028*^9, 3.519320767367241*^9}, 
   3.5193299099191504`*^9, {3.5193300315093203`*^9, 3.519330031709321*^9}, {
   3.5193323099968634`*^9, 3.5193324414114943`*^9}, {3.5193327317904043`*^9, 
   3.519332762148057*^9}, {3.519340455685931*^9, 3.519340463318947*^9}, 
   3.519340635545749*^9, {3.5193422478304834`*^9, 3.519342248704085*^9}, {
   3.519342492766513*^9, 3.519342493156514*^9}, {3.519345750301442*^9, 
   3.5193457505978427`*^9}, 3.519346263464343*^9, {3.519349307658399*^9, 
   3.519349310965605*^9}, {3.519352294781488*^9, 3.5193522950014887`*^9}, {
   3.5194134359455075`*^9, 3.519413446054325*^9}, {3.5194134908888044`*^9, 
   3.5194134927764077`*^9}, {3.5194141847624226`*^9, 
   3.5194142323581066`*^9}, {3.519414342899901*^9, 3.519414474720132*^9}, {
   3.5197939535480833`*^9, 3.5197939539380836`*^9}, {3.5197940297698174`*^9, 
   3.5197940299726176`*^9}, 3.519795153580191*^9, {3.5200103027397947`*^9, 
   3.5200103044997973`*^9}, {3.5232186878169007`*^9, 
   3.5232186881569014`*^9}, {3.5232193955778913`*^9, 
   3.5232193955778913`*^9}, {3.5233166487845354`*^9, 
   3.5233166487845354`*^9}, {3.5233167010346084`*^9, 
   3.5233167010346084`*^9}, {3.523318546315711*^9, 3.523318573368556*^9}, {
   3.5239220709492664`*^9, 3.5239220924617043`*^9}, {3.5239221279205666`*^9, 
   3.5239221365629816`*^9}, {3.523930270899587*^9, 3.523930270899587*^9}, {
   3.5240761703528385`*^9, 3.5240761703528385`*^9}, {3.52407843124921*^9, 
   3.5240784332304134`*^9}, {3.5240784932749186`*^9, 
   3.5240784932749186`*^9}, {3.524079230095413*^9, 3.524079246553442*^9}, {
   3.5240858499762287`*^9, 3.524085860178646*^9}, {3.524085945261196*^9, 
   3.524085945261196*^9}, {3.5242763399632483`*^9, 3.5242763411488504`*^9}, {
   3.5242771575294847`*^9, 3.5242771578102846`*^9}, {3.530500224433988*^9, 
   3.530500225113989*^9}, {3.5313467876910577`*^9, 3.5313467883119507`*^9}, {
   3.5332231066826243`*^9, 3.5332231066826243`*^9}, {3.5332248881251183`*^9, 
   3.533224890245121*^9}, {3.5361011385491977`*^9, 3.5361011394332485`*^9}, {
   3.5363501469745007`*^9, 3.5363501506345057`*^9}, {3.5363617411167684`*^9, 
   3.5363617534067855`*^9}, {3.536366615151597*^9, 3.5363666230616083`*^9}, {
   3.5363667754518213`*^9, 3.5363667821318307`*^9}, {3.536366906722005*^9, 
   3.536366922022027*^9}, {3.536596861742076*^9, 3.5365968701504908`*^9}, {
   3.5366042542749715`*^9, 3.536604254462172*^9}, {3.536625554425806*^9, 
   3.536625554775807*^9}, {3.5366276426139326`*^9, 3.536627642853933*^9}, {
   3.536629402336396*^9, 3.536629402746397*^9}, {3.5366940603006477`*^9, 
   3.536694061273703*^9}, {3.5366943323932104`*^9, 3.536694332661226*^9}, {
   3.536694390476533*^9, 3.536694402176202*^9}, {3.5368634751022315`*^9, 
   3.5368634756170325`*^9}, {3.536980474148706*^9, 3.5369804925387316`*^9}, {
   3.5369807910591497`*^9, 3.5369808136991816`*^9}, 3.538443999546241*^9, {
   3.538444040229568*^9, 3.5384440801737804`*^9}, {3.544471201864182*^9, 
   3.5444712020241823`*^9}, {3.5499343621399236`*^9, 
   3.5499343628887253`*^9}, {3.5693429012648573`*^9, 3.569342923814889*^9}, {
   3.5693432071452856`*^9, 3.5693432074652863`*^9}, {3.569344801387518*^9, 
   3.569344801387518*^9}, {3.5693533425947766`*^9, 3.5693533510147886`*^9}, {
   3.569353493264988*^9, 3.5693535115050135`*^9}, {3.5694290997903204`*^9, 
   3.569429100480322*^9}, {3.5718791895851946`*^9, 3.571879189775195*^9}, {
   3.572109875965929*^9, 3.572109876175929*^9}, {3.5722014327630053`*^9, 
   3.5722014341930075`*^9}, {3.5722014837230773`*^9, 
   3.5722014840130777`*^9}, {3.572205109828057*^9, 3.572205110068057*^9}, {
   3.5722827529728394`*^9, 3.57228275320284*^9}, {3.5728333915121145`*^9, 
   3.5728334104821405`*^9}, {3.5729228769791255`*^9, 3.572922877149126*^9}, {
   3.5729229234691906`*^9, 3.572922991489286*^9}, {3.5795507933078365`*^9, 
   3.5795508236878796`*^9}, {3.5795516549990435`*^9, 
   3.5795516630790544`*^9}, {3.5795525734903293`*^9, 
   3.5795525789003367`*^9}, {3.579558592186327*^9, 3.579558592186327*^9}, {
   3.579558853016692*^9, 3.579558853016692*^9}, {3.579560242101156*^9, 
   3.579560248491165*^9}, {3.579563052515091*^9, 3.579563061675103*^9}, {
   3.5795662562075777`*^9, 3.579566302467643*^9}, {3.5795665034316254`*^9, 
   3.5795665052116275`*^9}, {3.579566567021714*^9, 3.5795665773017282`*^9}, {
   3.5795676143031807`*^9, 3.5795676191331873`*^9}, 3.614448851898752*^9, 
   3.673810300156073*^9, {3.673815729357606*^9, 3.673815729357606*^9}, 
   3.6738158040318766`*^9, 3.674928983204632*^9, {3.6749290503057327`*^9, 
   3.6749290625597534`*^9}},
 FontSize->12,
 FontWeight->"Bold"],

Cell["Derivation: Piecewise constant extraction", "Subsubsection",
 CellChangeTimes->{
  3.4953915070313115`*^9, 3.495840214754669*^9, {3.4958483177760134`*^9, 
   3.495848318886015*^9}, {3.495925911922002*^9, 3.495925920122013*^9}, 
   3.4961847242848682`*^9, 3.496184957525194*^9, 3.496188449500083*^9, 
   3.4961908979261436`*^9, 3.4961924998683863`*^9, 3.4961928754389124`*^9, {
   3.51849075797458*^9, 3.5184907698562593`*^9}, {3.5747099413791857`*^9, 
