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 AiryBiPrime

 http://functions.wolfram.com/03.08.21.0013.01

 Input Form

 Integrate[(z^(\[Alpha] - 1) AiryBiPrime[a z])/E^((2/3) (a z)^(3/2)), z] == (1/(\[Alpha] Gamma[1/3])) (3^(1/6) z^\[Alpha] HypergeometricPFQ[ {-(1/6), (2 \[Alpha])/3}, {-(1/3), 1 + (2 \[Alpha])/3}, (-(4/3)) (a z)^(3/2)]) + (a^2 z^(2 + \[Alpha]) HypergeometricPFQ[{7/6, 4/3 + (2 \[Alpha])/3}, {7/3, 7/3 + (2 \[Alpha])/3}, (-(4/3)) (a z)^(3/2)])/ (3 3^(1/6) (2 + \[Alpha]) 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", " ", "z"]], ")"]], RowBox[List["3", "/", "2"]]]]]], RowBox[List["AiryBiPrime", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", RowBox[List["\[Alpha]", " ", RowBox[List["Gamma", "[", FractionBox["1", "3"], "]"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["3", RowBox[List["1", "/", "6"]]], " ", SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "6"]]], ",", FractionBox[RowBox[List["2", " ", "\[Alpha]"]], "3"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "3"]]], ",", RowBox[List["1", "+", FractionBox[RowBox[List["2", " ", "\[Alpha]"]], "3"]]]]], "}"]], ",", RowBox[List[RowBox[List["-", FractionBox["4", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", "z"]], ")"]], RowBox[List["3", "/", "2"]]]]]]], "]"]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", "+", "\[Alpha]"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["7", "6"], ",", RowBox[List[FractionBox["4", "3"], "+", FractionBox[RowBox[List["2", " ", "\[Alpha]"]], "3"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["7", "3"], ",", RowBox[List[FractionBox["7", "3"], "+", FractionBox[RowBox[List["2", " ", "\[Alpha]"]], "3"]]]]], "}"]], ",", RowBox[List[RowBox[List["-", FractionBox["4", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", "z"]], ")"]], RowBox[List["3", "/", "2"]]]]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List["3", " ", SuperscriptBox["3", RowBox[List["1", "/", "6"]]], " ", RowBox[List["(", RowBox[List["2", "+", "\[Alpha]"]], ")"]], " ", RowBox[List["Gamma", "[", FractionBox["5", "3"], "]"]]]], ")"]]]]]]]]]]

 MathML Form

 z α - 1 - 2 3 ( a z ) 3 / 2 Bi ( a z ) z 3 6 z α α Γ ( 1 3 ) 2 F 2 ( - 1 6 , 2 α 3 ; - 1 3 , 2 α 3 + 1 ; 1 3 ( - 4 ) ( a z ) 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]"]], "3"], 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]"]], "3"], "+", "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", " ", "z"]], ")"]], RowBox[List["3", "/", "2"]]]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] + a 2 z α + 2 3 3 6 ( α + 2 ) Γ ( 5 3 ) 2 F 2 ( 7 6 , 2 α 3 + 4 3 ; 7 3 , 2 α 3 + 7 3 ; 1 3 ( - 4 ) ( a z ) 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]"]], "3"], "+", 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]"]], "3"], "+", 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", " ", "z"]], ")"]], 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 2 3 a z 3 2 AiryBiPrime a z 3 1 6 z α α Gamma 1 3 -1 HypergeometricPFQ -1 1 6 2 α 3 -1 -1 1 3 2 α 3 -1 1 1 3 -4 a z 3 2 a 2 z α 2 3 3 1 6 α 2 Gamma 5 3 -1 HypergeometricPFQ 7 6 2 α 3 -1 4 3 7 3 2 α 3 -1 7 3 1 3 -4 a z 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_", " ", "z_"]], ")"]], RowBox[List["3", "/", "2"]]]]]], " ", RowBox[List["AiryBiPrime", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["3", RowBox[List["1", "/", "6"]]], " ", SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "6"]]], ",", FractionBox[RowBox[List["2", " ", "\[Alpha]"]], "3"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "3"]]], ",", RowBox[List["1", "+", FractionBox[RowBox[List["2", " ", "\[Alpha]"]], "3"]]]]], "}"]], ",", RowBox[List[FractionBox["1", "3"], " ", RowBox[List["(", RowBox[List["-", "4"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", "z"]], ")"]], RowBox[List["3", "/", "2"]]]]]]], "]"]]]], RowBox[List["\[Alpha]", " ", RowBox[List["Gamma", "[", FractionBox["1", "3"], "]"]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", "+", "\[Alpha]"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["7", "6"], ",", RowBox[List[FractionBox["4", "3"], "+", FractionBox[RowBox[List["2", " ", "\[Alpha]"]], "3"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["7", "3"], ",", RowBox[List[FractionBox["7", "3"], "+", FractionBox[RowBox[List["2", " ", "\[Alpha]"]], "3"]]]]], "}"]], ",", RowBox[List[FractionBox["1", "3"], " ", RowBox[List["(", RowBox[List["-", "4"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", "z"]], ")"]], RowBox[List["3", "/", "2"]]]]]]], "]"]]]], RowBox[List["3", " ", SuperscriptBox["3", RowBox[List["1", "/", "6"]]], " ", RowBox[List["(", RowBox[List["2", "+", "\[Alpha]"]], ")"]], " ", RowBox[List["Gamma", "[", FractionBox["5", "3"], "]"]]]]]]]]]]]

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

 2001-10-29