Let $X, Y, Z \sim U(0,1)$. How would one solve for $W$?
\begin{equation} W = \frac{XY}{(X+Y+Z)^2} \end{equation}
We can rewrite it as follows:
\begin{equation} \frac{1}{W} = \frac{X^2 + Y^2 + Z^2 + 2XY + 2XZ + 2YZ}{XY} \end{equation}
\begin{equation} \frac{1}{W} = \frac{X}{Y} + \frac{Y}{X} + \frac{Z^2}{XY} + 2 + 2\frac{Z}{Y} + 2 \frac{Z}{X} \end{equation}
The ratio of two uniform RVs is
\begin{equation} f(x) = \frac{1}{2}I_{[0,1)}(x) + \frac{1}{2x^2}I_{[1,\infty)}(x) \end{equation}
so we will use this in terms 1, 5, and 6 while we use the inverse distribution to $f$ in term 2.
This is as far as I have gotten. However, it looks like I may be off track.