Help solving this integral $\int_{-\infty}^{\infty} \frac{x^2 e^{-x^2/2}}{a+bx^2}dx$ So, I've got an integral in the following form:
$$\int_{-\infty}^{\infty} \frac{x^2 e^{-x^2/2}}{a+bx^2}dx$$
where $b<0$ and $a\in\mathbb{R}$.
I've tried substituting $y=x^2$ (after changing changing lower limit to 0 and multiplying by 2 of course) and $z=y+a$ but there is that pesky square root in the denominator...
Anyone with better ideas?  Is this thing even soluble?

For positive $a$ and negative $b$ the integral is divergent. However, it is possible to consider the principal value, for which mathematica gives 
$$
\frac{\sqrt{\pi } \left(2 \sqrt{-a b}
   F\left(\sqrt{-\frac{a}{2b}}\right)+\sqrt{2} b\right)}{b^2}
$$
where $F\;$ is the  Dawson integral.