   3.5747099436591883`*^9}, {3.5795667434919615`*^9, 3.5795667507119713`*^9}}],

Cell[TextData[{
 "Basis: ",
 Cell[BoxData[
  RowBox[{
   RowBox[{
    SubscriptBox["\[PartialD]", "x"], 
    FractionBox[
     SuperscriptBox[
      RowBox[{"(", 
       SuperscriptBox[
        RowBox[{"(", 
         RowBox[{"c", "+", 
          RowBox[{"d", " ", "x"}]}], ")"}], "q"], ")"}], "m"], 
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", " ", "x"}]}], ")"}], 
      RowBox[{"m", " ", "q"}]]]}], "\[Equal]", "0"}]],
  CellChangeTimes->{{3.5795666608818455`*^9, 3.579566680671873*^9}}]
}], "Subsubsection",
 CellChangeTimes->{
  3.4953915070313115`*^9, 3.495840214754669*^9, {3.4958483177760134`*^9, 
   3.495848318886015*^9}, {3.495925911922002*^9, 3.495925920122013*^9}, 
   3.4961847242848682`*^9, 3.496184957525194*^9, 3.496188449500083*^9, 
   3.4961908979261436`*^9, 3.4961924998683863`*^9, 3.4961928754389124`*^9, {
   3.51849075797458*^9, 3.5184907698562593`*^9}, {3.5747099413791857`*^9, 
   3.5747099436591883`*^9}, {3.5795666870818825`*^9, 3.579566694501893*^9}, {
   3.579566733451947*^9, 3.579566733451947*^9}, {3.6738157347579145`*^9, 
   3.673815735656966*^9}, {3.674929122271843*^9, 3.674929150940888*^9}}],

Cell[TextData[{
 "Rule: If  ",
 Cell[BoxData[
  RowBox[{
   RowBox[{"v", "\[Equal]", 
    RowBox[{"e", "+", 
     RowBox[{"f", " ", "x"}]}]}], " ", "\[And]", " ", 
   RowBox[{"u", "\[Equal]", 
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"c", "+", 
       RowBox[{"d", " ", "x"}]}], ")"}], "q"]}], " ", "\[And]", " ", 
   RowBox[{"m", "\[NotElement]", "\[DoubleStruckCapitalZ]"}]}]]],
 ", then"
}], "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.495391630941485*^9, 3.4953916682515373`*^9}, 
   3.495398892616804*^9, {3.495402251391507*^9, 3.4954022711315346`*^9}, {
   3.495488783827441*^9, 3.4954888500775337`*^9}, {3.4954889799777155`*^9, 
   3.495489009627757*^9}, {3.495489141407942*^9, 3.495489151717956*^9}, {
   3.495489235178073*^9, 3.495489239268079*^9}, {3.495489386068284*^9, 
   3.4954894212383337`*^9}, {3.495489464438394*^9, 3.4954894647383947`*^9}, {
   3.495489500148444*^9, 3.4954895034984484`*^9}, {3.4954897331687703`*^9, 
   3.495489780118836*^9}, {3.495493913565876*^9, 3.495493935715907*^9}, {
   3.495494084919718*^9, 3.4954941083997507`*^9}, 3.495494511560315*^9, {
   3.495494580840412*^9, 3.4954945834404163`*^9}, {3.495494615970462*^9, 
   3.495494740200636*^9}, {3.495494880710832*^9, 3.4954948880508423`*^9}, {
   3.4954949819649754`*^9, 3.495494999445*^9}, {3.495495046515066*^9, 
   3.495495072875103*^9}, {3.4954951239351745`*^9, 3.4954951241651745`*^9}, {
   3.4954953673051157`*^9, 3.4954953726551237`*^9}, {3.495495450474448*^9, 
   3.495495453173253*^9}, {3.495495542140209*^9, 3.495495583792282*^9}, {
   3.4954957431465626`*^9, 3.4954957746118174`*^9}, {3.495495914419263*^9, 
   3.495495917804469*^9}, {3.495495955322535*^9, 3.4954960459586945`*^9}, {
   3.495496189588146*^9, 3.4954961923181515`*^9}, {3.4954973384365644`*^9, 
   3.495497357577798*^9}, {3.5240858715198665`*^9, 3.52408591925595*^9}, {
   3.5331536712917595`*^9, 3.5331536971617956`*^9}, {3.533223063022563*^9, 
   3.5332230658125668`*^9}, {3.533224894045127*^9, 3.5332248954551287`*^9}, 
   3.5363501685345306`*^9, {3.538444353705845*^9, 3.5384443676658645`*^9}, {
   3.5499343655595293`*^9, 3.549934366379531*^9}, {3.5693429326149015`*^9, 
   3.5693429326149015`*^9}, {3.5693448172975397`*^9, 
   3.5693448172975397`*^9}, {3.5693533761948233`*^9, 
   3.5693533761948233`*^9}, {3.572923022359329*^9, 3.572923022359329*^9}, {
   3.579550848297914*^9, 3.579550848297914*^9}, {3.579551672029067*^9, 
   3.579551672029067*^9}, {3.5795525826803417`*^9, 3.5795525826803417`*^9}, {
   3.5795585969663343`*^9, 3.5795585969663343`*^9}, {3.5795588543766947`*^9, 
   3.5795588543766947`*^9}, {3.57956025200117*^9, 3.57956025200117*^9}, {
   3.5795630745851216`*^9, 3.5795630745851216`*^9}, {3.579566316937663*^9, 
   3.579566316937663*^9}, {3.5795665841317377`*^9, 3.5795665841317377`*^9}, {
   3.579567626853198*^9, 3.579567626853198*^9}, {3.6738157371580515`*^9, 
   3.6738157371580515`*^9}, {3.674929067899761*^9, 3.674929067899761*^9}}],

Cell[BoxData[
 RowBox[{"\[Integral]", 
  RowBox[{
   SuperscriptBox["u", "m"], " ", 
   SuperscriptBox[
    RowBox[{"(", 
     RowBox[{"a", "+", 
      RowBox[{"b", " ", 
       SuperscriptBox[
        RowBox[{"(", 
         SuperscriptBox["F", 
          RowBox[{"g", " ", "v"}]], ")"}], "n"]}]}], ")"}], "p"], 
   RowBox[{"\[DifferentialD]", 
    RowBox[{"x", "  ", "\[LongRightArrow]", "  ", 
     FractionBox[
      SuperscriptBox[
       RowBox[{"(", 
        SuperscriptBox[
         RowBox[{"(", 
          RowBox[{"c", "+", 
           RowBox[{"d", " ", "x"}]}], ")"}], "q"], ")"}], "m"], 
      SuperscriptBox[
       RowBox[{"(", 
        RowBox[{"c", "+", 
         RowBox[{"d", " ", "x"}]}], ")"}], 
       RowBox[{"m", " ", "q"}]]]}]}], 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", " ", "x"}]}], ")"}], 
      RowBox[{"m", " ", "q"}]], " ", 
