I'm trying to calculate the following integral: $\int_0^\infty \mathrm{BesselJ}[l_0,k_0r] \cdot \mathrm{BesselJ}[l_1,k_1r] \cdot \mathrm{BesselJ}[l_0-l_1,kr] \cdot r\,dr$

($\mathrm{BesselJ}[n,x]$ is the Bessel function of the first kind of order $n$) I assume that $l_0,l_1$ are integers, and that $k_0,k_1,k>0$.

The result has nice implications in quantum mechanics - to explain selection rules.

I was able to prove that the integral is zero when $k,k_0,k_1$ cannot be the lengths of a triangle. So there is a non-zero result when $|k_0-k_1| < k < k_0+k_1$ However, I don't know how to calculate the integral when it is not zero. Can anyone help?

I tried using Mathematica to get a numerical answer (which is fine for me), but I do not think I can count on it. Mathematica is giving a non-zero result for the regime where the integral should be zero.