I'm looking for a solution to the following integral :
$$ I(r)_{l_1, l_2, L} = \int_0^\infty j_{l_1}(\rho R_1)j_{l_2}(\rho R_2) j_L(\rho r) \frac{\rho^2}{\rho^2 + \kappa^2} d\rho $$
For the case $ r > R_1 + R_2$. Gradshteyn and Ryzhik (6.578) provide a solution for the case $\kappa = 0$ :
$$ \frac{\pi^{3/2}}{8} \frac{R_1^{l_1} R_2^{l_2} r^{-1 - l_1 - l_2}\Gamma\left(\frac{1 + l_1 + l_2 + L}{2}\right)}{\Gamma\left(l_1 + \frac{3}{2}\right) \Gamma\left(l_2+\frac{3}{2}\right)\Gamma\left(1+ \frac{L - l_1 - l_2}{2}\right)} F_4\left(\frac{l_1+l_2-L}{2}, \frac{1 + l_1 + l_2 + L}{2}; l_1 + \frac{3}{2}; l_2 + \frac{3}{2}; \frac{R_1^2}{r^2}, \frac{R_2^2}{r^2}\right) $$
Is there any known solution for the general case (arbitrary $\kappa$) in any table of integrals somewhere ?