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"a", "+", 
        RowBox[{"b", " ", 
         SuperscriptBox[
          RowBox[{"(", 
           SuperscriptBox["F", 
            RowBox[{"g", 
             RowBox[{"(", 
              RowBox[{"e", "+", 
               RowBox[{"f", " ", "x"}]}], ")"}]}]], ")"}], "n"]}]}], ")"}], 
      "p"], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579723816*^9, 3.4796580261689425`*^9}, 
   3.479661191290163*^9, {3.479686720164157*^9, 3.479686720204214*^9}, {
   3.479686831554328*^9, 3.479686834819022*^9}, {3.4940970712485504`*^9, 
   3.4940971130286083`*^9}, 3.4940971455886545`*^9, 3.4953915848214207`*^9, 
   3.495391696001576*^9, {3.495401987881138*^9, 3.49540199674115*^9}, {
   3.495402119591322*^9, 3.4954021482513623`*^9}, {3.4954887467773895`*^9, 
   3.4954887749874287`*^9}, 3.495488944397666*^9, 3.495489263778113*^9, 
   3.4954893636582527`*^9, 3.4954898177188883`*^9, 3.495489875358969*^9, 
   3.495493833545764*^9, 3.495493864175807*^9, 3.4954940303560395`*^9, 
   3.4954941516698112`*^9, 3.4954942700599775`*^9, 3.4954948473707857`*^9, {
   3.495495264965372*^9, 3.4954952782653904`*^9}, {3.4954956850364604`*^9, 
   3.4954957088577023`*^9}, {3.495495865341577*^9, 3.4954958884920177`*^9}, {
   3.495496084225561*^9, 3.4954961029143944`*^9}, {3.495496211459385*^9, 
   3.4954962213810024`*^9}, 3.495496440218187*^9, {3.495497026732417*^9, 
   3.4954970596328745`*^9}, {3.495497256957621*^9, 3.495497302338101*^9}, 
   3.533151648818928*^9, 3.5331516797189713`*^9, 3.533151717349024*^9, {
   3.5331518776292486`*^9, 3.533151970249378*^9}, {3.5331520176994443`*^9, 
   3.5331520227894516`*^9}, {3.533152054329496*^9, 3.5331520626995077`*^9}, 
   3.533152270349798*^9, {3.533153065130911*^9, 3.5331530971109557`*^9}, 
   3.53315335008131*^9, {3.5331535194015465`*^9, 3.5331535356615696`*^9}, 
   3.5331536017616625`*^9, {3.533221927890974*^9, 3.533221930830978*^9}, {
   3.533222264011444*^9, 3.533222283731472*^9}, {3.5332223304615374`*^9, 
   3.5332223353515444`*^9}, 3.5363501604045196`*^9, {3.536361767566806*^9, 
   3.5363617697768087`*^9}, {3.5363666287016163`*^9, 
   3.5363666313816195`*^9}, {3.536366784271834*^9, 3.5363667868618374`*^9}, {
   3.536366923782029*^9, 3.5363669254320316`*^9}, {3.5365968786057057`*^9, 
   3.536596881944112*^9}, {3.5366940930205193`*^9, 3.53669410440917*^9}, {
   3.5366942507975435`*^9, 3.536694251403578*^9}, {3.536694493325415*^9, 
   3.536694495295528*^9}, 3.536980387788585*^9, {3.5384441491255584`*^9, 
   3.538444158185571*^9}, 3.5384445157160716`*^9, {3.5693429405449123`*^9, 
   3.569342962434943*^9}, 3.5693487483783445`*^9, 3.5693533713048167`*^9, {
   3.56935351683502*^9, 3.5693535184850225`*^9}, {3.572833444812189*^9, 
   3.5728334591122093`*^9}, {3.572922997329294*^9, 3.5729229979692955`*^9}, {
   3.5729231499495077`*^9, 3.5729231561595163`*^9}, 3.5795508621679335`*^9, {
   3.579551676949074*^9, 3.579551678979077*^9}, 3.5795585977163353`*^9, 
   3.5795588554866962`*^9, {3.579560255261174*^9, 3.579560255591175*^9}, {
   3.5795630788551273`*^9, 3.579563079755129*^9}, {3.579566324877674*^9, 
   3.5795663330776854`*^9}, {3.579566627671799*^9, 3.5795666529918346`*^9}, {
   3.673815740953269*^9, 3.673815742225342*^9}, {3.673815806466016*^9, 
   3.6738158138034353`*^9}, {3.6749290802497787`*^9, 
   3.6749291038458166`*^9}, {3.6749291382188687`*^9, 3.674929139738871*^9}},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{"u_", "^", "m_."}], "*", 
     RowBox[{
      RowBox[{"(", 
       RowBox[{"a_.", "+", 
        RowBox[{"b_.", "*", 
         RowBox[{
          RowBox[{"(", 
           RowBox[{"F_", "^", 
            RowBox[{"(", 
             RowBox[{"g_.", "*", "v_"}], ")"}]}], ")"}], "^", "n_."}]}]}], 
       ")"}], "^", "p_."}]}], ",", "x_Symbol"}], "]"}], " ", ":=", "\n", "  ", 
  RowBox[{
   RowBox[{"Module", "[", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{
       RowBox[{"uu", "=", 
        RowBox[{"NormalizePowerOfLinear", "[", 
         RowBox[{"u", ",", "x"}], "]"}]}], ",", "z"}], "}"}], ",", "\n", "  ", 
     RowBox[{
      RowBox[{"z", "=", 
       RowBox[{"If", "[", 
        RowBox[{
         RowBox[{
          RowBox[{"PowerQ", "[", "uu", "]"}], " ", "&&", " ", 
          RowBox[{"FreeQ", "[", 
           RowBox[{
            RowBox[{"uu", "[", 
             RowBox[{"[", "2", "]"}], "]"}], ",", "x"}], "]"}]}], ",", " ", 
         RowBox[{
          RowBox[{"uu", "[", 
           RowBox[{"[", "1", "]"}], "]"}], "^", 
          RowBox[{"(", 
           RowBox[{"m", "*", 
            RowBox[{"uu", "[", 
             RowBox[{"[", "2", "]"}], "]"}]}], ")"}]}], ",", " ", 
         RowBox[{"uu", "^", "m"}]}], "]"}]}], ";", "\n", "  ", 
      RowBox[{
       RowBox[{
        RowBox[{"uu", "^", "m"}], "/", "z"}], "*", 
       RowBox[{"Int", "[", 
        RowBox[{
         RowBox[{"z", "*", 
          RowBox[{
           RowBox[{"(", 
            RowBox[{"a", "+", 
             RowBox[{"b", "*", 
              RowBox[{
               RowBox[{"(", 
                RowBox[{"F", "^", 
