Ok, so I am confused about the following;

When we have a polynomial, say $P(x)$, and we want to solve an integral where $P(x)$ is raised to a certain power, for example; $$\int (P(x))^adx$$Why can we not integrate it using the fact that integration is a reverse operation of differentiation? If we wanted to differentiate $(P(x))^a$, then it would result in the following; $$aP'(x)(P(x))^{(a-1)}$$ So my question is why can we not integrate the above function like so; $$\frac{(P(x))^{(a+1)}}{(a+1)(P'(x))}?$$