Let $f$ be a $C^1$ function with $f'>0$, and let $a\not= 0$ be a real number. Is there a closed form for the integral $$ \int f(x) f'(x)^a \mathrm dx? $$
Certainly if $a=1$, then the integral is simply $f^2/2 + c$, but I do not see a way of doing it for arbitrary $a$
No, there is no closed form for that. Indeed, try this one $f(x) = x\log x$, $a=1/2$: $$ \int x \log x \sqrt{1+\log x}\;dx $$
New example with $f$ defined on all of $\mathbb R$ and $f'>0$. Take $f(x) = x+e^x, a=1/2$. $$ \int(x+e^x)\sqrt{1+e^x}\;dx $$