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

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