Example 25 (2007 Bulgarian National Team Selection Test) Find all positive integers $x, y$ such that $\left(x^{2}+\right.$ $y)\left(y^{2}+x\right)$ is a fifth power of a prime.
Solution: Let $\left(x^{2}+y\right)\left(y^{2}+x\right)=p^{5}$, where $p$ is a prime number, then $x^{2}+y=p^{s}, y^{2}+x=p^{t}$, where $\{s, t\}=\{1,4\}$ or $\{2,3\}$.
In the first case, without loss of generality, assume $x^{2}+y=p^{4}$, which leads to a contradiction.
Therefore, let $x^{2}+y=p^{2}$ and $y^{2}+x=p^{3}$. Since $y^{2}+x=p^{3}>p^{2}(p-1)^{2}$, this also leads to a contradiction.
In conclusion, the solutions are $(2,5)$ and $(5,2)$.