\[ \boxed { -2\,xy \left ( x \right ) \left ( 1-x \right ) \left ( 1-y \left ( x \right ) \right ) \left ( x-y \left ( x \right ) \right ) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) +x \left ( 1-x \right ) \left ( x-2\,xy \left ( x \right ) -2\,y \left ( x \right ) +3\, \left ( y \left ( x \right ) \right ) ^{2} \right ) \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{2}+2\,y \left ( x \right ) \left ( 1-y \left ( x \right ) \right ) \left ( {x}^{2}+y \left ( x \right ) -2\,xy \left ( x \right ) \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) - \left ( y \left ( x \right ) \right ) ^{2} \left ( 1-y \left ( x \right ) \right ) ^{2}-f \left ( y \left ( x \right ) \left ( -1+y \left ( x \right ) \right ) \left ( y \left ( x \right ) -x \right ) \right ) ^{3/2}=0} \]
Mathematica: cpu = 18.424840 (sec), leaf count = 123 \[ \text {DSolve}\left [-f(x) ((y(x)-1) y(x) (y(x)-x))^{3/2}+2 (1-y(x)) \left (x^2-2 x y(x)+y(x)\right ) y(x) y'(x)-2 (1-x) x (1-y(x)) (x-y(x)) y(x) y''(x)+(1-x) x \left (3 y(x)^2-2 x y(x)-2 y(x)+x\right ) y'(x)^2-(1-y(x))^2 y(x)^2=0,y(x),x\right ] \]
Maple: cpu = 3.073 (sec), leaf count = 916 \[ \left \{ \left ( {\frac {3}{4}\int \!{\frac {1}{x-1}{{\rm e}^{\int \!{ \frac {1}{x \left ( x-1 \right ) }{\it EllipticE} \left ( \sqrt {x} \right ) \left ( {\it EllipticK} \left ( \sqrt {x} \right ) \right ) ^{- 1}}\,{\rm d}x}}\int \!{x}^{{\frac {3}{2}}}\int ^{y \left ( x \right ) } \!{\frac {1}{ \left ( -{\it \_a}+x \right ) ^{2}}{\frac {1}{\sqrt {{{ \it \_a}}^{3}-{{\it \_a}}^{2}x-{{\it \_a}}^{2}+{\it \_a}\,x}}}}{d{\it \_a}}{{\rm e}^{-{\frac {1}{2}\int \!{\frac {1}{x \left ( x-1 \right ) }{ \it EllipticE} \left ( \sqrt {x} \right ) \left ( {\it EllipticK} \left ( \sqrt {x} \right ) \right ) ^{-1}}\,{\rm d}x}}}\,{\rm d}x} \,{\rm d}x}+\int \!{\frac {1}{x-1}{{\rm e}^{\int \!{\frac {1}{x \left ( x-1 \right ) }{\it EllipticE} \left ( \sqrt {x} \right ) \left ( {\it EllipticK} \left ( \sqrt {x} \right ) \right ) ^{-1}}\,{\rm d}x}} \int \!\sqrt {x}\int ^{y \left ( x \right ) }\!{\frac {1}{{\it \_a}\, \left ( {\it \_a}-1 \right ) \left ( -{\it \_a}+x \right ) ^{2}}\sqrt {{ {\it \_a}}^{3}-{{\it \_a}}^{2}x-{{\it \_a}}^{2}+{\it \_a}\,x}}{d{\it \_a}}{{\rm e}^{-{\frac {1}{2}\int \!{\frac {1}{x \left ( x-1 \right ) }{ \it EllipticE} \left ( \sqrt {x} \right ) \left ( {\it EllipticK} \left ( \sqrt {x} \right ) \right ) ^{-1}}\,{\rm d}x}}}\,{\rm d}x} \,{\rm d}x-{\frac {3}{4}\int \!{\frac {1}{x-1}{{\rm e}^{\int \!{\frac {1}{x \left ( x-1 \right ) }{\it EllipticE} \left ( \sqrt {x} \right ) \left ( {\it EllipticK} \left ( \sqrt {x} \right ) \right ) ^{-1}} \,{\rm d}x}}\int \!\sqrt {x}\int ^{y \left ( x \right ) }\!{\frac {1}{ \left ( -{\it \_a}+x \right ) ^{2}}{\frac {1}{\sqrt {{{\it \_a}}^{3}-{{ \it \_a}}^{2}x-{{\it \_a}}^{2}+{\it \_a}\,x}}}}{d{\it \_a}}{{\rm e}^{- {\frac {1}{2}\int \!{\frac {1}{x \left ( x-1 \right ) }{\it EllipticE} \left ( \sqrt {x} \right ) \left ( {\it EllipticK} \left ( \sqrt {x} \right ) \right ) ^{-1}}\,{\rm d}x}}}\,{\rm d}x}\,{\rm d}x}-{\frac {1 }{2}\int \!{\frac {1}{x-1}{{\rm e}^{\int \!{\frac {1}{x \left ( x-1 \right ) }{\it EllipticE} \left ( \sqrt {x} \right ) \left ( {\it EllipticK} \left ( \sqrt {x} \right ) \right ) ^{-1}}\,{\rm d}x}}\int \! {1\int ^{y \left ( x \right ) }\!{\frac {1}{{\it \_a}\, \left ( {\it \_a} -1 \right ) \left ( -{\it \_a}+x \right ) ^{2}}\sqrt {{{\it \_a}}^{3}-{{ \it \_a}}^{2}x-{{\it \_a}}^{2}+{\it \_a}\,x}}{d{\it \_a}}{{\rm e}^{-{ \frac {1}{2}\int \!