$\frac{a^{2}-1}{b+1}+\frac{b^{2}-1}{a+1}$ an integer $\Rightarrow \frac{a^{2}-1}{b+1}$ and $\frac{b^{2}-1}{a+1}$ are integers.
$\frac{a^{3}+1}{b+1}+\frac{b^{3}+1}{a+1}$ an integer $\Rightarrow \frac{a^{3}+1}{b+1}$ and $\frac{b^{3}+1}{a+1}$ are integers.