\[ y'(x)-\frac {\sqrt {| (1-y(x)) y(x) (1-a y(x))| }}{\sqrt {| (1-x) x (1-a x)| }}=0 \] ✓ Mathematica : cpu = 0.174545 (sec), leaf count = 67
DSolve[-(Sqrt[Abs[(1 - y[x])*y[x]*(1 - a*y[x])]]/Sqrt[Abs[(1 - x)*x*(1 - a*x)]]) + Derivative[1][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\sqrt {| (1-K[1]) K[1] (1-a K[1])| }}dK[1]\& \right ]\left [\int _1^x\frac {1}{\sqrt {| (1-K[2]) K[2] (1-a K[2])| }}dK[2]+c_1\right ]\right \}\right \}\] ✓ Maple : cpu = 0.233 (sec), leaf count = 40
dsolve(diff(y(x),x)-abs(y(x)*(-1+y(x))*(-1+a*y(x)))^(1/2)/abs(x*(x-1)*(a*x-1))^(1/2) = 0,y(x))
\[\int \frac {1}{\sqrt {{| x \left (x -1\right ) \left (a x -1\right )|}}}d x -\left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {{| \textit {\_a} \left (\textit {\_a} -1\right ) \left (\textit {\_a} a -1\right )|}}}d \textit {\_a} \right )+c_{1} = 0\]