Answer
$\frac{2x}{x^{2}+y^{2}}+x+\frac{2z}{y^{2}+z^{2}}$
Work Step by Step
div $\textbf{F}=\nabla\cdot \textbf{F}=\nabla\cdot\langle \ln(x^{2}+y^{2}),xy,\ln(y^{2}+z^{2})\rangle$
$=\frac{\partial}{\partial x}(\ln(x^{2}+y^{2}))+\frac{\partial}{\partial y}(xy)+\frac{\partial}{\partial z}(\ln(y^{2}+z^{2}))$
$=\frac{2x}{x^{2}+y^{2}}+x+\frac{2z}{y^{2}+z^{2}}$
