${\displaystyle \int \frac{x^{5m-1}+2x^{4m-1}}{{\left(x^{2m}+x^m+1\right)}^3}dx=f\left(x\right)}+c$ then$f\left(x\right)=$

${\displaystyle \int \frac{x^{5m-1}+2x^{4m-1}}{x^{6m}{\left(1+x^{-m}+x^{-2m}\right)}^3}={\displaystyle \int \frac{x^{-m-1}+2x^{-2m-1}}{{\left(1+x^{-m}+x^{-2m}\right)}^3}dx}}$

$\begin{array}{c}putt=1+x^{-m}+x^{-2m}\\ \text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\frac{dt}{dx}=-m{x}^{-m-1}-2m{x}^{-2m-1}\\ -\frac{dt}{m}=\left(x^{-m-1}+2x^{-2m-1}\right)dx\end{array}$

$\therefore {\displaystyle \int \frac{x^{5m-1}+2x^{4m-1}}{{\left(x^{2m}+x^m+1\right)}^3}=-\frac{1}{m}{\displaystyle \int t^{-3}dt=\frac{1}{2m{t}^2}+c}}$

$=\frac{1}{2m{\left(1+x^{-m}+x^{-2m}\right)}^2}+c$

$=\frac{x^{4m}}{2m{\left(x^{2m}+x^m+1\right)}^2}+c$