I want to solve the following integral
$$ \int _{0}^{\infty} xJ_0(x)\exp(-q^2x^2-bx) \,\mathrm dx .$$
Firstly I used the expansion of Bessel function $J_0(ax)$
$$ \sum_{n=0}^{\infty}\frac{(-1)^n(x)^{2n}}{(n!)^24^n} $$
and expansion of the exponential term
$$ \sum_{k=0}^{\infty}\frac{(-b)^k(x)^k}{k!} .$$
The final answer is
$$ =\frac{1}{p^2} \sum_{n=0}^{\infty}\frac{(-1)^n}{(n!)^2 4^np^{2n}} \sum_{k=0}^{\infty}\frac{(-b)^k(n+\frac{k}{2})!}{k!p^k}.$$
Also, I tried to simplify the above answer to convert it to the $ \exp(-x^2)$ form, but I could not. Is there any alternative suggestion to solve the integral or convert the form? I need help, please.
The closest thing I could find is DLMF 10.22.51 which is the general integral
$$\int_0^\infty t^{\nu+1}\exp(-p^2t^2) J_\nu(at)\mathrm dt=\frac{a^\nu}{(2p^2)^{\nu+1}}\exp\left(-\frac{a^2}{4p^2}\right)$$ So at least in the case $b=0$ your integral is $$\frac{1}{2q^2}\exp\left(\frac{-1}{4q^2}\right)$$ I don't know if there are any known expressions for $b\neq 0$ though.