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 Exp

 http://functions.wolfram.com/01.03.21.0794.01

 Input Form

 Integrate[z^3 (a + b c^(d z))^\[Beta], z] == (1/(d^4 \[Beta]^4 Log[c]^4)) (((a + b c^(d z))^\[Beta] (-6 HypergeometricPFQ[{-\[Beta], -\[Beta], -\[Beta], -\[Beta], -\[Beta]}, {1 - \[Beta], 1 - \[Beta], 1 - \[Beta], 1 - \[Beta]}, -(a/(c^(d z) b))] + d z \[Beta] Log[c] (6 HypergeometricPFQ[{-\[Beta], -\[Beta], -\[Beta], -\[Beta]}, {1 - \[Beta], 1 - \[Beta], 1 - \[Beta]}, -(a/(c^(d z) b))] + d z \[Beta] Log[c] (-3 HypergeometricPFQ[{-\[Beta], -\[Beta], -\[Beta]}, {1 - \[Beta], 1 - \[Beta]}, -(a/(c^(d z) b))] + d z \[Beta] Hypergeometric2F1[-\[Beta], -\[Beta], 1 - \[Beta], -(a/(c^(d z) b))] Log[c]))))/(1 + a/(c^(d z) b))^\[Beta])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "3"], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox["c", RowBox[List["d", " ", "z"]]]]]]], ")"]], "\[Beta]"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List[SuperscriptBox["d", "4"], " ", SuperscriptBox["\[Beta]", "4"], " ", SuperscriptBox[RowBox[List["Log", "[", "c", "]"]], "4"]]]], RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["a", " ", SuperscriptBox["c", RowBox[List[RowBox[List["-", "d"]], " ", "z"]]]]], "b"]]], ")"]], RowBox[List["-", "\[Beta]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox["c", RowBox[List["d", " ", "z"]]]]]]], ")"]], "\[Beta]"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "6"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", "\[Beta]"]], ",", RowBox[List["1", "-", "\[Beta]"]], ",", RowBox[List["1", "-", "\[Beta]"]], ",", RowBox[List["1", "-", "\[Beta]"]]]], "}"]], ",", RowBox[List["-", FractionBox[RowBox[List["a", " ", SuperscriptBox["c", RowBox[List[RowBox[List["-", "d"]], " ", "z"]]]]], "b"]]]]], "]"]]]], "+", RowBox[List["d", " ", "z", " ", "\[Beta]", " ", RowBox[List["Log", "[", "c", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["6", " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", "\[Beta]"]], ",", RowBox[List["1", "-", "\[Beta]"]], ",", RowBox[List["1", "-", "\[Beta]"]]]], "}"]], ",", RowBox[List["-", FractionBox[RowBox[List["a", " ", SuperscriptBox["c", RowBox[List[RowBox[List["-", "d"]], " ", "z"]]]]], "b"]]]]], "]"]]]], "+", RowBox[List["d", " ", "z", " ", "\[Beta]", " ", RowBox[List["Log", "[", "c", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", "\[Beta]"]], ",", RowBox[List["1", "-", "\[Beta]"]]]], "}"]], ",", RowBox[List["-", FractionBox[RowBox[List["a", " ", SuperscriptBox["c", RowBox[List[RowBox[List["-", "d"]], " ", "z"]]]]], "b"]]]]], "]"]]]], "+", RowBox[List["d", " ", "z", " ", "\[Beta]", " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["1", "-", "\[Beta]"]], ",", RowBox[List["-", FractionBox[RowBox[List["a", " ", SuperscriptBox["c", RowBox[List[RowBox[List["-", "d"]], " ", "z"]]]]], "b"]]]]], "]"]], " ", RowBox[List["Log", "[", "c", "]"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]], ")"]]]]]]]]

