# expectation involving normal pdf and Rayleigh distribution

I need to calculate following definite integral

\begin{equation*} \frac{1}{2\pi }\int_0^\infty \frac{x^2 e^{-x^2/\sigma^2 } }{\sigma} \frac{e^{-\frac{\lambda}{{ax^2+b}}}}{\sqrt{ax^2+b}} ~~dx. \end{equation*}

It is actually finding the expected value of xϕ(λ/√(ax^2+b)) , where ϕ(.) is pdf of a standard normal distribution and x is a random variable with Rayleigh distribution with parameter σ.

• any solution will help, even in terms of numerical functions or even approximations – Alireza May 6 '15 at 20:52