$\int \frac{5x^8+7x^6}{{(x^2+1+2x^7)}^2}dx=$

$$\begin{gathered} I=\int \frac{5x^8+7x^6}{{(x^2+1+2x^7)}^2}dx=\int \frac{5x^8+7x^6}{{x^{14}({\displaystyle \frac{1}{x^5}}+{\displaystyle \frac{1}{x^7}}+2)}^2}dx \\ =\int \frac{5x^{-6}+7x^{-8}}{{\left(\frac{1}{x^7}+\frac{1}{x^5}+2\right)}^2}dx \\ put \frac{1}{x^7}+\frac{1}{x^5}+2=t\Rightarrow \left(-7x^{-8}-5x^{-6}\right)dx=dt \\ I=\int \frac{-dt}{t^2}=\frac{1}{t}+c \\ =\frac{1}{2+\frac{1}{x^5}+\frac{1}{x^7}}+c \end{gathered}$$

=$\frac{x^7}{2x^7+x^2+1}$+c