I try to find an closed form answer the following integral, $$\int_{0}^{\infty}dx\,x^{m}J_l(Qx)e^{-x^2},\quad m\in\mathbb{Z}, l\in\mathbb{Z},Q>0$$ where $J_l(Qx)$ is Bessel function of the first kind, but I am not sure that it exists. Can anybody help me?
Wolfram Mathematica says that the answer (with assumption $\mathrm{Re}\,(l+m)>-1$) is $$2^{-l-1}Q^l\Gamma\left(\frac{m+l+1}{2}\right){}_1\tilde{F}_1\left(\frac{m+l+1}{2},l+1,-\frac{Q^2}{4}\right),$$ where ${}_1\tilde{F}_1$ is regularized confluent hypergeometric function. However, I feel that due to $m,l\in\mathbb{Z}$ more simple answer can exist.