I need to compute the following integral
$ \int_{0}^{\infty}x\,\left\{\vphantom{\LARGE A}% 1- \left[\vphantom{\Large A}1- \exp(-a\,x^{\alpha}) \right]^M \right\}\,{\rm d}x \qquad \mbox{with}\quad \alpha > 0\quad\mbox{and}\quad a,M > 0. $
Is this a known integral ( if possible, without approximating the exponential function ) ?.