Since, a rank-3 tensor has 10 components and a rank-1 tensor has 3 components in $SU(3)$, I know that we are searching for the different irreducible representations of the tensor $v_{ijk}w_{l}$.
The fully symmetric part is equivalent to a rank-4 totally symmetric tensor with $v_{(ijk}w_{l)} = x_{ijkl}$. This will have 15 components.
But, I am not sure how to account for the rest 15.
$$\mathbf{10 \otimes 3 = 15 \ \oplus \ ? }$$