I am trying to solve a couple integrals of the form:
\begin{equation} I(g)=\int_{0}^{\infty} x \, e^{-a(gx-b)^{2}}\,e^{-\beta_{1}x}\, {_{1}}F_{1}(-\alpha_{1};-\alpha_{3};\beta_{3} x) \ \mathrm{d}x \end{equation}
where, $a>0$, $g\in\mathbb{R}$, $b\in\mathbb{R}$, $0<\beta_{1}<\beta_{3}$, and $0<\alpha_{1}<\alpha_{3}$.
$\alpha_{1}$, and $\alpha_{3}$ are either integers or of the form $\frac{2n+1}{2}$ where $n=0,1,2,\dots$
The shifting of the quadratic in the first exponential term is really giving me issues. Any thoughts on how to approach this?
Update 08/16/16
I posted a solution to the question for $\alpha_{1}\in \mathbb{Z}^{+}$. If anyone can find a closed-form solution for the integral when $\alpha_{1}$ is not an integer, I will accept that answer instead.