The integral $\int \frac{2x^{12}+5x^9}{{(x^5+x^3+1)}^3}dx$ is equal to

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

$=\int \frac{2x^{12}+5x^9}{{\left(x^5\right)}^3{\left[1+{\displaystyle \frac{1}{x^2}}+{\displaystyle \frac{1}{x^5}}\right]}^3}dx$

$=\int \frac{2x^{-3}+5x^{-6}}{{(1+x^{-2}+x^{-5})}^3}dx$

$put 1+x^{-2}+x^{-5}=t$

$\left(-2x^{-3}-5x^{-6}\right)dx=dt$

$\left(2x^{-3}+5x^{-6}\right)dx=dt$

$=\int \frac{-dt}{t^3}$

$$\begin{gathered} =-\int t^{-3} dt \\ =-\frac{t^{-2}}{-2}+C \\ =\frac{1}{2t^2}+C \\ =\frac{1}{2{\left(1+x^{-2}+x^{-5}\right)}^2}+C=\frac{1}{2\left({\displaystyle \frac{x^5+x^3+1}{x^5}}\right)}+C \end{gathered}$$