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http://functions.wolfram.com/03.08.21.0032.01
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Integrate[z^(\[Alpha] - 1) Cosh[b + (2/3) (a z)^(3/2)] AiryBiPrime[a z],
z] ==
(z^\[Alpha] (9 (2 + \[Alpha]) Gamma[5/3] HypergeometricPFQ[
{-(1/6), (2 \[Alpha])/3}, {-(1/3), 1 + (2 \[Alpha])/3},
(-(4/3)) (a z)^(3/2)] + 9 E^(2 b) (2 + \[Alpha]) Gamma[5/3]
HypergeometricPFQ[{-(1/6), (2 \[Alpha])/3},
{-(1/3), 1 + (2 \[Alpha])/3}, (4/3) (a z)^(3/2)] +
3^(2/3) a^2 z^2 \[Alpha] Gamma[1/3]
(HypergeometricPFQ[{7/6, 4/3 + (2 \[Alpha])/3},
{7/3, 7/3 + (2 \[Alpha])/3}, (-(4/3)) (a z)^(3/2)] +
E^(2 b) HypergeometricPFQ[{7/6, 4/3 + (2 \[Alpha])/3},
{7/3, 7/3 + (2 \[Alpha])/3}, (4/3) (a z)^(3/2)])))/E^b/
(6 3^(5/6) \[Alpha] (2 + \[Alpha]) Gamma[1/3] Gamma[5/3])
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