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
$curl F=-yz(x+yz)^{-2}+(x+yz)^{-1}+yz(x+yz)^{-2}-(x+yz)^{-1}-y(x+yz)^{-2}+y (x+yz)^{-2}+z (x+yz)^{-2}-z (x+yz)^{-2}= 0$
Therefore, the given field is conservative.