He divided, or simplified, the original LHS, but he did not do that in the original RHS?
Then his final "equation or identity", even if the last term in the new RHS is corrected, is wrong.
Because if he multiplied the original RHS by (x^2)(x^3 +1), then he should have multiplied the original LHS by the same amount.
Clearly, (-x^2 -1) is not [(x^5 -1) / (x^2)(x^3 +1)] * [(x^2)(x^3 +1)].
You lost me there.
So the final decomposition might be:
x/x + [the decomposition of (-x^2 -1) / (x^2)(x^3 +1)]?
or, (x^2 / x^2) + [the decomposition of (-x^2 -1) / (x^2)(x^3 +1)]?
Etc....
And so the the final integral would be
= x +{INT[the decomposition of (-x^2 -1) / (x^2)(x^3 +1)]dx} +C?
Umm...