$\int \frac{2x^{12}+5x^9}{(1+x^3+x^5)}dx=$

$$\begin{gathered} \int \frac{2{\mathrm{x}}^{12}+5{\mathrm{x}}^9}{{(1+{\mathrm{x}}^3+{\mathrm{x}}^5)}^3}\mathrm{dx} \\ =\int \frac{2{\mathrm{x}}^{12}+5{\mathrm{x}}^9}{{\mathrm{x}}^{15}{({\mathrm{x}}^{-5}+{\mathrm{x}}^{-2}+1)}^3} \\ =\int \frac{{\displaystyle \frac{2}{{\mathrm{x}}^3}}+{\displaystyle \frac{5}{{\mathrm{x}}^6}}}{{\left({\mathrm{x}}^{-5}+{\mathrm{x}}^{-2}+1\right)}^3}\mathrm{dx} \\ \mathrm{Put} {\mathrm{x}}^{-5}+{\mathrm{x}}^{-2}+1=\mathrm{t} \\ (-5{\mathrm{x}}^{-6}-2{\mathrm{x}}^{-3})\mathrm{dx}=\mathrm{dt} \\ =\int \frac{-\mathrm{dt}}{{\mathrm{t}}^3} \\ =\frac{1}{2{\mathrm{t}}^2}=\frac{1}{2{({\mathrm{x}}^{-5}+{\mathrm{x}}^{-2}+1)}^2} \\ =\frac{{\mathrm{x}}^{10}}{2{(1+{\mathrm{x}}^3+{\mathrm{x}}^5)}^2}+\mathrm{C} \end{gathered}$$