If m is a non-zero number and ∫x5m + 2x4m/(x2m + xm + 1)3 dx = f(x) + c, then f(x) is :

∫x5m + 2x4m/(x2m + xm + 1)3 dx
Rearrange the equation
∫2x4m(x2m + xm + 1) - x4m(x2m + xm + 1)/(x2m + xm + 1)3 dx
= ∫2x4m/(x2m + xm + 1)2 - x4m/(x2m + xm + 1)2 dx
Let u = x2m + xm + 1
du/dx = 2mx2m-1 + mxm-1
du = m(2x2m-1 + xm-1) dx
∫2x4m/(x2m + xm + 1)2 dx = x5m/2m(x2m + xm + 1)2 + C
∫x4m/(x2m + xm + 1)2 dx = x5m - x4m/2m(x2m + xm + 1)2 + C
Therefore,
f(x) = (x5m - x4m)/2m(x2m + xm + 1)2