Let $\Omega=(0,1)^3$ be a cube in $\mathbb{R}^3$ and $0<\beta<3/2$.
Do there exist two constants $C>0$ and $d\geq 1$ such that the following inequality holds $$ \left(\int_\Omega \left\vert D^\alpha \frac{1}{|x|}\right\vert^2|x|^{2(\beta+k)}dx\right)^{1/2}\leq C(d)^kk!,\quad\forall |\alpha|=k\in \mathbb{N} $$ where $\alpha=(\alpha_1,\alpha_2,\alpha_3)$ and $$ D^\alpha f=\frac{\partial^{|\alpha|}f}{\partial x^{\alpha_1}_1\partial x_2^{\alpha_2}\partial x_3^{\alpha_3}} $$
I try to calculate the integration directly and find the explicit expression for the integrations under any $k$. But I can not find the explicit formula after calculating the integrations for $k= 1,2,3$.