\int_{0}^{\infty}\frac{t^{\nu+1}J_{\nu}\left(at\right)}{(t^{2}+b^{2})^{\mu+1}}\mathrm{d}t=\frac{a^{\mu}b^{\nu-\mu}}{2^{\mu}\Gamma\left(\mu+1\right)}K_{\nu-\mu}\left(ab\right)