I tried many things for this problem, like Simon's Favorite Factoring Trick which gave me $(x+1)(y+1)=6$ and $(x^3+1)(y^3+1)=18$ and substituting $u=x+y$ and $v=xy$ to get

$\begin{cases}  u(u^2-3v)+v^3=17 \\ u+v=5 \end{cases}$

Both didn't help that much however. Is there something that I'm missing?