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 AiryBi

 http://functions.wolfram.com/03.06.21.0018.01

 Input Form

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

 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["AiryBi", "[", 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[SuperscriptBox["3", RowBox[List["2", "/", "3"]]], " ", RowBox[List["(", RowBox[List["r", "+", "\[Alpha]"]], ")"]], " ", RowBox[List["Gamma", "[", FractionBox["4", "3"], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "6"], ",", FractionBox[RowBox[List["2", " ", "\[Alpha]"]], RowBox[List["3", " ", "r"]]]]], "}"]], ",", 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["a", " ", SuperscriptBox["z", "r"], " ", "\[Alpha]", " ", RowBox[List["Gamma", "[", FractionBox["2", "3"], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["5", "6"], ",", RowBox[List[FractionBox["2", "3"], "+", FractionBox[RowBox[List["2", " ", "\[Alpha]"]], RowBox[List["3", " ", "r"]]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["5", "3"], ",", RowBox[List[FractionBox["5", "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[SuperscriptBox["3", RowBox[List["5", "/", "6"]]], " ", "\[Alpha]", " ", RowBox[List["(", RowBox[List["r", "+", "\[Alpha]"]], ")"]], " ", RowBox[List["Gamma", "[", FractionBox["2", "3"], "]"]], " ", RowBox[List["Gamma", "[", FractionBox["4", "3"], "]"]]]], ")"]]]]]]]]

 MathML Form

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

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

 2001-10-29