Let $f(x)$ function define by
$$ f(x)=x^m e^{-bx}K_{n+1}(ax)^{'} $$ where $K_v(⋅)$ is the $v$-th order modified Bessel function of the second kind and $L^{'}(x)$ is the derivation of function $L(x)$.
I would like to compute the following integral integral
$$ I=\int_{0}^{\infty}f(x)dx=\int_{0}^{\infty}x^m e^{-bx} K_{n+1}(ax)^{'}dx. $$
So my question is what is the derivation of $K_{n+1}(ax)$.
I found the following formula to evaluate the integral $$ \int_{0}^{\infty}x^{\mu-1}e^{-\alpha x}K_{v}(bx)dx= $$ $$\frac{\sqrt\pi (2b)^v}{(\alpha+b)^{\mu+v}} \frac{\Gamma(\mu+v)\Gamma(\mu-v)}{\Gamma(\mu+\frac{1}{2})} F\left( \mu+v,v+\frac{1}{2},\mu+\frac{1}{2};\frac{\alpha-b}{\alpha+b} \right) $$ where $F(a,b,c;z)$ is is the generalized hypergeometric function.
So what is the final result of Integral $I$.