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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 ...

In this paper (Maroun's PhD dissertation, 2013) at page 46 the following formula is given (apparently without a reference): $$\int_0^{\infty } e^{i a x^s+i b x^p} \, dx=\sum _{n=0}^{\infty } \frac{\...