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I am trying to compute the following integral:

$$ \int _0^\infty x\exp[-\pi b x^2]{_1F_1}[n,1,\frac{-\pi b x^2}{2}]{_2F_1}[1,\frac{2}{aq},1+\frac{2}{aq},\frac{-sx^{-a}}{2}]dx$$

Is there any general formula for computing integrals involving the confluent and the Gauss Hypergeometric functions with exponential functions?

Thank you in advance

PS: Note that this integral is equivalent to the following one:

$$ \int _{x=0}^\infty x\exp[-\pi b x^2]{_1F_1}[n,1,\frac{-\pi b x^2}{2}][\int_{t=0}^\infty e^{-t}t^{\frac{2}{aq}-1}{_1F_1}[1,1+\frac{2}{aq},\frac{-stx^{-a}}{2}]dt]dx$$ $$= \int _{t=0}^\infty e^{-t}t^{\frac{2}{aq}-1} [\int_{x=0}^\infty x\exp[-\pi b x^2]{_1F_1}[n,1,\frac{-\pi b x^2}{2}] {_1F_1}[1,1+\frac{2}{aq},\frac{-stx^{-a}}{2}]dx]dt$$

,i.e., by solving the inside integral, perhaps, we can reach to a simpler version of the original integral. Note also that the second confluent hypergeometric function can be further simplified to simple Gamma and incomplete Gamma functions.

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Are you trying to find a definite integral? If so, what are the bounds of integration? –  Antonio Vargas Jul 22 '13 at 17:02
I have updated the question. x=0->$\infty$ –  dioxen Jul 22 '13 at 17:04

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