An integral involving Bessel functions and exponentials is given in Gradshteyn & Ryzhik, Equation 6.653 2.) :
$$ \int_0^{\infty} \exp \left[-\frac{1}{2} x -\frac{1}{2x}(z^2+w^2)\right] K_{\nu}\left(\frac{zw}{x}\right)\frac{dx}{x} = 2K_{\nu}(z) 2K_{\nu}(w) \\ |\text{arg}z|,|\text{arg}w|<\pi, \text{arg}(z+w)<\frac{1}{4}\pi $$
I am interested in how one might obtain this formula, and whether it could be generalized for arbitrary positive or negative integer powers of $x$ beyond $x^{-1}$.