Expectation of X over Y plus 1 for a joint density

The pair $(X,Y)$ has joint density $f(x,y)=x\,e^{-x(y+1)},\qquad x>0,\ y>0,$ and zero elsewhere. Compute $\mathbb{E}\!\left[\dfrac{X}{Y+1}\right]$.

$\tfrac23$. Integrate over $x$ first: $\int_0^\infty x^2e^{-x(y+1)}\,dx=\tfrac{2}{(y+1)^3}$ (a Gamma integral). That leaves $\int_0^\infty \tfrac{2}{(y+1)^4}\,dy=\tfrac23$.