Working on asymptotic expansion of certain integrals, it appear the following family of integrals
$$F(m,p,n,\alpha,\sigma)=\int_{\alpha}^{\infty}e^{-A\,x^m-B\,x^p}x^n(x-\alpha)^{\sigma}\,dx$$
with $m,n,p\in\mathbb{N}$ , $m\neq p$ , $\alpha\in(0,\infty)$ and $\sigma\in[0,1)$. Also $A,B>0$.
I wonder if is possible write $F(m,p,n,\alpha)$ in a closed form in terms of (probably) special functions.
Any help is welcomed.