# integration – Definite Gaussian integral of error function with even powers in the argument

I am trying to solve the following definite integral,

$$int_{-infty}^{infty} dx , phi(x) x^2 Phi( a + bx)$$

Where,

$$phi (x) = frac{1}{sqrt{2 pi}} exp(-x^2/2)$$ and $$Phi(x)$$ is defined as,
$$Phi(x) = int_{-infty}^{x} phi(x) = frac{1}{2}left(1 + erfc left(frac{x}{sqrt{2}} right) right)$$

I have looked in tables, but I don’t seem to find an answer.

Could someone help?

Thanks