I want to perform the following integration wrt $x$:
$$\int_{-\infty}^{\infty}\frac{1}{\sqrt(2\pi\sigma^2)}e^(\frac{-(y-hx)^2}{2\sigma^2})[(1+\frac{x^2}{b})^{-(\frac{b+1}{2})}]dx$$

Here first part is a gaussian random variable (treating $y,h,\sigma$ as constants and the second part is student-t distribution with parameter $b$.
Does this have any closed form or represents any other kind of distribution. I am unable to figure out.