upper bound for $\int \frac{1}{|x-y|}$

Let $\alpha>0$, determine some upper bound for this integral for $x\in B_{1}$, $$\int_{B_{1}\cap B_{(1+1/\alpha)|x|}}\frac{1}{|x-y|}\,dy.$$

My approach: If $x\in B_{1}$ then $|x|\leq 1$ and $(1+1/\alpha)|x|\leq (1+1/\alpha)$, so $$\int_{B_{1}\cap B_{(1+1/\alpha)|x|}}\frac{1}{|x-y|}\,dy \leq \int_{ B_{(1+1/\alpha)|x|}}\frac{1}{|x-y|}\,dy \leq \int_{ B_{(1+1/\alpha)|x|}}\frac{1}{|x|-|y|}\,dy$$

but now, $B_{(1+1/\alpha)|x|}:=\{y\in \mathbb{R}^n: |y|\leq(1+1/\alpha)|x|\} = \{y\in \mathbb{R}^n: |y| - |x|\leq \frac{|x|}{\alpha}\}$.Any hint will be appreciated