separating equation $$
\begin{align}
\frac {dy}{dx} &= \frac{(y-1)(x+2)}{(y+1)(x-3)}\\
\frac{y+1}{y-1}dy &= \frac{x+2}{x-3}dx
\end{align}
$$
Integrate both sides :
$$
\begin{align}
\int\frac{y+1}{y-1}dy & =\int\frac{x+2}{x-3}dx\\
\int 1+\frac{2}{y-1}dy &=\int 1+\frac{5}{x-3}dx\\
y+2\ln|y-1|&=x+5\ln|x-3|+C\\
\end{align}
$$
trapped here because I cannot get rid of $2\ln|y-1|$, any techniques I should use to solve this?

You are not trapped and the solution cannot be expressed in terms of elementary functions.
If fact, there is a solution in terms of Lambert function you will learn at a time and the solution would be $$y=1+2 W\left(\pm\frac{c \sqrt{(x-3)^5 e^{x}}}{2 \sqrt{e}}\right)$$ where $c$ is the integration constant you missed.
You could be interested knowing that any equation which can write $$A+B x+C \log(D+Ex)=0$$ has solutions in terms of Lambert function.
