When is $ (1+ |x+y|^2)^{\alpha}(x_i + y_i)(x_j + y_j) \leq C \frac{1}{(|x|+|y|)^2}$ true?

Question

For $x=(x_1,x_2,x_3,...,x_n),y=(y_1,y_2,y_3,...y_n)$ are two vectors in $\mathbb R^n$, is the following inequality always true for any $\alpha >0$? $$ (1+ |x+y|^2)^{\alpha}(x_i + y_i)(x_j + y_j) \leq C \frac{1}{(|x|+|y|)^2}$$ for some constant $C$ and for any $i,j$.