How to simplify a geometric calculus equation?

Question

In 3-dimensional vector space, let $\vec{r}=[x,y,z]^T, \vec{v} \in R^3$ and $\nabla$ is gradient: $$\iiint_Q\nabla\times[(\vec{v}\times\vec{r})z]dQ=?$$ I tried to use triple product and Levi-Civita, but without success. The integrand becomes $$\nabla\times[(\vec{v}\times\vec{r})z]=z[\nabla\times(\vec{v}\times\vec{r})]+\nabla z\times(\vec{v}\times\vec{r})=\\ z\epsilon_{ijk}\epsilon_{klm}\nabla_jv_lr_m+\epsilon_{ijk}\epsilon_{klm}\delta_{j3}v_lr_m=\\(\delta_{il}\delta_{jm}-\delta_{jl}\delta_{im})(z\nabla_jv_lr_m+\delta_{j3}v_lr_m)=...$$

Answer 1

$$(\delta_{il}\delta_{jm}-\delta_{jl}\delta_{im})(z\nabla_jv_lr_m+\delta_{j3}v_lr_m)\\=z\delta_{il}\delta_{jm}z\nabla_jv_lr_m+\delta_{il}\delta_{jm}\delta_{j3}v_lr_m-z\delta_{jl}\delta_{im}\nabla_jv_lr_m-\delta_{jl}\delta_{im}\delta_{j3}v_lr_m\\=3zv_i-v_3r_i$$.