                 RowBox[{"(", 
                  RowBox[{"g", "*", 
                   RowBox[{"ExpandToSum", "[", 
                    RowBox[{"v", ",", "x"}], "]"}]}], ")"}]}], ")"}], "^", 
               "n"}]}]}], ")"}], "^", "p"}]}], ",", "x"}], "]"}]}]}]}], "]"}],
    " ", "/;", " ", "\n", 
   RowBox[{
    RowBox[{"FreeQ", "[", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{
       "F", ",", "a", ",", "b", ",", "g", ",", "m", ",", "n", ",", "p"}], 
       "}"}], ",", "x"}], "]"}], " ", "&&", " ", 
    RowBox[{"LinearQ", "[", 
     RowBox[{"v", ",", "x"}], "]"}], " ", "&&", " ", 
    RowBox[{"PowerOfLinearQ", "[", 
     RowBox[{"u", ",", "x"}], "]"}], " ", "&&", " ", 
    RowBox[{"Not", "[", 
     RowBox[{
      RowBox[{"LinearMatchQ", "[", 
       RowBox[{"v", ",", "x"}], "]"}], " ", "&&", " ", 
      RowBox[{"PowerOfLinearMatchQ", "[", 
       RowBox[{"u", ",", "x"}], "]"}]}], "]"}], " ", "&&", " ", "\n", "  ", 
    RowBox[{"Not", "[", 
     RowBox[{"IntegerQ", "[", "m", "]"}], "]"}]}]}]}]], "Code",
 CellChangeTimes->{{3.494097165958683*^9, 3.4940971894087152`*^9}, {
   3.4940972668388243`*^9, 3.4940972723188314`*^9}, 3.49539148440128*^9, 
   3.4954018196509027`*^9, {3.495401867050969*^9, 3.4954018672609687`*^9}, {
   3.4954022872215567`*^9, 3.4954022882115583`*^9}, 3.495488713187342*^9, 
   3.4954889210576334`*^9, 3.4954892270180616`*^9, 3.4954893399982195`*^9, {
   3.4954894833184204`*^9, 3.495489485938424*^9}, 3.4954897194087505`*^9, 
   3.495493755131653*^9, {3.4954938826158323`*^9, 3.495493884975836*^9}, 
   3.4954939962159915`*^9, {3.4954942766399865`*^9, 3.4954942945800114`*^9}, 
   3.4954948208307486`*^9, {3.495495087335123*^9, 3.495495087705124*^9}, 
   3.4954951805652537`*^9, 3.495495346275087*^9, 3.495495665489626*^9, 
   3.4954958533919563`*^9, {3.49549609483358*^9, 3.495496110792408*^9}, 
   3.4954961789333277`*^9, 3.495497173544275*^9, {3.495681286888199*^9, 
   3.4956812892482023`*^9}, {3.5240859566180162`*^9, 3.52408598201486*^9}, {
   3.524086057533393*^9, 3.524086058937395*^9}, {3.5242757161181526`*^9, 
   3.5242757216717625`*^9}, {3.5242763445964565`*^9, 
   3.5242763477320623`*^9}, {3.5331525961202545`*^9, 3.533152618700286*^9}, {
   3.5331527018704023`*^9, 3.5331528058905478`*^9}, {3.533153282981216*^9, 
   3.5331533342812877`*^9}, {3.533153565251611*^9, 3.533153599281659*^9}, {
   3.5332224159616575`*^9, 3.533222435081684*^9}, {3.5332224656717267`*^9, 
   3.533222494301767*^9}, {3.533223281652869*^9, 3.53322331799292*^9}, {
   3.5332234651831264`*^9, 3.5332234684731307`*^9}, {3.533223854473671*^9, 
   3.5332238609936805`*^9}, {3.5332239572838154`*^9, 3.5332239599938188`*^9}, 
   3.533224847425061*^9, {3.5332249144151554`*^9, 3.533224915905157*^9}, 
   3.533246982034763*^9, 3.5332605459462523`*^9, {3.536350185774555*^9, 
   3.536350207334585*^9}, {3.5363506584052167`*^9, 3.5363506914352627`*^9}, {
   3.5363508662655077`*^9, 3.5363509090255675`*^9}, {3.5363510097557087`*^9, 
   3.5363510569557743`*^9}, {3.5363512834660916`*^9, 3.536351309536128*^9}, {
   3.5363515550764723`*^9, 3.5363516176765594`*^9}, {3.5363516722566357`*^9, 
   3.5363516724366364`*^9}, {3.536356352083194*^9, 3.536356353383196*^9}, {
   3.5363617881968346`*^9, 3.536361833056897*^9}, {3.536366667801671*^9, 
   3.5363667247217503`*^9}, {3.5363667904918423`*^9, 3.536366799971856*^9}, {
   3.536366941382054*^9, 3.536366980882109*^9}, {3.5365968943773336`*^9, 
   3.536596936123007*^9}, {3.536596976558278*^9, 3.536597025589164*^9}, 
   3.536602461258152*^9, {3.5366940777176437`*^9, 3.5366940881722417`*^9}, 
   3.5366943020244737`*^9, {3.536694516327731*^9, 3.5366945346767807`*^9}, {
   3.5369801281282215`*^9, 3.5369801381582355`*^9}, 3.536980329338503*^9, {
   3.53698041236862*^9, 3.5369804166186256`*^9}, {3.5369807322990675`*^9, 
   3.536980746969088*^9}, 3.538444106556738*^9, {3.5384444438259706`*^9, 
   3.5384444863560305`*^9}, 3.544471282097495*^9, {3.549934369376335*^9, 
   3.549934370546336*^9}, {3.555274643817338*^9, 3.555274658697359*^9}, {
   3.5693429690149527`*^9, 3.569343004485002*^9}, {3.569343115935158*^9, 
   3.5693431274951744`*^9}, {3.5693438869262376`*^9, 
   3.5693438923862453`*^9}, {3.5693448264475527`*^9, 3.56934483149756*^9}, {
   3.5693449570477357`*^9, 3.5693449590377383`*^9}, {3.569345094427928*^9, 
   3.569345098257933*^9}, {3.569345161418022*^9, 3.5693451624280233`*^9}, {
   3.5693487017082796`*^9, 3.5693487044182835`*^9}, {3.5693492595090604`*^9, 
   3.5693493605292015`*^9}, {3.569349958830039*^9, 3.5693499638300467`*^9}, {
   3.5693533958348513`*^9, 3.5693534053048644`*^9}, {3.569353530255039*^9, 
   3.5693535588850794`*^9}, {3.5728334684022217`*^9, 3.572833498292264*^9}, {
   3.572922999249297*^9, 3.5729230018393*^9}, {3.5729230511593695`*^9, 
   3.5729231183194637`*^9}, {3.579550879927958*^9, 3.5795508902279725`*^9}, {
   3.5795509383780403`*^9, 3.5795509459380503`*^9}, {3.579551686859088*^9, 
   3.5795516944490986`*^9}, {3.5795602638411865`*^9, 3.579560288641221*^9}, {
   3.57956309471515*^9, 3.5795631845052757`*^9}, {3.579563216945321*^9, 
