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 Exp

 http://functions.wolfram.com/01.03.21.0796.01

 Input Form

 Integrate[z^2 e^(f z) (a + b c^(d z))^\[Beta], z] == (1/(f^3 Log[e]^3)) (((a + b c^(d z))^\[Beta] e^(f z) (2 HypergeometricPFQ[{-\[Beta], (f Log[e])/(d Log[c]), (f Log[e])/(d Log[c]), (f Log[e])/(d Log[c])}, {1 + (f Log[e])/(d Log[c]), 1 + (f Log[e])/(d Log[c]), 1 + (f Log[e])/(d Log[c])}, -((b c^(d z))/a)] + f z Log[e] (-2 HypergeometricPFQ[{-\[Beta], (f Log[e])/(d Log[c]), (f Log[e])/(d Log[c])}, {1 + (f Log[e])/(d Log[c]), 1 + (f Log[e])/(d Log[c])}, -((b c^(d z))/a)] + f z Hypergeometric2F1[-\[Beta], (f Log[e])/(d Log[c]), 1 + (f Log[e])/(d Log[c]), -((b c^(d z))/a)] Log[e])))/ (1 + (b c^(d z))/a)^\[Beta])

 Standard Form

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 MathML Form

 z 2 e f z ( a + b c d z ) β z 1 f 3 log 3 ( e ) ( ( b c d z + a ) β ( b c d z a + 1 ) - β e f z ( 2 4 F 3 ( - β , f log ( e ) d log ( c ) , f log ( e ) d log ( c ) , f log ( e ) d log ( c ) ; f log ( e ) d log ( c ) + 1 , f log ( e ) d log ( c ) + 1 , f log ( e ) d log ( c ) + 1 ; - b c d z a ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["4", TraditionalForm]], SubscriptBox["F", FormBox["3", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", "\[Beta]"]], HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List["f", " ", RowBox[List["log", "(", "e", ")"]]]], RowBox[List["d", " ", RowBox[List["log", "(", "c", ")"]]]]], HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List["f", " ", RowBox[List["log", "(", "e", ")"]]]], RowBox[List["d", " ", RowBox[List["log", "(", "c", ")"]]]]], HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List["f", " ", RowBox[List["log", "(", "e", ")"]]]], RowBox[List["d", " ", RowBox[List["log", "(", "c", ")"]]]]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[RowBox[List["f", " ", RowBox[List["log", "(", "e", ")"]]]], RowBox[List["d", " ", RowBox[List["log", "(", "c", ")"]]]]], "+", "1"]], HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List["f", " ", RowBox[List["log", "(", "e", ")"]]]], RowBox[List["d", " ", RowBox[List["log", "(", "c", ")"]]]]], "+", "1"]], HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List["f", " ", RowBox[List["log", "(", "e", ")"]]]], RowBox[List["d", " ", RowBox[List["log", "(", "c", ")"]]]]], "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["c", RowBox[List["d", " ", "z"]]]]], "a"]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] + f z log ( e ) ( f z 2 F 1 ( - β , f log ( e ) d log ( c ) ; f log ( e ) d log ( c ) + 1 ; - b c d z a ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", "\[Beta]"]], Hypergeometric2F1], ",", TagBox[FractionBox[RowBox[List["f", " ", RowBox[List["log", "(", "e", ")"]]]], RowBox[List["d", " ", RowBox[List["log", "(", "c", ")"]]]]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox[RowBox[List["f", " ", RowBox[List["log", "(", "e", ")"]]]], RowBox[List["d", " ", RowBox[List["log", "(", "c", ")"]]]]], "+", "1"]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["c", RowBox[List["d", " ", "z"]]]]], "a"]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] log ( e ) - 2 3 F 2 ( - β , f log ( e ) d log ( c ) , f log ( e ) d log ( c ) ; f log ( e ) d log ( c ) + 1 , f log ( e ) d log ( c ) + 1 ; - b c d z a ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", "\[Beta]"]], HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List["f", " ", RowBox[List["log", "(", "e", ")"]]]], RowBox[List["d", " ", RowBox[List["log", "(", "c", ")"]]]]], HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List["f", " ", RowBox[List["log", "(", "e", ")"]]]], RowBox[List["d", " ", RowBox[List["log", "(", "c", ")"]]]]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[RowBox[List["f", " ", RowBox[List["log", "(", "e", ")"]]]], RowBox[List["d", " ", RowBox[List["log", "(", "c", ")"]]]]], "+", "1"]], HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List["f", " ", RowBox[List["log", "(", "e", ")"]]]], RowBox[List["d", " ", RowBox[List["log", "(", "c", ")"]]]]], "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["c", RowBox[List["d", " ", "z"]]]]], "a"]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] ) ) ) z