Common-intercept partial fraction identity ($\frac{1}{(A+mx)(A+nx)}=\frac{1}{A(m-n)}\left(\frac{m}{A+mx}-\frac{n}{A+nx}\right)$)

For $A\ne 0$ and $m\ne n$, combining the right-hand fractions gives numerator $A(m-n)$ and proves the identity. Integration on an interval avoiding the poles gives $\ln |(A+mx)/(A+nx)|/[A(m-n)]$ plus a constant. The overall $1/A$ cannot be dropped unless the variable has already been normalized to $A=1$.