I have the following integration:
$$ \vec I = \int d^3 \vec r F(\vec r) \vec{G}(\vec{r}) \left(({\vec{r}-\vec{r}_c})\cdot \vec{\nabla}(\delta(\vec{r}-\vec{r}_c))\right) $$ here $F(\vec{r})$ is a scalar function, and $\vec{G}(\vec{r})$ is a vector function. And we can assume that both functions $F(\vec{r})$ and $\vec{G}(\vec{r})$ goes to zero as $\vec r\to \pm\infty$.
I know that for 1D integration with all scalar function, if we have $$ I = \int dx F(x) G(x) (x-x_c) \nabla_x(\delta(x-x_c)) $$ then we can use distributive derivative and write $$ I = -\int dx \nabla_x\left[F(x) G(x) (x-x_c)\right] \delta(x-x_c)\\ = -\nabla_x\left[F(x) G(x) (x-x_c)\right] |_{x=x_c} $$ I am not sure how to generalize this rule if the functions inside integral are 3D and vector quantities.