I'd like to show the following equality (at least Mathematica claims it is an equality): \begin{multline*} \int_0^\infty x^p \exp(-(ax - b)^2)\, dx = \frac{1}{2} e^{-b^2} a^{-p-1} \left(\Gamma \left(\frac{p+1}{2}\right) \, _1F_1\left(\frac{p+1}{2};\frac{1}{2};b^2\right)+\ b p \Gamma \left(\frac{p}{2}\right) \, _1F_1\left(\frac{p}{2}+1;\frac{3}{2};b^2\right)\right) \end{multline*} Here $a,b>0$ and $p > 1$ (if it matters).
This looks a lot like the expression given on Wikipedia for the uncentered moments of a Gaussian, except the integral is over $[0,\infty)$ rather than all of $\mathbb{R}.$
Any suggestions on how to proceed? I haven't been able to find this in any table of integrals.