If X and Y are independent exponential random variables with parameters λ1 and λ2,How do I express the density function of Z = XY?

Question

My solution is:

1.calculating the cdf(Cumulative Distribution Function) of Z using the F(Z<z)=F(XY<z), written as F(z).

2.the derivation of F(Z) is just f(z), namely the pdf of Z.

But I failed to get F(z), because the integrand is unintegrable.

And unfortunately, I cannot find other solutions.