$\begin{array}{l}\phantom{\phantom{;}x^{2}+1;}{\phantom{;}x^{3}\phantom{-;x^n}-5x\phantom{;}\phantom{-;x^n}}\\\phantom{;}x^{2}+1\overline{\smash{)}\phantom{;}x^{5}\phantom{-;x^n}-4x^{3}\phantom{-;x^n}+x\phantom{;}-1\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}+1;}\underline{-x^{5}\phantom{-;x^n}-x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{5}-x^{3};}-5x^{3}\phantom{-;x^n}+x\phantom{;}-1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+1-;x^n;}\underline{\phantom{;}5x^{3}\phantom{-;x^n}+5x\phantom{;}\phantom{-;x^n}}\\\phantom{;\phantom{;}5x^{3}+5x\phantom{;}-;x^n;}\phantom{;}6x\phantom{;}-1\phantom{;}\phantom{;}\\\end{array}$

$x^{3}-5x+\frac{6x-1}{x^2+1}$