   3.579563227765336*^9}, {3.5795661043473654`*^9, 3.57956613599741*^9}, {
   3.5795661679974546`*^9, 3.579566168317455*^9}, {3.57956635769772*^9, 
   3.579566373517742*^9}, {3.5795665865317416`*^9, 3.5795665907417474`*^9}, {
   3.5795668980921774`*^9, 3.579566946962246*^9}, {3.5795669955923142`*^9, 
   3.579567077432429*^9}, {3.5795671101224747`*^9, 3.5795671237724934`*^9}, {
   3.5795676953932943`*^9, 3.579567703443305*^9}, 3.579569767068694*^9, 
   3.579571058200502*^9, {3.6144493986070223`*^9, 3.614449402011217*^9}, {
   3.61445008461726*^9, 3.6144500965059395`*^9}, {3.673815837214775*^9, 
   3.673815899915361*^9}, {3.67458480050288*^9, 3.674584804451106*^9}, {
   3.6749290271757*^9, 3.674929027695701*^9}, {3.674929174757927*^9, 
   3.674929209105978*^9}, 3.674929343328185*^9},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{
  3.4796643211106243`*^9, {3.536351324436149*^9, 3.536351326496152*^9}, {
   3.5363515481064625`*^9, 3.536351548866463*^9}}]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{Cell[TextData[StyleBox["X:",
   FontFamily->"Arial",
   FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
  RowBox[{"\[Integral]", 
   RowBox[{
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"c", "+", 
       RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"a", "+", 
       RowBox[{"b", " ", 
        SuperscriptBox[
         RowBox[{"(", 
          SuperscriptBox["F", 
           RowBox[{"g", 
            RowBox[{"(", 
             RowBox[{"e", "+", 
              RowBox[{"f", " ", "x"}]}], ")"}]}]], ")"}], "n"]}]}], ")"}], 
     "p"], 
    RowBox[{"\[DifferentialD]", "x"}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193422443984776`*^9, 3.5193422447260776`*^9}, 
   3.5193424488212357`*^9, {3.5193434708254333`*^9, 3.519343473103037*^9}, {
   3.5193435097631016`*^9, 3.51934353169674*^9}, {3.519345739319023*^9, 
   3.519345739599823*^9}, 3.5193462601259375`*^9, {3.5193475061060295`*^9, 
   3.5193475233908596`*^9}, {3.519351391880224*^9, 3.5193513931402254`*^9}, {
   3.519403773021052*^9, 3.5194037798564425`*^9}, 3.5194106793250647`*^9, {
   3.519537578341393*^9, 3.5195375792305946`*^9}, {3.5195376277778797`*^9, 
   3.51953767927357*^9}, {3.5197586260636144`*^9, 3.5197586260636144`*^9}, {
   3.51978053295651*^9, 3.519780542862528*^9}, {3.5197806526087203`*^9, 
   3.5197806657595434`*^9}, {3.5197807490168896`*^9, 3.5197807642581167`*^9}, 
   3.5197809004619555`*^9, {3.519782653514635*^9, 3.5197826623286505`*^9}, {
   3.5200207810164795`*^9, 3.520020785046485*^9}, {3.5213126833374643`*^9, 
   3.5213126833374643`*^9}, {3.521328235411002*^9, 3.521328235411002*^9}, {
   3.523315567893022*^9, 3.523315567893022*^9}, {3.5233157569332867`*^9, 
   3.5233157613032923`*^9}, {3.5323023198133526`*^9, 3.532302355723403*^9}, {
   3.5323026417638035`*^9, 3.5323026417638035`*^9}, {3.5323026980138817`*^9, 
   3.5323026980138817`*^9}, {3.532636287104328*^9, 3.532636287104328*^9}, {
   3.5330803686035347`*^9, 3.533080368863535*^9}, {3.533248909289961*^9, 
   3.5332489110299635`*^9}, {3.5418136182858934`*^9, 3.541813646495933*^9}, {
   3.541885900975107*^9, 3.541885900975107*^9}, {3.5418899389143834`*^9, 
   3.5418899662244215`*^9}, 3.5418900751945744`*^9, {3.5462974735235267`*^9, 
   3.546297473743527*^9}, {3.560128588234247*^9, 3.560128589104248*^9}, {
   3.560128670294362*^9, 3.560128673044366*^9}, 3.560216297553895*^9, 
   3.5606251400716968`*^9, {3.561749988629304*^9, 3.56175000739933*^9}, {
   3.5617574489046125`*^9, 3.561757453934619*^9}, 3.5617578683751993`*^9, {
   3.561758400818445*^9, 3.5617584075484543`*^9}, {3.561758717288888*^9, 
   3.5617587174988885`*^9}, {3.5617603061411123`*^9, 3.5617603105011187`*^9}, 
   3.5617605991515226`*^9, {3.561760630041566*^9, 3.561760630041566*^9}, {
   3.5617653654281955`*^9, 3.561765368188199*^9}, {3.562108393456187*^9, 
   3.5621083944701886`*^9}, 3.5622940305792723`*^9, {3.5622942446095715`*^9, 
   3.562294244919572*^9}, {3.564341532826086*^9, 3.5643415472093115`*^9}, {
   3.5643415952261963`*^9, 3.564341621543442*^9}, {3.5653712427559214`*^9, 
   3.5653712662959547`*^9}, {3.567536711771888*^9, 3.5675367305719137`*^9}, {
   3.567536861132097*^9, 3.5675368736121144`*^9}, {3.5675373448827744`*^9, 
   3.567537359022794*^9}, 3.567537618133157*^9, {3.5675449022087917`*^9, 
   3.567544938338858*^9}, 3.5675454535133247`*^9, 3.568756731398242*^9, {
   3.568835895390687*^9, 3.568835896342289*^9}, {3.5688359541215906`*^9, 
   3.5688359541215906`*^9}, {3.568851625271058*^9, 3.5688516293410635`*^9}, {
   3.569204001072007*^9, 3.569204004992013*^9}, {3.569204303412431*^9, 
   3.569204307292436*^9}, {3.5692045761028123`*^9, 3.569204587352828*^9}, {
   3.569205883784643*^9, 3.5692058902746525`*^9}, {3.5692059270447035`*^9, 
   3.569205949874736*^9}, {3.5692065112355213`*^9, 3.569206514985527*^9}, {
   3.569208580617423*^9, 3.5692085822774253`*^9}, 3.569699853809244*^9, {
   3.5697004398877664`*^9, 3.5697004398877664`*^9}, 3.673470295682214*^9, {
   3.6734808068368773`*^9, 3.673480812856886*^9}, {3.6734812118905478`*^9, 
