How to Find the Integral of Modified Bessel Function of 2nd Type?

Question

How can I find a closed form for this integral

\begin{align} I_1=\int\limits_{0}^{x_{out}}x^{M+2} K_{M-1}(x)dx, M>0~\text{and}~ x_{out}>0 \end{align}

Answer 1

$\int_0^{x_{out}}x^{M+2}K_{M-1}(x)~dx$

$=-\int_0^{x_{out}}x^2~d(x^MK_M(x))$ (according to http://people.math.sfu.ca/~cbm/aands/page_484.htm)

$=-[x^{M+2}K_M(x)]_0^{x_{out}}+\int_0^{x_{out}}x^MK_M(x)~d(x^2)$

$=-x_{out}^{M+2}K_M(x_{out})+2\int_0^{x_{out}}x^{M+1}K_M(x)~dx$

$=2^{M+1}\Gamma(M+1)-2x_{out}^{M+1}K_{M+1}(x_{out})-x_{out}^{M+2}K_M(x_{out})$ (according to http://people.math.sfu.ca/~cbm/aands/page_484.htm)