TITLE: Evaluate the integral, $\int_0^{\infty} \frac{x^2}{(x+c)^{3/2}}\,e^{-x^2} dx$, where c>0
QUESTION [5 upvotes]: This integral came up in a research problem I'm working on, but I haven't had much luck calculating it. I suspect that the integral doesn't have a very clean form, but if anyone knows of an easy substitution or some elementary way to evaluate this integral, it would be very helpful. 
Thanks!

REPLY [2 votes]: Denote $b[x]:=\; \text{BesselI}\left(x,\frac{c^2}{2}\right)$ and $h:=\text{HypergeometricPFQ}\left[\left\{\frac{3}{2},2\right\},\left\{\frac{5}{4},\frac{7}{4}\right\},-c^2\right]$. 
Then (according to mathematica)
$$\int\limits_0^\infty\frac{x^2}{(x+c)^{3/2}}e^{-x^2}=\\
\frac{e^{-\frac{c^2}{2}} \left(3 \pi  \left(c^2 \left(-3+4 c^2\right) b[-1/4]+\left(1-7 c^2+4 c^4\right) b[1/4]-c^2 \left(-1+4 c^2\right) \left(b[3/4]+b[5/4]\right)\right)+64 c^2 e^{\frac{c^2}{2}} h\right)}{12 \sqrt{c}}
$$
I am absolutely sure this does not help you at all. I am very sorry to disappoint you, but I dont know how this can be done. I know improper Integrals are usually solved by integrating over semi-circles in complex space, however that is not easily done for this function.