ODE
\[ \sqrt {(1-x) x (1-a x)} y'(x)=\sqrt {(1-y(x)) y(x) (1-a y(x))} \] ODE Classification
[_separable]
Book solution method
Separable ODE, Neither variable missing
Mathematica ✓
cpu = 0.683616 (sec), leaf count = 100
\[\left \{\left \{y(x)\to \text {ns}\left (\frac {1}{2} i \sqrt {a} c_1-F\left (i \sinh ^{-1}\left (\frac {1}{\sqrt {x-1}}\right )|\frac {a-1}{a}\right )|\frac {a-1}{a}\right ){}^2 \left (-1+\text {sn}\left (\frac {1}{2} i \sqrt {a} c_1-F\left (i \sinh ^{-1}\left (\frac {1}{\sqrt {x-1}}\right )|\frac {a-1}{a}\right )|\frac {a-1}{a}\right ){}^2\right )\right \}\right \}\]
Maple ✓
cpu = 0.021 (sec), leaf count = 38
\[\left [\int \frac {1}{\sqrt {x \left (x -1\right ) \left (a x -1\right )}}d x -\left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {\textit {\_a} \left (\textit {\_a} -1\right ) \left (a \textit {\_a} -1\right )}}d \textit {\_a} \right )+\textit {\_C1} = 0\right ]\] Mathematica raw input
DSolve[Sqrt[(1 - x)*x*(1 - a*x)]*y'[x] == Sqrt[(1 - y[x])*y[x]*(1 - a*y[x])],y[x],x]
Mathematica raw output
{{y[x] -> JacobiNS[(I/2)*Sqrt[a]*C[1] - EllipticF[I*ArcSinh[1/Sqrt[-1 + x]], (-1 + a)/a], (-1 + a)/a]^2*(-1 + JacobiSN[(I/2)*Sqrt[a]*C[1] - EllipticF[I*ArcSinh[1/Sqrt[-1 + x]], (-1 + a)/a], (-1 + a)/a]^2)}}
Maple raw input
dsolve(diff(y(x),x)*(x*(1-x)*(-a*x+1))^(1/2) = (y(x)*(1-y(x))*(1-a*y(x)))^(1/2), y(x))
Maple raw output
[Int(1/(x*(x-1)*(a*x-1))^(1/2),x)-Intat(1/(_a*(_a-1)*(_a*a-1))^(1/2),_a = y(x))+_C1 = 0]