Let $B(0,a)$ denotes the ball of center $0$ and radius $a$ in $\mathbb{R}^3$.
It is well-known, using divergence theorem and spherical coordinates,how to compute the integrals of the form
$$I(\alpha,\beta,\gamma,k) := \int_{\partial B(0,a)} \frac{x_1^\alpha x_2^\beta x_3^\gamma}{|x|^k} \, \mathrm{d}x \text{ for } \alpha, \beta, \gamma, k \in \mathbb{N}.$$
For exemple, we have
$$I(2,2,2,0)=\frac{4 \pi a^8}{105}, \ I(4,0,0,0)= \frac{4 \pi a^6}{5},$$ and we can show that if any of the coefficient $\alpha$, $\beta$ or $\gamma$ is odd, then $I(\alpha,\beta,\gamma,k)=0$.
I would like to know if there exists a method on how to calculate the integrals of the form
$$J(\alpha,\beta,\gamma,k,b):= \int_{\partial B(b,a)} \frac{x_1^\alpha x_2^\beta x_3^\gamma}{|x|^k} \, \mathrm{d}x $$ where $b$ is a point outside of $B(0,a)$. It is more complicated since from the start we have $|x| \neq a$ but $|x-b|=a$. I tried passing to spherical coordinates but the denominator is too complicated to handle.
Any ideas or advices on how to compute these integrals are welcomed. I am more particularly interested in the cases :
$\alpha=1$, $\beta=1$, $\gamma = 0$ and $k=3$
$\alpha=2$, $\beta=1$, $\gamma = 0$ and $k=3$
$\alpha=1$, $\beta=1$, $\gamma = 1$ and $k=3$