$\frac{a^2b+a}{a^2+b}-b=\frac{a-b^2}{a^2+b}$ and $\frac{ab^2+b}{b^2-a}-a=\frac{a^2+b}{b^2-a}$ Thus, either $a-b^2=a^2+b\implies b^2+b=a-a^2$, which is impossible since $a$, $b$ are positive integers, or $b^2-a=a^2+b \implies b^2-a^2=a+b \implies b-a=1$ Thus, all $(a,a+1)$ work.