Basically, this question can be considered to be an extenstion to my other question.
What I wanted to do was this integral as homework (it is indefinite BTW so no approximations using Simpson's Rule or Boole's Rule)
$$\int(x^{3m}+x^{2m}+x^{m})(2x^{2m}+3x^{m}+6)^{\frac1{m}}dx$$
So using Mathematica's Integrate
function the answer was
Apparently, after rigorous substitutions and transformations the answer was found to be correct.
What I wanted to know was how Mathematica integrates these functions that require a human tons of intuition to compute, within seconds, and often in the most simple way and also presents them in the most humanly computable form.
(Even differentiation for that matter)