z 2 e f z a b c d z β 1 f 3 e 3 -1 b c d z a β b c d z a -1 1 -1 β e f z 2 HypergeometricPFQ -1 β f e d c -1 f e d c -1 f e d c -1 f e d c -1 1 f e d c -1 1 f e d c -1 1 -1 b c d z a -1 f z e f z Hypergeometric2F1 -1 β f e d c -1 f e d c -1 1 -1 b c d z a -1 e -1 2 HypergeometricPFQ -1 β f e d c -1 f e d c -1 f e d c -1 1 f e d c -1 1 -1 b c d z a -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", "2"], " ", SuperscriptBox["e_", RowBox[List["f_", " ", "z_"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a_", "+", RowBox[List["b_", " ", SuperscriptBox["c_", RowBox[List["d_", " ", "z_"]]]]]]], ")"]], "\[Beta]_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox["c", RowBox[List["d", " ", "z"]]]]]]], ")"]], "\[Beta]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["b", " ", SuperscriptBox["c", RowBox[List["d", " ", "z"]]]]], "a"]]], ")"]], RowBox[List["-", "\[Beta]"]]], " ", SuperscriptBox["e", RowBox[List["f", " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "\[Beta]"]], ",", FractionBox[RowBox[List["f", " ", RowBox[List["Log", "[", "e", "]"]]]], RowBox[List["d", " ", RowBox[List["Log", "[", "c", "]"]]]]], ",", FractionBox[RowBox[List["f", " ", RowBox[List["Log", "[", "e", "]"]]]], RowBox[List["d", " ", RowBox[List["Log", "[", "c", "]"]]]]], ",", FractionBox[RowBox[List["f", " ", RowBox[List["Log", "[", "e", "]"]]]], RowBox[List["d", " ", RowBox[List["Log", "[", "c", "]"]]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", FractionBox[RowBox[List["f", " ", RowBox[List["Log", "[", "e", "]"]]]], RowBox[List["d", " ", RowBox[List["Log", "[", "c", "]"]]]]]]], ",", RowBox[List["1", "+", FractionBox[RowBox[List["f", " ", RowBox[List["Log", "[", "e", "]"]]]], RowBox[List["d", " ", RowBox[List["Log", "[", "c", "]"]]]]]]], ",", RowBox[List["1", "+", FractionBox[RowBox[List["f", " ", RowBox[List["Log", "[", "e", "]"]]]], RowBox[List["d", " ", RowBox[List["Log", "[", "c", "]"]]]]]]]]], "}"]], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["c", RowBox[List["d", " ", "z"]]]]], "a"]]]]], "]"]]]], "+", RowBox[List["f", " ", "z", " ", RowBox[List["Log", "[", "e", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "\[Beta]"]], ",", FractionBox[RowBox[List["f", " ", RowBox[List["Log", "[", "e", "]"]]]], RowBox[List["d", " ", RowBox[List["Log", "[", "c", "]"]]]]], ",", FractionBox[RowBox[List["f", " ", RowBox[List["Log", "[", "e", "]"]]]], RowBox[List["d", " ", RowBox[List["Log", "[", "c", "]"]]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", FractionBox[RowBox[List["f", " ", RowBox[List["Log", "[", "e", "]"]]]], RowBox[List["d", " ", RowBox[List["Log", "[", "c", "]"]]]]]]], ",", RowBox[List["1", "+", FractionBox[RowBox[List["f", " ", RowBox[List["Log", "[", "e", "]"]]]], RowBox[List["d", " ", RowBox[List["Log", "[", "c", "]"]]]]]]]]], "}"]], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["c", RowBox[List["d", " ", "z"]]]]], "a"]]]]], "]"]]]], "+", RowBox[List["f", " ", "z", " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", "\[Beta]"]], ",", FractionBox[RowBox[List["f", " ", RowBox[List["Log", "[", "e", "]"]]]], RowBox[List["d", " ", RowBox[List["Log", "[", "c", "]"]]]]], ",", RowBox[List["1", "+", FractionBox[RowBox[List["f", " ", RowBox[List["Log", "[", "e", "]"]]]], RowBox[List["d", " ", RowBox[List["Log", "[", "c", "]"]]]]]]], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["c", RowBox[List["d", " ", "z"]]]]], "a"]]]]], "]"]], " ", RowBox[List["Log", "[", "e", "]"]]]]]], ")"]]]]]], ")"]]]], RowBox[List[SuperscriptBox["f", "3"], " ", SuperscriptBox[RowBox[List["Log", "[", "e", "]"]], "3"]]]]]]]]

 Date Added to functions.wolfram.com (modification date)

 2002-12-18