I am trying to solve an integral like
$F(q,\alpha)=\int_0^\infty dx~ x~ e^{-(x-\alpha)^2/\beta^2}J_0(qx)$ where $\alpha,\beta>0,\in \mathbb{R}$; $q\geq 0,\in \mathbb{R}$
For a similar result, it is well known that $\int_0^\infty dx~ x ~e^{-x^2/\beta^2}J_0(qx)=\frac{\beta^2}{2}e^{-q^2\beta^2/4}$. However, I am unable to find any standard tricks (like one does with a normal Gaussian when shifted) in order to find the closed form expression for $F(q,\alpha)$.
I am able to verify numerically that this integral exists for a given $q,\alpha$ on the intervals specified, and have tried to compute this symbolically in Mathematica to no avail. My first thought was to expand the exponential and then do the integral, where I find the Taylor expansion for the solution,
$F(q,\alpha)\approx -\frac{0.735759}{\alpha q^3} -\frac{2.20728}{\alpha^3 q^5}+\frac{5.51819}{\alpha^5 q^7} +\frac{373.398}{\alpha^7 q^9}+ \frac{9705.12}{\alpha^9 q^{11}} + \frac{229893.}{\alpha^{11} q^{13}} +\frac{5.37139\times 10^6}{\alpha^{13} q^{15}}+... $
(Note that these numerial coefficients are to high precision consistent with $-\frac{2}{e}$,$-\frac{6}{e}$,$\frac{15}{e}$,$\frac{1015}{e}$,...)
Then my thinking was that if I can guess the solution, which from the solved example above is potentially composed of exponetials, and confirm this by Taylor expand my guess. This program has been unsuccessful so far.
A proof is not necessary, maybe someone knows a reference where this is done, or a means to finding the solution. However, a closed form expression is what I am after, unfortunately the numerical solution is not sufficient.