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 Exp

 http://functions.wolfram.com/01.03.21.0509.01

 Input Form

 Integrate[(a + b E^(2 c z))^(5/2) (E^(c z))^\[Nu], z] == ((E^(c z))^\[Nu] Sqrt[a + b E^(2 c z)] (2 a b E^(2 c z) \[Nu] (4 + \[Nu]) Hypergeometric2F1[-(1/2), 1 + \[Nu]/2, 2 + \[Nu]/2, -((b E^(2 c z))/a)] + (2 + \[Nu]) (b^2 E^(4 c z) \[Nu] Hypergeometric2F1[-(1/2), 2 + \[Nu]/2, 3 + \[Nu]/2, -((b E^(2 c z))/a)] + a^2 (4 + \[Nu]) Hypergeometric2F1[-(1/2), \[Nu]/2, 1 + \[Nu]/2, -((b E^(2 c z))/a)])))/ (c Sqrt[1 + (b E^(2 c z))/a] \[Nu] (2 + \[Nu]) (4 + \[Nu]))

 Standard Form

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 MathML Form

 ( a + b 2 c z ) 5 / 2 ( c z ) ν z ( ( c z ) ν a + b 2 c z ( ( ν + 2 ) ( ( ν + 4 ) 2 F 1 ( - 1 2 , ν 2 ; ν 2 + 1 ; - b 2 c 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["-", FractionBox["1", "2"]]], Hypergeometric2F1], ",", TagBox[FractionBox["\[Nu]", "2"], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", "1"]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "a"]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] a 2 + b 2 4 c z ν 2 F 1 ( - 1 2 , ν 2 + 2 ; ν 2 + 3 ; - b 2 c 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["-", FractionBox["1", "2"]]], Hypergeometric2F1], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", "2"]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", "3"]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "a"]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] ) + 2 a b 2 c z ν ( ν + 4 ) 2 F 1 ( - 1 2 , ν 2 + 1 ; ν 2 + 2 ; - b 2 c 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["-", FractionBox["1", "2"]]], Hypergeometric2F1], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", "1"]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", "2"]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "a"]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] ) ) / ( c 2 c z b a + 1 ν ( ν + 2 ) ( ν + 4 ) ) z a b 2 c z 5 2 c z ν c z ν a b 2 c z 1 2 ν 2 ν 4 Hypergeometric2F1 -1 1 2 ν 2 -1 ν 2 -1 1 -1 b 2 c z a -1 a 2 b 2 4 c z ν Hypergeometric2F1 -1 1 2 ν 2 -1 2 ν 2 -1 3 -1 b 2 c z a -1 2 a b 2 c z ν ν 4 Hypergeometric2F1 -1 1 2 ν 2 -1 1 ν 2 -1 2 -1 b 2 c z a -1 c 2 c z b a -1 1 1 2 ν ν 2 ν 4 -1 [/itex]

 Rule Form

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

 2002-12-18