**TLDR:** trying to solve,
$$\int_1^\infty \exp\left(-\frac{x^2}{2\omega^2}\right) \frac{1}{\sqrt{ax^2+bx-1}}dx$$

After doing some reading and looking at some other questions 1, 2 (and even going through a few integral tables) I realized there's probably no closed-form solution here, and that expressing the integral as a series of special functions is probably the best approach here,

$$ \sum_{k=0}^\infty \frac{(2\omega^2)^{-k}}{k!} \int_{1}^{\infty}\frac{x^{2k}}{\sqrt{(c-x)(x-d)}}dx$$

However, I wasn't able to find a solution even in this reduced form. Any ideas?

**More details:**

I encountered this integral during my research, trying to find the product of two functions $y_1, y_2$ of two independent random variables $p,q\sim N(\mu,\sigma^2)$, where

$$y_1(p,q) = \frac{1}{\sqrt{1+p^2+q^2}}$$ and $$y_2(p,q) = ap+bq+c$$