Change of Variables & the Jacobian · Superchallenge problem

$X$ and $Y$ are independent random variables, each with density $e^{-t}$ for $t > 0$ (so their joint density is $e^{-(x+y)}$ on the first quadrant). Find $E\!\left[\dfrac{(XY)^{3/2}}{(X + Y)^{2}}\right]$. Give an exact answer.