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

Let $x=\frac{1}{y}$, we get
$$f(y)+2 y^{2} f\left(\frac{1}{y}\right)=\frac{3 y^{3}+4 y^{2}-y+3}{y+1} .$$

Substitute $y$ with $x$ to get
$$f(x)+2 x^{2} f\left(\frac{1}{x}\right)=\frac{3 x^{3}+4 x^{2}-x+3}{x+1} \text {. }$$

Also, $2 f(x)+x^{2} f\left(\frac{1}{x}\right)=\frac{3 x^{3}-x^{2}+4 x+3}{x+1}$.
By eliminating $f\left(\frac{1}{x}\right)$ from the above two equations, we obtain $f(x)$.