   3.6734812121005483`*^9}, {3.673485207378795*^9, 3.6734852076887956`*^9}, 
   3.6734852379518394`*^9, {3.67348528819391*^9, 3.673485311610944*^9}, {
   3.673485373452033*^9, 3.673485373452033*^9}, 3.673544572185848*^9, 
   3.673544743054061*^9, {3.6735447765981197`*^9, 3.6735447897021427`*^9}, {
   3.6735451069205027`*^9, 3.6735451216841288`*^9}, 3.673571609358359*^9, 
   3.674054419613327*^9, 3.67405499278094*^9, 3.674055340543049*^9, 
   3.674582605319323*^9},
 FontSize->12,
 FontWeight->"Bold"],

Cell["Rule:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.4964417379912624`*^9, 3.4964417471112747`*^9}, {
   3.496441910111503*^9, 3.4964419275315275`*^9}, 3.496451213764926*^9, 
   3.496528833243797*^9, {3.4975777080419827`*^9, 3.4975777244120054`*^9}, 
   3.4976397903561964`*^9, {3.497662273761881*^9, 3.497662314743153*^9}, {
   3.49766243823297*^9, 3.4976624525849953`*^9}, {3.4976629473554645`*^9, 
   3.49766295624748*^9}, {3.4982405753732615`*^9, 3.4982405793132668`*^9}, {
   3.499020266797401*^9, 3.499020267357402*^9}, {3.673480815126889*^9, 
   3.673480815126889*^9}, {3.673485324113962*^9, 3.673485324113962*^9}, {
   3.6734853756320357`*^9, 3.6734853756320357`*^9}, {3.6735448067267733`*^9, 
   3.6735448067267733`*^9}, {3.6735451425901656`*^9, 3.673545143962968*^9}}],

Cell[BoxData[
 RowBox[{"\[Integral]", 
  RowBox[{
   SuperscriptBox[
    RowBox[{"(", 
     RowBox[{"c", "+", 
      RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
   SuperscriptBox[
    RowBox[{"(", 
     RowBox[{"a", "+", 
      RowBox[{"b", " ", 
       SuperscriptBox[
        RowBox[{"(", 
         SuperscriptBox["F", 
          RowBox[{"g", 
           RowBox[{"(", 
            RowBox[{"e", "+", 
             RowBox[{"f", " ", "x"}]}], ")"}]}]], ")"}], "n"]}]}], ")"}], 
    "p"], 
   RowBox[{"\[DifferentialD]", 
    RowBox[{"x", "  ", "\[LongRightArrow]", "  ", 
     RowBox[{"\[Integral]", 
      RowBox[{
       SuperscriptBox[
        RowBox[{"(", 
         RowBox[{"c", "+", 
          RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
       SuperscriptBox[
        RowBox[{"(", 
         RowBox[{"a", "+", 
          RowBox[{"b", " ", 
           SuperscriptBox[
            RowBox[{"(", 
             SuperscriptBox["F", 
              RowBox[{"g", 
               RowBox[{"(", 
                RowBox[{"e", "+", 
                 RowBox[{"f", " ", "x"}]}], ")"}]}]], ")"}], "n"]}]}], ")"}], 
        "p"], " ", 
       RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579732027807`*^9, 3.479658002474872*^9}, {
   3.4796580599975853`*^9, 3.4796580609489536`*^9}, {3.479658295195784*^9, 
   3.47965829684816*^9}, 3.4796611938939075`*^9, 3.4796613924794593`*^9, {
   3.479661443342597*^9, 3.479661446647349*^9}, {3.479661576554146*^9, 
   3.479661589302477*^9}, 3.479661847553824*^9, {3.4796622294729967`*^9, 
   3.4796622450654173`*^9}, {3.479686720424531*^9, 3.4796867204345455`*^9}, {
   3.4796874646646957`*^9, 3.4796874745589232`*^9}, {3.492826573513727*^9, 
   3.492826578630536*^9}, {3.4928267070863624`*^9, 3.492826707460763*^9}, 
   3.4928267903497095`*^9, {3.492829799151373*^9, 3.4928299617216005`*^9}, {
   3.4940972077187414`*^9, 3.4940972347887793`*^9}, {3.496441851501421*^9, 
   3.496441872881451*^9}, {3.496528916553913*^9, 3.4965289272239285`*^9}, {
   3.497576165747217*^9, 3.4975761801372375`*^9}, {3.497577761382057*^9, 
   3.4975777938921027`*^9}, 3.49763959300025*^9, 3.497639634480723*^9, {
   3.4976623833052735`*^9, 3.4976624037569094`*^9}, {3.497663070736081*^9, 
   3.4976631028721375`*^9}, {3.4990190901657534`*^9, 
   3.4990190901657534`*^9}, {3.499020281147421*^9, 3.499020315247469*^9}, {
   3.673470299122218*^9, 3.673470309116233*^9}, {3.6734808261669044`*^9, 
   3.6734808320589128`*^9}, {3.6734852411818438`*^9, 
   3.6734852630578747`*^9}, {3.6735448134191847`*^9, 3.673544853919856*^9}, 
   3.673545135068952*^9, {3.673571621535055*^9, 3.673571622023083*^9}, {
   3.6740544196445274`*^9, 3.6740544196445274`*^9}, {3.6740549927965403`*^9, 
   3.6740549927965403`*^9}, {3.674055340558649*^9, 3.674055340558649*^9}, {
   3.674582852613467*^9, 3.674582855560636*^9}},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{"c_.", "+", 
        RowBox[{"d_.", "*", "x_"}]}], ")"}], "^", "m_."}], "*", 
     RowBox[{
      RowBox[{"(", 
       RowBox[{"a_", "+", 
        RowBox[{"b_.", "*", 
         RowBox[{
          RowBox[{"(", 
           RowBox[{"F_", "^", 
            RowBox[{"(", 
             RowBox[{"g_.", "*", 
              RowBox[{"(", 
               RowBox[{"e_.", "+", 
                RowBox[{"f_.", "*", "x_"}]}], ")"}]}], ")"}]}], ")"}], "^", 
          "n_."}]}]}], ")"}], "^", "p_."}]}], ",", "x_Symbol"}], "]"}], " ", ":=",
   "\n", "  ", 
  RowBox[{
   RowBox[{"Unintegrable", "[", 
    RowBox[{
     RowBox[{
      RowBox[{
       RowBox[{"(", 
        RowBox[{"c", "+", 
         RowBox[{"d", "*", "x"}]}], ")"}], "^", "m"}], "*", 
      RowBox[{
       RowBox[{"(", 
        RowBox[{"a", "+", 
         RowBox[{"b", "*", 
          RowBox[{
