Answer
Conservative
Work Step by Step
A vector field will be conservative when $curl F=\nabla \times F=0$
$curl F= \nabla \times F=(\dfrac{\partial z}{\partial y}-\dfrac{\partial y}{\partial z})i+(\dfrac{\partial x}{\partial z}-\dfrac{\partial z}{\partial x})j+(\dfrac{\partial y}{\partial x}-\dfrac{\partial x}{\partial y})k$
This implies that
$curl F=-3yz(x^2+y^2+z^2)^{-5/2}+3yz(x^2+y^2+z^2)^{-5/2}--3xz(x^2+y^2+z^2)^{-5/2}+3xz(x^2+y^2+z^2)^{-5/2}-3xy(x^2+y^2+z^2)^{-5/2}+3xy(x^2+y^2+z^2)^{-5/2}=0$
Therefore, the given field is conservative.