$\begin{array}{l}\phantom{\phantom{;}3x^{2}+1;}{\frac{1}{3}x^{7}\phantom{-;x^n}+\frac{-\frac{1}{3}}{3}x^{5}\phantom{-;x^n}+\frac{\frac{1}{9}}{3}x^{3}\phantom{-;x^n}+\frac{-\frac{1}{27}}{3}x\phantom{;}\phantom{-;x^n}}\\\phantom{;}3x^{2}+1\overline{\smash{)}\phantom{;}x^{9}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{\phantom{;}3x^{2}+1;}\underline{-x^{9}\phantom{-;x^n}-\frac{1}{3}x^{7}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{9}-\frac{1}{3}x^{7};}-\frac{1}{3}x^{7}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\\\phantom{\phantom{;}3x^{2}+1-;x^n;}\underline{\phantom{;}0.3333333x^{7}\phantom{-;x^n}+\frac{1}{9}x^{5}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;\phantom{;}0.3333333x^{7}+\frac{1}{9}x^{5}-;x^n;}\frac{1}{9}x^{5}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\\\phantom{\phantom{;}3x^{2}+1-;x^n-;x^n;}\underline{-0.1111111x^{5}\phantom{-;x^n}-\frac{1}{27}x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;-0.1111111x^{5}-\frac{1}{27}x^{3}-;x^n-;x^n;}-\frac{1}{27}x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\\\phantom{\phantom{;}3x^{2}+1-;x^n-;x^n-;x^n;}\underline{\phantom{;}0.037037x^{3}\phantom{-;x^n}+\frac{1}{81}x\phantom{;}\phantom{-;x^n}}\\\phantom{;;;\phantom{;}0.037037x^{3}+\frac{1}{81}x\phantom{;}-;x^n-;x^n-;x^n;}\frac{1}{81}x\phantom{;}\phantom{-;x^n}\\\end{array}$

$\frac{1}{3}x^{7}-\frac{1}{9}x^{5}+\frac{1}{27}x^{3}-\frac{1}{81}x+\frac{\frac{1}{81}x}{3x^2+1}$