Can you prove that the integral below, with a vectorial field, is zero?

Question

If $\vec{J}(\vec{r})$ is a vector field limited in infinity. Prove that the integral below is zero: \begin{equation} \int{\left(\vec{J}\left(\vec{r}'\right)\cdot\nabla\right)\cdot\left(\nabla\frac{1}{\vert\vec{r} - \vec{r}'\vert}\right)}d^3r' \end{equation} where the integral is over all space. I have tried to use the condition $\nabla^2(\frac{1}{r}) = -4\pi \delta(\vec{r})$ of the Dirac delta. You can use integration by parts and then the Divergent Theorem. These are the tip's that my friends told me, but I still couldn't answer with a good proof.