I am faced with an integral which I have no clue how to approach. It is essentially this:
$$ \int_0^\infty dq J_0(qr_{12}) F_{ij}^{kl}(q) $$ where $$ F_{ij}^{kl}(q) = e^{-qd} \frac{P_k^{(10)}(q)}{R_k^{(12)}(q)} + \frac{Q_k^{(11)}(q)}{R_k^{(12)}(q)} $$ where the P,Q,R all denote polynomials of the respective power, and $J_0$ is the Bessel Function of the first kind. I have tried everything I can think of, I don't even know how to approximate this, since I cannot Taylor expand the $J_0$ for small argument. Any ideas?