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 AiryAiPrime

 http://functions.wolfram.com/03.07.21.0017.01

 Input Form

 Integrate[(z^(\[Alpha] - 1) AiryAiPrime[a z^r])/E^((2/3) (a z^r)^(3/2)), z] == (z^\[Alpha] (-3 3^(1/3) (2 r + \[Alpha]) Gamma[5/3] HypergeometricPFQ[{-(1/6), (2 \[Alpha])/(3 r)}, {-(1/3), 1 + (2 \[Alpha])/(3 r)}, (-(4/3)) (a z^r)^(3/2)] + a^2 z^(2 r) \[Alpha] Gamma[1/3] HypergeometricPFQ[ {7/6, 4/3 + (2 \[Alpha])/(3 r)}, {7/3, 7/3 + (2 \[Alpha])/(3 r)}, (-(4/3)) (a z^r)^(3/2)]))/(3 3^(2/3) \[Alpha] (2 r + \[Alpha]) Gamma[1/3] Gamma[5/3])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["2", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["3", "/", "2"]]]]]], RowBox[List["AiryAiPrime", "[", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", SuperscriptBox["3", RowBox[List["1", "/", "3"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "r"]], "+", "\[Alpha]"]], ")"]], " ", RowBox[List["Gamma", "[", FractionBox["5", "3"], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "6"]]], ",", FractionBox[RowBox[List["2", " ", "\[Alpha]"]], RowBox[List["3", " ", "r"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "3"]]], ",", RowBox[List["1", "+", FractionBox[RowBox[List["2", " ", "\[Alpha]"]], RowBox[List["3", " ", "r"]]]]]]], "}"]], ",", RowBox[List[RowBox[List["-", FractionBox["4", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["3", "/", "2"]]]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]], " ", "\[Alpha]", " ", RowBox[List["Gamma", "[", FractionBox["1", "3"], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["7", "6"], ",", RowBox[List[FractionBox["4", "3"], "+", FractionBox[RowBox[List["2", " ", "\[Alpha]"]], RowBox[List["3", " ", "r"]]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["7", "3"], ",", RowBox[List[FractionBox["7", "3"], "+", FractionBox[RowBox[List["2", " ", "\[Alpha]"]], RowBox[List["3", " ", "r"]]]]]]], "}"]], ",", RowBox[List[RowBox[List["-", FractionBox["4", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["3", "/", "2"]]]]]]], "]"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List["3", " ", SuperscriptBox["3", RowBox[List["2", "/", "3"]]], " ", "\[Alpha]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "r"]], "+", "\[Alpha]"]], ")"]], " ", RowBox[List["Gamma", "[", FractionBox["1", "3"], "]"]], " ", RowBox[List["Gamma", "[", FractionBox["5", "3"], "]"]]]], ")"]]]]]]]]

 MathML Form

 z α - 1 1 3 ( - 2 ) ( a z r ) 3 / 2 Ai ( a z r ) z 1 3 3 2 / 3 α ( 2 r + α ) Γ ( 1 3 ) Γ ( 5 3 ) ( z α ( a 2 z 2 r α Γ ( 1 3 ) 2 F 2 ( 7 6 , 2 α 3 r + 4 3 ; 7 3 , 2 α 3 r + 7 3 ; 1 3 ( - 4 ) ( a z r ) 3 / 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox["7", "6"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox[RowBox[List["2", " ", "\[Alpha]"]], RowBox[List["3", " ", "r"]]], "+", FractionBox["4", "3"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["7", "3"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox[RowBox[List["2", " ", "\[Alpha]"]], RowBox[List["3", " ", "r"]]], "+", FractionBox["7", "3"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List[FractionBox["1", "3"], " ", RowBox[List["(", RowBox[List["-", "4"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["3", "/", "2"]]]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] - 3 3 3 ( 2 r + α ) Γ ( 5 3 ) 2 F 2 ( - 1 6 , 2 α 3 r ; - 1 3 , 2 α 3 r + 1 ; 1 3 ( - 4 ) ( a z r ) 3 / 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", FractionBox["1", "6"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["2", " ", "\[Alpha]"]], RowBox[List["3", " ", "r"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", FractionBox["1", "3"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox[RowBox[List["2", " ", "\[Alpha]"]], RowBox[List["3", " ", "r"]]], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List[FractionBox["1", "3"], " ", RowBox[List["(", RowBox[List["-", "4"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["3", "/", "2"]]]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] ) ) z z α -1 1 3 -2 a z r 3 2 AiryAiPrime a z r 1 3 3 2 3 α 2 r α Gamma 1 3 Gamma 5 3 -1 z α a 2 z 2 r α Gamma 1 3 HypergeometricPFQ 7 6 2 α 3 r -1 4 3 7 3 2 α 3 r -1 7 3 1 3 -4 a z r 3 2 -1 3 3 1 3 2 r α Gamma 5 3 HypergeometricPFQ -1 1 6 2 α 3 r -1 -1 1 3 2 α 3 r -1 1 1 3 -4 a z r 3 2 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "3"], " ", RowBox[List["(", RowBox[List["-", "2"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a_", " ", SuperscriptBox["z_", "r_"]]], ")"]], RowBox[List["3", "/", "2"]]]]]], " ", RowBox[List["AiryAiPrime", "[", RowBox[List["a_", " ", SuperscriptBox["z_", "r_"]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", SuperscriptBox["3", RowBox[List["1", "/", "3"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "r"]], "+", "\[Alpha]"]], ")"]], " ", RowBox[List["Gamma", "[", FractionBox["5", "3"], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "6"]]], ",", FractionBox[RowBox[List["2", " ", "\[Alpha]"]], RowBox[List["3", " ", "r"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "3"]]], ",", RowBox[List["1", "+", FractionBox[RowBox[List["2", " ", "\[Alpha]"]], RowBox[List["3", " ", "r"]]]]]]], "}"]], ",", RowBox[List[FractionBox["1", "3"], " ", RowBox[List["(", RowBox[List["-", "4"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["3", "/", "2"]]]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]], " ", "\[Alpha]", " ", RowBox[List["Gamma", "[", FractionBox["1", "3"], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["7", "6"], ",", RowBox[List[FractionBox["4", "3"], "+", FractionBox[RowBox[List["2", " ", "\[Alpha]"]], RowBox[List["3", " ", "r"]]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["7", "3"], ",", RowBox[List[FractionBox["7", "3"], "+", FractionBox[RowBox[List["2", " ", "\[Alpha]"]], RowBox[List["3", " ", "r"]]]]]]], "}"]], ",", RowBox[List[FractionBox["1", "3"], " ", RowBox[List["(", RowBox[List["-", "4"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["3", "/", "2"]]]]]]], "]"]]]]]], ")"]]]], RowBox[List["3", " ", SuperscriptBox["3", RowBox[List["2", "/", "3"]]], " ", "\[Alpha]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "r"]], "+", "\[Alpha]"]], ")"]], " ", RowBox[List["Gamma", "[", FractionBox["1", "3"], "]"]], " ", RowBox[List["Gamma", "[", FractionBox["5", "3"], "]"]]]]]]]]]

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

 2001-10-29