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http://functions.wolfram.com/03.07.21.0014.01
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Integrate[(Sqrt[z] AiryAiPrime[a z])/E^((2/3) (a z)^(3/2)), z] ==
(1/(21 a^2 Sqrt[z] Gamma[1/3])) ((6 a^2 z^2 AiryAiPrime[a z] Gamma[1/3] +
2 Sqrt[a z] (-2 3^(2/3) E^((2/3) (a z)^(3/2)) -
(a^3 z^3 BesselI[-(5/3), (2/3) a^(3/2) z^(3/2)] Gamma[1/3])/
(a^(3/2) z^(3/2))^(1/3) + a^2 z^2 (a^(3/2) z^(3/2))^(1/3)
BesselI[5/3, (2/3) a^(3/2) z^(3/2)] Gamma[1/3]))/E^((2/3) (a z)^(3/2)))
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</mfrac> <mo> ⁢ </mo> <msup> <mi> a </mi> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 2 <sep /> 3 </cn> </apply> <apply> <power /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <ci> AiryAiPrime </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 21 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Gamma </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='rational'> 1 <sep /> 3 </cn> <cn type='integer'> -2 </cn> <apply> <power /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> AiryAiPrime </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <ci> Gamma </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> BesselI </ci> <cn type='rational'> 5 <sep /> 3 </cn> <apply> <times /> <cn type='rational'> 2 <sep /> 3 </cn> <apply> <power /> <ci> a </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 3 </cn> </apply> </apply> <apply> <times /> <cn 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Date Added to functions.wolfram.com (modification date)
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