I've been trying to find out the closed form result of that integration in order to get a closed form CDF. Can anyone help me on that? Some asymptotic simplification can also be considered.
\begin{equation} F_{W}(\rho)=\int_{w=0}^{w=\rho}\int_{x=0}^{x=1} x^{(\frac{M+1}{2})}(1-x)^{(\frac{M-4}{2})}w^{(\frac{M-2}{2})} \{ {(x+wc)}^{-M}+ {(w+cx)}^{-M}\}dxdw \end{equation}