I'd like to calculate the following integral:
$\int^{\infty}_{0} \mathrm{erf}\left(\frac{\alpha}{\sqrt{1+x}} - \frac{\sqrt{1+x}}{\beta}\right) \exp\left(-\frac{x}{\gamma}\right)\, dx,$
where $\beta > 0$, $\gamma > 0$ and $\alpha \in \mathbb{R}$.
I've tried a few approaches, but with no success.
The form is similar to Equation 12 on page 177 of Erdelyi's Tables of Integral Transforms (Vol. 1):
$\int^{\infty}_{0} \mathrm{erf}\left(\frac{\alpha}{\sqrt{t}} - \frac{\sqrt{t}}{\beta}\right) \exp\left(-\frac{t}{\gamma}\right)\, dt$
but the change of variables requires a change in limits.
Any advice would be greatly appreciated!