           RowBox[{"(", 
            RowBox[{"F", "^", 
             RowBox[{"(", 
              RowBox[{"g", "*", 
               RowBox[{"(", 
                RowBox[{"e", "+", 
                 RowBox[{"f", "*", "x"}]}], ")"}]}], ")"}]}], ")"}], "^", 
           "n"}]}]}], ")"}], "^", "p"}]}], ",", "x"}], "]"}], " ", "/;", "\n", 
   RowBox[{"FreeQ", "[", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{
      "a", ",", "b", ",", "c", ",", "d", ",", "e", ",", "f", ",", "g", ",", 
       "m", ",", "n", ",", "p"}], "}"}], ",", "x"}], "]"}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, 
   3.496441834031397*^9, 3.496442095051762*^9, 3.496528791863739*^9, 
   3.496528888643874*^9, 3.497575939976901*^9, 3.4975761146971455`*^9, 
   3.4975775509417624`*^9, 3.497577752792045*^9, 3.4975834953320856`*^9, 
   3.4976395861674376`*^9, 3.4976620452370796`*^9, 3.497662356878827*^9, {
   3.497662464394216*^9, 3.497662468965024*^9}, 3.497662899853381*^9, {
   3.4990190901657534`*^9, 3.4990190901757536`*^9}, 3.499020260597392*^9, {
   3.569701527794991*^9, 3.569701549864253*^9}, {3.673470334807271*^9, 
   3.673470343827284*^9}, {3.673472101348009*^9, 3.6734721101550217`*^9}, {
   3.6734808382869225`*^9, 3.673480849636938*^9}, 3.6734809989872427`*^9, {
   3.673485355797007*^9, 3.6734853977560673`*^9}, 3.673485499780302*^9, {
   3.6735449135999613`*^9, 3.6735449901426964`*^9}, {3.673545154948387*^9, 
   3.673545192577654*^9}, {3.673546145558842*^9, 3.6735461464012437`*^9}, {
   3.6735716224641085`*^9, 3.673571622933135*^9}, 3.6740544424557676`*^9, 
   3.674054603219452*^9, 3.674055018803786*^9, 3.674055053564647*^9, {
   3.6740553582490807`*^9, 3.674055384241726*^9}, 3.6740561192246227`*^9, {
   3.674582868520377*^9, 3.6745829028233395`*^9}, 3.712336218534007*^9, 
   3.715541357056416*^9},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9}]
}, Closed]]
},
AutoGeneratedPackage->Automatic,
CellGrouping->Manual,
WindowSize->{1908, 946},
WindowMargins->{{0, Automatic}, {Automatic, 0}},
PrintingCopies->1,
PrintingStartingPageNumber->1,
PrintingPageRange->{1, 32000},
PageHeaders->{{
   Cell[
    TextData[{
      StyleBox[
       CounterBox["Page"], "PageNumber"], " ", "|", " ", 
      StyleBox[
       ValueBox["FileName"], "Header"]}], "Header", 
    CellMargins -> {{0, Inherited}, {Inherited, Inherited}}], None, None}, {
   Cell[
    TextData[{
     "Rules", " ", "for", " ", "integrands", " ", "of", " ", "the", " ", 
      "form", " ", "(", "c", "+", "d", " ", "x", ")", "^", "m", " ", "(", "a",
       "+", "b", " ", "(", "F", "^", "(", "g", " ", "(", "e", "+", "f", " ", 
      "x", ")", ")", ")", "^", "n", ")", "^", "p"}], "Header", 
    CellMargins -> {{0, Inherited}, {Inherited, Inherited}}], None, 
   Cell[
    TextData[
     CounterBox["Page", CounterFunction :> Identity]], "Header", 
    CellMargins -> {{Inherited, 0}, {Inherited, Inherited}}]}},
PageFooters->{{None, None, None}, {None, None, None}},
PageHeaderLines->{False, False},
PageFooterLines->{False, False},
PrintingOptions->{"FacingPages"->False,
"FirstPageFace"->Right,
"FirstPageFooter"->False,
"FirstPageHeader"->False,
"Magnification"->1.,
"PaperOrientation"->"Landscape",
"PaperSize"->{783., 601.8},
"PrintCellBrackets"->False,
"PrintMultipleHorizontalPages"->False,
"PrintRegistrationMarks"->False,
"PrintingMargins"->28.8},
Magnification->1.399999976158142,
FrontEndVersion->"8.0 for Microsoft Windows (64-bit) (October 6, 2011)",
StyleDefinitions->"Default.nb"
]
(* End of Notebook Content *)

(* Internal cache information *)
(*CellTagsOutline
CellTagsIndex->{}
*)
(*CellTagsIndex
CellTagsIndex->{}
*)
(*NotebookFileOutline
Notebook[{
Cell[545, 20, 2537, 51, 45, "Subsubtitle"],
Cell[CellGroupData[{
Cell[3107, 75, 3541, 61, 53, "Subsubtitle"],
Cell[6651, 138, 9239, 150, 60, "Subsubsection"],
Cell[15893, 290, 2669, 38, 53, "Code"],
Cell[CellGroupData[{
Cell[18587, 332, 6443, 107, 53, "Subsubtitle"],
Cell[25033, 441, 176, 2, 35, "Subsubsection"],
Cell[25212, 445, 1636, 38, 47, "Subsubsection"],
Cell[26851, 485, 2094, 35, 26, "Subsubsection"],
Cell[28948, 522, 3622, 84, 68, "Subsubtitle"],
Cell[32573, 608, 817, 11, 35, "Subsubsection"],
Cell[33393, 621, 5680, 122, 129, "Code"],
Cell[39076, 745, 89, 2, 35, "Subsubsection"]
}, Closed]],
Cell[CellGroupData[{
Cell[39202, 752, 6427, 107, 53, "Subsubtitle"],
Cell[45632, 861, 176, 2, 35, "Subsubsection"],
Cell[45811, 865, 1299, 30, 44, "Subsubsection"],
Cell[47113, 897, 1638, 29, 26, "Subsubsection"],
Cell[48754, 928, 3605, 86, 68, "Subsubtitle"],
Cell[52362, 1016, 771, 11, 35, "Subsubsection"],
Cell[53136, 1029, 5425, 122, 129, "Code"],
Cell[58564, 1153, 89, 2, 35, "Subsubsection"]
}, Closed]],
Cell[CellGroupData[{
Cell[58690, 1160, 3433, 58, 53, "Subsubtitle"],
Cell[CellGroupData[{
Cell[62148, 1222, 3774, 67, 53, "Subsubtitle"],
Cell[CellGroupData[{
Cell[65947, 1293, 3450, 59, 64, "Subsubtitle"],
Cell[69400, 1354, 462, 12, 50, "Subsubsection"],
Cell[69865, 1368, 1236, 17, 25, "Subsubsection"],
Cell[71104, 1387, 2389, 51, 64, "Subsubtitle"],
