\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) -{\frac {\sqrt { \left | y \left ( x \right ) \left ( 1-y \left ( x \right ) \right ) \left ( 1-ay \left ( x \right ) \right ) \right | }}{\sqrt { \left | x \left ( 1-x \right ) \left ( -ax+1 \right ) \right | }}}=0} \]
Mathematica: cpu = 877.251897 (sec), leaf count = 65 \[ \left \{\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}} \frac {1}{\sqrt {\left | (1-K[1]) K[1] (1-a K[1])\right | }} \, dK[1]\& \right ]\left [\int _1^x \frac {1}{\sqrt {\left | (1-K[2]) K[2] (1-a K[2])\right | }} \, dK[2]+c_1\right ]\right \}\right \} \]
Maple: cpu = 0.062 (sec), leaf count = 40 \[ \left \{ \int \!{\frac {1}{\sqrt { \left | x \left ( x-1 \right ) \left ( ax-1 \right ) \right | }}}\,{\rm d}x-\int ^{y \left ( x \right ) }\!{\frac {1}{\sqrt { \left | {\it \_a}\, \left ( {\it \_a}-1 \right ) \left ( {\it \_a}\,a-1 \right ) \right | }}}{d{\it \_a}}+{\it \_C1}=0 \right \} \]
Sage: cpu = 1.572 (sec), leaf count = 0 \[ \left [{\left (y\left (x\right ) - 1\right )} \mathrm {sgn}\left (y\left (x\right ) - 1\right ) = {\left (x - 1\right )} \mathrm {sgn}\left (x - 1\right ) + c, \text {\texttt {separable}}\right ] \]