The integral $\int \frac{2 x^{12}+5 x^9}{\left(x^5+x^3+1\right)^3} d x$ is equal to (where C is a constant of integration)

$\frac{x^{10}}{2\left(x^5+x^3+1\right)^2}+C$

$I=\int \frac{2 x^{12}+5 x^9}{\left(x^5+x^3+1\right)^3} d x=\int \frac{\frac{2}{x^3}+\frac{5}{x^6}}{\left(1+\frac{1}{x^2}+\frac{1}{x^5}\right)^3} d x$               [Dividing numerator and denominator by $x^{15}$]

let $y=1+\frac{1}{x^2}+\frac{1}{x^5}$

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

$⇒I=-\int\frac{dy}{y^3}=\frac{1}{2y^2}+C$

$=\frac{1}{2(1+\frac{1}{x^2}+\frac{1}{x^5})^2}+C=\frac{x^{10}}{2\left(x^5+x^3+1\right)^2}+C$