Let $(X, Y)$ and $(Z, W)$ be independent random vectors with the same density: $$f_{X,Y}(x,y) = e^{-x}\frac{1}{\sqrt{2\pi x}}e^{-\frac{(y-x)^2}{2x}}$$
for $x > 0$ and $y \in \mathbb{R}$. Find the density of random vector $(X + Z, Y + W)$.
I've tried using the transformation formula for the map $h(x, y, z, w) = (x + z, y + w, z, w)$ but I got a hard integral. Then I tried calculating conditional probabilities with the condition $Z = z, W = w$ and then using the formula for total probability but I got the same integral. Is there any way to solve this problem more efficiently?