# Integral of Bessel function products times Gaussian times algebraic function

I am searching for a closed form solution to the following integral:

$$\int_0^\infty J_m(\rho x ) J_m(a x) \frac{x} {x^2 + c^2} e^{-d x^2} \, dx.$$

The tables from Gradshteyn and Ryzhik provide solutions for similar integrals, but not this one in particular.

Does anyone know what is the solution to the integral? Also, how would one go about solving this integral?