I would like to calculate the closed-form expression for the following integral:
$$ I = \int_{0}^{\infty} x^{M}\exp(-\frac{x}{a})K_{\nu}(b\sqrt{1+x})\mathrm{d}x,$$ where $M$, $a$, and $b$ are all positive constants. Note that $K_{\nu}(\cdot)$ is a modified Bessel's function of the second kind.
I have unsuccessfully tried substituting $\sqrt{1+x}$ by a variable $u$.