Question

Find the integral of $\int \frac{2x^{12}+5x^9}{(x^5+x^3+1{)}^3}dx$.

Answer 1

$I=\int \frac{2x^{12}+5x^9}{(x^5+x^3+1{)}^3}dx=\int \frac{2x^{12}+5x^9}{x^{15}{(1+\frac{1}{x^2}+\frac{1}{x^5})}^3}dx=\int \frac{\frac{2}{x^3}+\frac{5}{x^6}}{{(1+\frac{1}{x^2}+\frac{1}{x^5})}^3}dx=\int \frac{-d(1+\frac{1}{x^2}+\frac{1}{x^5})}{(1+\frac{1}{x^2}+\frac{1}{x^5})}=\frac{1}{2}{(1+\frac{1}{x^2}+\frac{1}{x^5})}^2=\frac{x^{10}}{2}(x^5+x^3+1{)}^2+C$