Given the identity $$ \int^\infty_0 K_v\left(\alpha\sqrt{x^2+z^2}\right) \frac{x^{2\mu+1}}{\left(\sqrt{x^2+z^2}\right)^v}\:\mathrm{d}x = \frac{2^\mu \Gamma(\mu+1)}{\alpha^{\mu+1}z^{v-\mu-1}} K_{v-\mu-1}(\alpha z), \quad \alpha>0,\quad \Re[\mu]>-1$$

how can I find a closed form for the integral:

$$ \int^\infty_0 \exp\left(-\beta x^2\right) K_v\left(\alpha\sqrt{x^2+z^2}\right) \frac{x^{2\mu+1}}{\left(\sqrt{x^2+z^2}\right)^v}\:\mathrm{d}x $$

I tried using series representation of the exponential function, but I got an infinite series.