Evaluating a difficult integral

Question

I need to evaluate this integral $$\int_{0}^{1}\mathrm dx\, x^{m}\exp\left[-k\frac{x}{(1-x^2)^{2/3}}\right],$$ where the real number $k>0$ and $m$ is an integer, $m\geq 0$. Is there any way to get dependence of this integral on $k$. Any rough approximation would be good. I already know rough dependence for $m=0$. I am interested in the case when $m\neq0$. Any hints would be appreciated.