How to Calculate the following integration?

Question

I need to calculate the following integration.

$$\int_0^\infty \!\,xe^{-tx^2}\, j_\alpha ( u_1 x)j_{\alpha }(u_2 x)dx$$

where $j_\alpha ( u_1 x)$ and $j_{\alpha }(u_2 x)$ are Bessel functions of the first kind of an integer order $\alpha$.

Answer 1

Identity 10.22.67 on DLMF says that $$\int_0^\infty x\exp(-p^2x^2)J_{\nu}(ax)J_\nu(bx)\mathrm{d}x=\frac{1}{2p^2}\exp\left(-\frac{a^2+b^2}{4p^2}\right)I_\nu\left(\frac{ab}{2p^2}\right)$$ With $I_\nu$ being the modified Bessel function of the first kind of order $\nu$. Plug in $p=\sqrt{t}$ for your answer.