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.