in the context of a physics problem, I encountered the following integral: $$ \int_0^{\infty} d x J_{N_1+N_2}\left(q x\right) \cdot x^{\left|N_1\right|+\left|N_2\right|} e^{-\frac{x^2}{2}} L_{a_1}^{\left|N_1\right|}\left(\frac{x^2}{2}\right) L_{a_2}^{\left|N_2\right|}\left(\frac{x^2}{2}\right) $$
There is an expression in terms of a finite summation involving ${}_1F_1(\dots,\dots,q^2/2)$ given in [Phys Rev B 30 4392]. I can obtain it by plugging in the explicit sum representation for the Laguerre polynomials and then using Gradshtyen and Ryzhik for the resulting integral that only involves the Bessel function, powers and exponentials.
However: I suspect there's a simpler way to represent the final result, and can possibly be found by deriving the result without the use of lookup tables. Even if not, it would be very interesting to see if someone knows an alternative derivation.