I have been looking as crazy for a closed-form expression of integrals of the following nature
$$\int_0^\infty\text{d}x\,e^{-(a+x)^2}\frac{x^m}{(x^2+b)^n}$$
where $a,b>0$ and $n$ and $m$ are positive integers. Its form looks simple enough to have a closed-form expression, but I have failed in every thing I tried. Does there exist a closed form for arbitrary $n,m$ or, at least, for $n=8$ and $m=3,5,7$?
Thank you in advance.