Is there a closed form for the integral $$\int_{B_r(x_0)} |x|^\alpha \,dx$$ where $B_r(x_0)$ denotes the a ball of radius $r$ and centre $x_0 \in \mathbb{R}^n$ and $\alpha > -n$ and $x_0 \neq 0$?
After rescaling and orthogonal change of coordinates it reduces to $\displaystyle r^{n + \alpha}\int_{B_1(0)} |x + (\lambda,0')|^\alpha\,dx$ where, $\lambda = r^{-1}|x_0|$ and $0' \in \mathbb{R}^{n-1}$. Is there a way of evaluating this integral or perhaps finding the asymptotic behavior in terms of $\lambda$?