I have two continuous random variables X and Y. X follows an Exponential distribution with rate $\displaystyle $\lambda$$, and Y follows a Normal distribution with $\displaystyle $\mu$$ and $\displaystyle $\sigma^2$$. The domain of X is $\displaystyle $[0,\infty)$$, and that of Y is $\displaystyle $(-\infty,\infty)$$.

I have come up with the following expression for the pdf of V=XY.

Using the result of Rohatgi (1976, p. 141)

$\displaystyle f_V(v)=\frac{\lambda}{\sigma_\alpha\sqrt{2\pi}} \int_0^\infty e^{-\left(\lambda x+\frac{(\frac{v}{x}-\mu_\alpha)^2}{2\sigma_\alpha^2}\right)}\frac{1}{x }\,dx$

Do you have any suggestions on how to solve this (analytically)?