While I was elaborating on a flow problem, I came across the following non-trivial improper integral: $$ \int_0^\infty k^3 \left( k + \sqrt{k^2+\alpha^2} \right) e^{-\beta \left( k + \sqrt{k^2+\alpha^2} \right)} J_0 \left( \rho k \right) \mathrm{d} k $$ wherein $\alpha$, $\beta$, and $\rho$ are positive real numbers. Here, $J_0$ denotes the zeroth-order Bessel function of the first kind.
For $\alpha = 0$ and for $\rho = 0$, the corresponding analytical expressions can easily be determined.
For the general case, I tried to use various integral definitions of the Bessel function. Unfortunately this did not seem to help.
I was wondering if someone here, who has expertise in those types of problems, could be of help and provide a few hints or suggestions to evaluate the above integral, that would be highly appreciated.
Thank you!