I'm having some trouble computing the following integral/Hankel Transform: $$\int_0^\infty \frac{ue^{-u}}{u^2 + a^2} J_0(2\sqrt{u}) du$$ where $a$ is some real number.
The actual integral I want to compute is $$\int_0^\infty \frac{ue^{-u}}{u^2 + (2\pi n)^2} I_0(2\sqrt{bui}) du$$ where $n$ is a positive integer, and $b$ is a real, positive number. I'm sure the computations are symmetric, so the first integral (probably) is sufficient.
An integral I have encountered in my search that could help are
$$\int_0^\infty \frac{x \sin(ax)}{x^2 + b^2} J_0(yx) dx = \frac{\pi}{2} e^{-ab} I_0(by)$$ for $y \leq a$
and $$\int_0^\infty \frac{x \cos(ax)}{x^2 + b^2} J_0(yx) dx = \cosh(ab) K_0(by)$$ for $y \geq a$
from the link: Double integral with Hankel transform
Since $\sin,\cos$ is just related to the exponential, this could help.
Thank you so much!