Inspired by Ron Gordon's answer here I attempted to integrate $$ g(w)=h(w)\ln w=\frac{\ln w}{(w+p)\sqrt{(w+x)(w+y)(w+z)}}, \quad (x,y,z,-p>0) $$ over the contour $\gamma$ shown below.
Description of the contour
Line segment joining $P_{k},P_{j}$ is denoted by $L_{k,j}$. Each line segment is parallel to and at a distance $\delta$ from the real line. Small circle centered at $-\alpha$ is denoted by $C_\alpha$ and has radius $r_\alpha$. The large circle centered at $0$ has radius $R$ and is denoted by $C_R$.
We shall consider the limiting case where $$\begin{align}&r_\alpha\to0,\quad \alpha=0,x,y,z,p\\ &R\to+\infty\\&\delta\to0\end{align}$$
Question
Integrals over $C_\alpha(\alpha=x,y,z,0), C_R$ vanish. Integral over $C_p$ can be computed using residue but how can the branch cuts be defined so that $g$ remains continuous inside the contour?