What is the antiderivative of $f(x)=(a(K-x)^{1/n}+b)\times x^d$

Question

What is the antiderivative of the following function of x ? $$\forall x>0, f(x)=(a(K-x)^{1/n}+b)\times x^d $$

When $d=1$, there is an easy solution but in the case where $d$ is a positive real, i don't know.

Answer 1

Thanks to Blitzer in the comments, here is the antiderivative of the function. first, b has no importance as $b x^d$ is trivial to integral. The question comes down to integrate $(K-x)^{1/n}\times x^d$. The result is in terms of the Beta function like: $$k^{1/n+d+1}\times B_{x/k}(d+1,1/n+1)$$ with $B$ being the incomplete Beta function such that: $$B_{x}(a,b)=\int_0^x t^{a-1}(1-t)^{b-1}dt$$