Cell[73496, 1440, 771, 11, 35, "Subsubsection"],
Cell[74270, 1453, 2794, 61, 104, "Code"],
Cell[77067, 1516, 89, 2, 35, "Subsubsection"]
}, Closed]],
Cell[CellGroupData[{
Cell[77193, 1523, 5882, 96, 53, "Subsubtitle"],
Cell[83078, 1621, 1374, 23, 36, "Subsubsection"],
Cell[84455, 1646, 3194, 72, 79, "Subsubtitle"],
Cell[87652, 1720, 771, 11, 35, "Subsubsection"],
Cell[88426, 1733, 3786, 86, 104, "Code"],
Cell[92215, 1821, 89, 2, 35, "Subsubsection"]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell[92353, 1829, 3809, 68, 53, "Subsubtitle"],
Cell[CellGroupData[{
Cell[96187, 1901, 3641, 62, 70, "Subsubtitle"],
Cell[99831, 1965, 408, 6, 35, "Subsubsection"],
Cell[100242, 1973, 3748, 74, 53, "Subsubsection"],
Cell[103993, 2049, 1236, 17, 25, "Subsubsection"],
Cell[105232, 2068, 4588, 84, 70, "Subsubtitle"],
Cell[109823, 2154, 771, 11, 35, "Subsubsection"],
Cell[110597, 2167, 3207, 70, 104, "Code"],
Cell[113807, 2239, 89, 2, 35, "Subsubsection"]
}, Closed]],
Cell[CellGroupData[{
Cell[113933, 2246, 5539, 91, 53, "Subsubtitle"],
Cell[119475, 2339, 1242, 22, 36, "Subsubsection"],
Cell[120720, 2363, 3809, 87, 90, "Subsubtitle"],
Cell[124532, 2452, 771, 11, 35, "Subsubsection"],
Cell[125306, 2465, 4416, 103, 129, "Code"],
Cell[129725, 2570, 89, 2, 35, "Subsubsection"]
}, Closed]]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell[129875, 2579, 3584, 63, 53, "Subsubtitle"],
Cell[133462, 2644, 353, 5, 130, "Subsubsection"],
Cell[133818, 2651, 2359, 46, 167, "Subsubsection"],
Cell[136180, 2699, 1236, 17, 90, "Subsubsection"],
Cell[137419, 2718, 3013, 68, 250, "Subsubtitle"],
Cell[140435, 2788, 771, 11, 130, "Subsubsection"],
Cell[141209, 2801, 3410, 81, 394, "Code"],
Cell[144622, 2884, 89, 2, 130, "Subsubsection"]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell[144760, 2892, 5625, 97, 53, "Subsubtitle"],
Cell[150388, 2991, 280, 6, 130, "Subsubsection"],
Cell[150671, 2999, 1578, 26, 96, "Subsubsection"],
Cell[152252, 3027, 2926, 66, 201, "Subsubtitle"],
Cell[155181, 3095, 771, 11, 130, "Subsubsection"],
Cell[155955, 3108, 2849, 71, 394, "Code"],
Cell[158807, 3181, 89, 2, 130, "Subsubsection"]
}, Closed]],
Cell[CellGroupData[{
Cell[158933, 3188, 5745, 97, 69, "Subsubtitle"],
Cell[164681, 3287, 302, 6, 48, "Subsubsection"],
Cell[164986, 3295, 1117, 26, 54, "Subsubsection"],
Cell[166106, 3323, 1504, 25, 36, "Subsubsection"],
Cell[167613, 3350, 3525, 84, 102, "Subsubtitle"],
Cell[171141, 3436, 771, 11, 48, "Subsubsection"],
Cell[171915, 3449, 3638, 94, 145, "Code"],
Cell[175556, 3545, 89, 2, 48, "Subsubsection"]
}, Closed]],
Cell[CellGroupData[{
Cell[175682, 3552, 5799, 98, 69, "Subsubtitle"],
Cell[181484, 3652, 247, 4, 48, "Subsubsection"],
Cell[181734, 3658, 1608, 41, 88, "Subsubsection"],
Cell[183345, 3701, 1504, 25, 36, "Subsubsection"],
Cell[184852, 3728, 3708, 90, 96, "Subsubtitle"],
Cell[188563, 3820, 771, 11, 48, "Subsubsection"],
Cell[189337, 3833, 4195, 106, 180, "Code"],
Cell[193535, 3941, 89, 2, 48, "Subsubsection"]
}, Closed]],
Cell[CellGroupData[{
Cell[193661, 3948, 5954, 102, 53, "Subsubtitle"],
Cell[199618, 4052, 280, 6, 48, "Subsubsection"],
Cell[199901, 4060, 930, 27, 58, "Subsubsection"],
Cell[200834, 4089, 1802, 31, 36, "Subsubsection"],
Cell[202639, 4122, 3889, 96, 149, "Subsubtitle"],
Cell[206531, 4220, 771, 11, 48, "Subsubsection"],
Cell[207305, 4233, 4134, 110, 180, "Code"],
Cell[211442, 4345, 89, 2, 48, "Subsubsection"]
}, Closed]],
Cell[CellGroupData[{
Cell[211568, 4352, 5835, 100, 53, "Subsubtitle"],
Cell[217406, 4454, 221, 3, 48, "Subsubsection"],
Cell[217630, 4459, 6779, 112, 41, "Subsubsection"],
Cell[224412, 4573, 2613, 59, 74, "Subsubtitle"],
Cell[227028, 4634, 771, 11, 48, "Subsubsection"],
Cell[227802, 4647, 3459, 90, 180, "Code"],
Cell[231264, 4739, 89, 2, 48, "Subsubsection"]
}, Closed]],
Cell[CellGroupData[{
Cell[231390, 4746, 5887, 99, 53, "Subsubtitle"],
Cell[CellGroupData[{
Cell[237302, 4849, 6670, 113, 201, "Subsubtitle"],
Cell[243975, 4964, 480, 7, 130, "Subsubsection"],
Cell[244458, 4973, 3547, 57, 96, "Subsubsection"],
Cell[248008, 5032, 4167, 78, 201, "Subsubtitle"],
Cell[252178, 5112, 771, 11, 130, "Subsubsection"],
Cell[252952, 5125, 6493, 116, 394, "Code"],
Cell[259448, 5243, 190, 4, 130, "Subsubsection"]
}, Closed]],
Cell[CellGroupData[{
Cell[259675, 5252, 6597, 112, 201, "Subsubtitle"],
Cell[266275, 5366, 536, 7, 130, "Subsubsection"],
Cell[266814, 5375, 1170, 28, 167, "Subsubsection"],
Cell[267987, 5405, 3500, 56, 96, "Subsubsection"],
Cell[271490, 5463, 4459, 90, 257, "Subsubtitle"],
Cell[275952, 5555, 771, 11, 130, "Subsubsection"],
Cell[276726, 5568, 7568, 145, 683, "Code"],
Cell[284297, 5715, 190, 4, 130, "Subsubsection"]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell[284536, 5725, 5443, 89, 53, "Subsubtitle"],
Cell[289982, 5816, 1341, 18, 130, "Subsubsection"],
Cell[291326, 5836, 2933, 66, 201, "Subsubtitle"],
Cell[294262, 5904, 771, 11, 130, "Subsubsection"],
Cell[295036, 5917, 2830, 68, 394, "Code"],
Cell[297869, 5987, 89, 2, 130, "Subsubsection"]
}, Closed]]
}
]
*)

(* End of internal cache information *)