If m is a non-zero number and $$\int \frac{x^{5m-1}+2x^{4m-1}}{(x^{2m}+x^m+1)^3} dx = f(x) + c$$, then $$f(x)$$ is equal to:

$$I = \int \frac{x^{5m-1} + 2x^{4m-1}}{(x^{2m} + x^m + 1)^3} \, dx$$

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

$$(x^{2m} + x^m + 1)^3 = \left[ x^{2m} (1 + x^{-m} + x^{-2m}) \right]^3 = x^{6m} (1 + x^{-m} + x^{-2m})^3$$

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

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

$$I = \int \frac{x^{-m-1} + 2x^{-2m-1}}{(1 + x^{-m} + x^{-2m})^3} \, dx$$

$$t = 1 + x^{-m} + x^{-2m}$$

$$dt = \left( 0 - m \cdot x^{-m-1} - 2m \cdot x^{-2m-1} \right) dx$$

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

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

$$I = \int \frac{-\frac{dt}{m}}{t^3} = -\frac{1}{m} \int t^{-3} \, dt$$

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

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

$$1 + x^{-m} + x^{-2m} = 1 + \frac{1}{x^m} + \frac{1}{x^{2m}} = \frac{x^{2m} + x^m + 1}{x^{2m}}$$

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

$$I = \frac{1}{2m \cdot \frac{(x^{2m} + x^m + 1)^2}{(x^{2m})^2}} + c$$

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

$$f(x) = \frac{1}{2m} \cdot \frac{x^{4m}}{(x^{2m} + x^m + 1)^2}$$