 MathML Form

 z 3 ( a + b c d z ) β z 1 d 4 β 4 log 4 ( c ) ( ( a c - d z b + 1 ) - β ( b c d z + a ) β ( d z β log ( c ) ( 6 4 F 3 ( - β , - β , - β , - β ; 1 - β , 1 - β , 1 - β ; - a c - d z b ) 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[RowBox[List["-", "\[Beta]"]], HypergeometricPFQ], ",", TagBox[RowBox[List["-", "\[Beta]"]], HypergeometricPFQ], ",", TagBox[RowBox[List["-", "\[Beta]"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", "\[Beta]"]], HypergeometricPFQ], ",", TagBox[RowBox[List["1", "-", "\[Beta]"]], HypergeometricPFQ], ",", TagBox[RowBox[List["1", "-", "\[Beta]"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", FractionBox[RowBox[List["a", " ", SuperscriptBox["c", RowBox[List[RowBox[List["-", "d"]], " ", "z"]]]]], "b"]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] + d z β log ( c ) ( d z β 2 F 1 ( - β , - β ; 1 - β ; - a c - d z b ) 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[RowBox[List["-", "\[Beta]"]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List["1", "-", "\[Beta]"]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", FractionBox[RowBox[List["a", " ", SuperscriptBox["c", RowBox[List[RowBox[List["-", "d"]], " ", "z"]]]]], "b"]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] log ( c ) - 3 3 F 2 ( - β , - β , - β ; 1 - β , 1 - β ; - a c - d z b ) 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[RowBox[List["-", "\[Beta]"]], HypergeometricPFQ], ",", TagBox[RowBox[List["-", "\[Beta]"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", "\[Beta]"]], HypergeometricPFQ], ",", TagBox[RowBox[List["1", "-", "\[Beta]"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", FractionBox[RowBox[List["a", " ", SuperscriptBox["c", RowBox[List[RowBox[List["-", "d"]], " ", "z"]]]]], "b"]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] ) ) - 6 5 F 4 ( - β , - β , - β , - β , - β ; 1 - β , 1 - β , 1 - β , 1 - β ; - a c - d z b ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["5", TraditionalForm]], SubscriptBox["F", FormBox["4", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", "\[Beta]"]], HypergeometricPFQ], ",", TagBox[RowBox[List["-", "\[Beta]"]], HypergeometricPFQ], ",", TagBox[RowBox[List["-", "\[Beta]"]], HypergeometricPFQ], ",", TagBox[RowBox[List["-", "\[Beta]"]], HypergeometricPFQ], ",", TagBox[RowBox[List["-", "\[Beta]"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", "\[Beta]"]], HypergeometricPFQ], ",", TagBox[RowBox[List["1", "-", "\[Beta]"]], HypergeometricPFQ], ",", TagBox[RowBox[List["1", "-", "\[Beta]"]], HypergeometricPFQ], ",", TagBox[RowBox[List["1", "-", "\[Beta]"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", FractionBox[RowBox[List["a", " ", SuperscriptBox["c", RowBox[List[RowBox[List["-", "d"]], " ", "z"]]]]], "b"]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] ) ) z z 3 a b c d z β 1 d 4 β 4 c 4 -1 a c -1 d z b -1 1 -1 β b c d z a β d z β c 6 HypergeometricPFQ -1 β -1 β -1 β -1 β 1 -1 β 1 -1 β 1 -1 β -1 a c -1 d z b -1 d z β c d z β Hypergeometric2F1 -1 β -1 β 1 -1 β -1 a c -1 d z b -1 c -1 3 HypergeometricPFQ -1 β -1 β -1 β 1 -1 β 1 -1 β -1 a c -1 d z b -1 -1 6 HypergeometricPFQ -1 β -1 β -1 β -1 β -1 β 1 -1 β 1 -1 β 1 -1 β 1 -1 β -1 a c -1 d z b -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", "3"], " ", 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["1", "+", FractionBox[RowBox[List["a", " ", SuperscriptBox["c", RowBox[List[RowBox[List["-", "d"]], " ", "z"]]]]], "b"]]], ")"]], RowBox[List["-", "\[Beta]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox["c", RowBox[List["d", " ", "z"]]]]]]], ")"]], "\[Beta]"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "6"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", "\[Beta]"]], ",", RowBox[List["1", "-", "\[Beta]"]], ",", RowBox[List["1", "-", "\[Beta]"]], ",", RowBox[List["1", "-", "\[Beta]"]]]], "}"]], ",", RowBox[List["-", FractionBox[RowBox[List["a", " ", SuperscriptBox["c", RowBox[List[RowBox[List["-", "d"]], " ", "z"]]]]], "b"]]]]], "]"]]]], "+", RowBox[List["d", " ", "z", " ", "\[Beta]", " ", RowBox[List["Log", "[", "c", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["6", " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", "\[Beta]"]], ",", RowBox[List["1", "-", "\[Beta]"]], ",", RowBox[List["1", "-", "\[Beta]"]]]], "}"]], ",", RowBox[List["-", FractionBox[RowBox[List["a", " ", SuperscriptBox["c", RowBox[List[RowBox[List["-", "d"]], " ", "z"]]]]], "b"]]]]], "]"]]]], "+", RowBox[List["d", " ", "z", " ", "\[Beta]", " ", RowBox[List["Log", "[", "c", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", "\[Beta]"]], ",", RowBox[List["1", "-", "\[Beta]"]]]], "}"]], ",", RowBox[List["-", FractionBox[RowBox[List["a", " ", SuperscriptBox["c", RowBox[List[RowBox[List["-", "d"]], " ", "z"]]]]], "b"]]]]], "]"]]]], "+", RowBox[List["d", " ", "z", " ", "\[Beta]", " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["1", "-", "\[Beta]"]], ",", RowBox[List["-", FractionBox[RowBox[List["a", " ", SuperscriptBox["c", RowBox[List[RowBox[List["-", "d"]], " ", "z"]]]]], "b"]]]]], "]"]], " ", RowBox[List["Log", "[", "c", "]"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]], RowBox[List[SuperscriptBox["d", "4"], " ", SuperscriptBox["\[Beta]", "4"], " ", SuperscriptBox[RowBox[List["Log", "[", "c", "]"]], "4"]]]]]]]]

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

 2002-12-18

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