Integral involving bessel function/gaussian/rational function

Question

I'd like to solve:

$$\int_0^{\infty}\quad J_1(ak)\,\frac{b+k^2}{(k-\alpha_1)(k-\alpha_2)(k-\alpha_3)(k-\alpha_4)}\,\exp(-ck^2)\,\,dk$$

Is there any specific rule for it? Thanks!

Answer 1

Perhaps this is the most simple approach:

$\int_0^\infty\dfrac{(b+k^2)\exp(-ck^2)J_1(ak)}{(k-\alpha_1)(k-\alpha_2)(k-\alpha_3)(k-\alpha_4)}dk$

$=\int_0^\infty\sum\limits_{n=0}^\infty\dfrac{(-1)^na^{2n+1}k^{2n+1}(b+k^2)\exp(-ck^2)}{(k-\alpha_1)(k-\alpha_2)(k-\alpha_3)(k-\alpha_4)n!(n+1)!}dk$

Note that $\int_0^\infty\dfrac{e^{-wk^2}}{\alpha k+\beta}dk=\dfrac{1}{2\pi\beta\sqrt{w}}G^{3,2}_{2,3}\left({\dfrac{\beta^2w}{\alpha^2}}\Bigg\vert\,^{\frac{1}{2},1}_{1,\frac{1}{2},\frac{1}{2}}\right)$ , where $G^{3, 2}_{2, 3}$ is the Meijer G-function.

The above integral can be expressed in terms of the infinite series of this result.