Rewrite $\int\limits_{y=0}^p{2{{({\frac{a}{y}})}^{\frac{M}{2}}}{K_M}({2\sqrt{\frac{b}{y}}})\exp({-\frac{y}{c}})dy}$ as non-elementary function? How to rewrite this integral as some non-elementary function or any possible close-form $?$:
$$
I \equiv
\int_{0}^{p}2\left(a \over y\right)^{M/2}\
\operatorname{K}_{M}\left(2\,\sqrt{\,{b \over y}}\,\right)
\exp\left(-\,{y \over c}\right)\,\mathrm{d}y
$$
where $a,b,c,p$ are all positive integer and
$\operatorname{K}_{M}\left( {} \right)$ is the modified Bessel function of the second kind with $M$ is a positive integer that can be $1,2\ \mbox{or}\ 3$.
Currently, I am not considering the general case for all positive integer $M$ but any contribution is great.

The closest identity I could find that might be useful is:
$$\int_0^\infty t^{\mu-1}~e^{-at}~K_{\nu}(t)~\mathrm{d}t=\begin{cases}
\sqrt{\frac{\pi }{2}} \Gamma ( \mu -\nu ) \Gamma ( \mu +\nu )\left( 1-a^{2}\right)^{-\mu /2+1/4} P^{-\mu +1/2}_{-\nu -1/2}( a) & -1< a< 1\\
\sqrt{\frac{\pi }{2}} \Gamma ( \mu -\nu ) \Gamma ( \mu +\nu )\left( 1-a^{2}\right)^{-\mu /2+1/4} P^{-\mu +1/2}_{\nu -1/2}( a) & \operatorname{Re}( a) \geq 0\ ;\ a\neq 1
\end{cases}$$
With $P^\alpha_\beta$ the associated Legendre function.
REFERENCE: https://dlmf.nist.gov/10.43
