2.2.4.5 ✓ Mathematica. Time used: 3.168 (sec). Leaf size: 82

ode=D[x[t],t]==k*(A-n*x[t])*(M-m*x[t]); 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {A e^{M n (k t+c_1)}-M e^{A m (k t+c_1)}}{n e^{M n (k t+c_1)}-m e^{A m (k t+c_1)}}\\ x(t)&\to \frac {M}{m}\\ x(t)&\to \frac {A}{n} \end{align*}