I have two random variables, X and Y. The variable X has exponential distribution: $$ X \sim Exp(\lambda).$$ The variable Y is normally distributed conditional on $X$ as follows: $$Y | X \sim N(\mu + k \sigma^2 X, \sigma^2)$$where $k$ is a constant. I am looking for the distribution of their product, $Z = XY$.
I have tried the following: $$f_z(z) = \int_0^\infty f_{Y|X}\left(\frac{z}{x} | x\right)f_X(x)\frac{1}{x} \, dx$$ where $f_{Y|X}$ is the pdf of the normally distributed $Y|X$ and $f_X$ is the pdf of exponentially distributed $X$, but struggled to solve the integral.
Is there a tractable form for this distribution?
Alternatively, is there some other distribution(s) for $X$ which would give a tractable distribution for $Z$? Lognormal? Pareto?
Thanks!