Is there an analytical solution for $\operatorname{E}[x^m\exp(kx)]$ when $k,m$ are positive integers and $x$ has skew normal distribution with parameters $\xi,\omega,\alpha$?
Wiki says that $\operatorname{E}[x]=\xi + \omega\delta\sqrt{\frac{2}{\pi}}$ where $\delta = \frac{\alpha}{\sqrt{1+\alpha^2}}$ and $\operatorname{E}[\exp(kx)]=2\exp\left(k\xi +\frac{k^2\omega^2}{2}\right)\Phi\left(k\omega\delta\right)$.
But how to find $\operatorname{E}[x^m\exp(kx)]$??? It involves rather complex integral over PDF. Wolfram Mathematica can't take the integral.