Probability distribution: exponential of a quartic

[vvaibhav08]

TL;DR

I've been trying to work out the normalization of this probability distribution but have found very few resources that have dealt with this. (one being: https://www.tandfonline.com/doi/abs/10.1080/00401706.1978.10489702). Can anyone help me in solving this? Additionally, I also wanted to know the astronomical relevance of this distribution (Where in astrophysics/cosmology does this distribution come up?)

Any help is much appreciated.

$$\int_{-\infty}^{+\infty} exp (-[ax + bx^2]^2) dx$$
$$a\&b\in R$$

 

[Gaussian97]

Well, doing some change of variables (a shift to complete square and then define the exponent as $x$) I've reduced the integral to $$\int_{-\frac{a^2}{4b}}^{\infty}\frac{e^{-x^2}}{\sqrt{bx+\frac{a^2}{4}}}\text{d}x$$ Which seems to be solvable using the Bessel functions, unfortunately, the solution that WolfralAlpha give is a complex number, so I don't know how to solve it, maybe from here you can work something more.