Simplify $\int_0^{\infty}\,dk\,\exp{(-\delta^2k^2)}\,J_1(kR)\frac{1}{1+bk^2}$

Question

EDIT:

I would love to find an analytical solution for this definite integral:

$$\int_0^{\infty}\,dk\,\exp{(-\delta^2k^2)}\,J_1(kR)\frac{1}{1+bk^2}$$

with $\delta>0,\, R>0,\,b>0$.

Does anyone can help me? I tried to have a look at Gradshteyn and Ryzhik, without any success.

Thanks