This is an integral which I want to solve but could not manage to find any analytical solution in the tables.
$$\int_0^{\infty} x^{\lambda} e^{-x^2} K_\mu(a x)K_\nu(b x)dx$$
If there is any way to perform the integral, please let me know.
I arrived at the integral while trying to evaluate an apparently more complicated integral involving the Whittaker functions
$$\int_0^{\infty} x^{\gamma} W_{\mu,\alpha}(x)W_{\mu,\beta}(x) dx $$
and then using the integral representation of Whittaker W. Any known formula to help on this Whittaker integral bypassing the BesselK integral stated above?