{\frac {1}{x \left ( x-1 \right ) }{\it EllipticE} \left ( \sqrt {x} \right ) \left ( {\it EllipticK} \left ( \sqrt {x} \right ) \right ) ^{-1}}\,{\rm d}x}}}{\frac {1}{\sqrt {x}}}}\,{\rm d}x }\,{\rm d}x}+{\frac {1}{4}\int \!{\frac {1}{x-1}{{\rm e}^{\int \!{ \frac {1}{x \left ( x-1 \right ) }{\it EllipticE} \left ( \sqrt {x} \right ) \left ( {\it EllipticK} \left ( \sqrt {x} \right ) \right ) ^{- 1}}\,{\rm d}x}}\int \!{1\int ^{y \left ( x \right ) }\!{\frac {1}{\sqrt {{{\it \_a}}^{3}-{{\it \_a}}^{2}x-{{\it \_a}}^{2}+{\it \_a}\,x}}}{d{ \it \_a}}{{\rm e}^{-{\frac {1}{2}\int \!{\frac {1}{x \left ( x-1 \right ) }{\it EllipticE} \left ( \sqrt {x} \right ) \left ( {\it EllipticK} \left ( \sqrt {x} \right ) \right ) ^{-1}}\,{\rm d}x}}}{ \frac {1}{\sqrt {x}}}}\,{\rm d}x}\,{\rm d}x}+{\frac {f}{2}\int \!{ \frac {1}{x-1}{{\rm e}^{\int \!{\frac {1}{x \left ( x-1 \right ) }{\it EllipticE} \left ( \sqrt {x} \right ) \left ( {\it EllipticK} \left ( \sqrt {x} \right ) \right ) ^{-1}}\,{\rm d}x}}\int \!{1{{\rm e}^{-{ \frac {1}{2}\int \!{\frac {1}{x \left ( x-1 \right ) }{\it EllipticE} \left ( \sqrt {x} \right ) \left ( {\it EllipticK} \left ( \sqrt {x} \right ) \right ) ^{-1}}\,{\rm d}x}}}{\frac {1}{\sqrt {x}}}}\,{\rm d}x }\,{\rm d}x}+{\frac { \left ( y \left ( x \right ) \right ) ^{4}}{2}\int ^{x}\!{\frac {1}{{\it \_f}-1}{{\rm e}^{\int \!{\frac {1}{{\it \_f}\, \left ( {\it \_f}-1 \right ) }{\it EllipticE} \left ( \sqrt {{\it \_f}} \right ) \left ( {\it EllipticK} \left ( \sqrt {{\it \_f}} \right ) \right ) ^{-1}}\,{\rm d}{\it \_f}}}\int \!{1{{\rm e}^{-{\frac {1}{2} \int \!{\frac {1}{{\it \_f}\, \left ( {\it \_f}-1 \right ) }{\it EllipticE} \left ( \sqrt {{\it \_f}} \right ) \left ( {\it EllipticK} \left ( \sqrt {{\it \_f}} \right ) \right ) ^{-1}}\,{\rm d}{\it \_f}}}} {\frac {1}{\sqrt {{\it \_f}}}} \left ( -{\it \_f}\, \left ( y \left ( x \right ) \right ) ^{2}+ \left ( y \left ( x \right ) \right ) ^{3}+{\it \_f}\,y \left ( x \right ) - \left ( y \left ( x \right ) \right ) ^{2} \right ) ^{-{\frac {3}{2}}}}\,{\rm d}{\it \_f}}{d{\it \_f}}}- \left ( y \left ( x \right ) \right ) ^{3}\int ^{x}\!{\frac {1}{{\it \_f}-1}{ {\rm e}^{\int \!{\frac {1}{{\it \_f}\, \left ( {\it \_f}-1 \right ) }{ \it EllipticE} \left ( \sqrt {{\it \_f}} \right ) \left ( {\it EllipticK } \left ( \sqrt {{\it \_f}} \right ) \right ) ^{-1}}\,{\rm d}{\it \_f}}} \int \!{1{{\rm e}^{-{\frac {1}{2}\int \!{\frac {1}{{\it \_f}\, \left ( {\it \_f}-1 \right ) }{\it EllipticE} \left ( \sqrt {{\it \_f}} \right ) \left ( {\it EllipticK} \left ( \sqrt {{\it \_f}} \right ) \right ) ^{-1 }}\,{\rm d}{\it \_f}}}}{\frac {1}{\sqrt {{\it \_f}}}} \left ( -{\it \_f }\, \left ( y \left ( x \right ) \right ) ^{2}+ \left ( y \left ( x \right ) \right ) ^{3}+{\it \_f}\,y \left ( x \right ) - \left ( y \left ( x \right ) \right ) ^{2} \right ) ^{-{\frac {3}{2}}}}\,{\rm d}{ \it \_f}}{d{\it \_f}}+{\frac { \left ( y \left ( x \right ) \right ) ^{2} }{2}\int ^{x}\!{\frac {1}{{\it \_f}-1}{{\rm e}^{\int \!{\frac {1}{{ \it \_f}\, \left ( {\it \_f}-1 \right ) }{\it EllipticE} \left ( \sqrt {{ \it \_f}} \right ) \left ( {\it EllipticK} \left ( \sqrt {{\it \_f}} \right ) \right ) ^{-1}}\,{\rm d}{\it \_f}}}\int \!{1{{\rm e}^{-{ \frac {1}{2}\int \!{\frac {1}{{\it \_f}\, \left ( {\it \_f}-1 \right ) } {\it EllipticE} \left ( \sqrt {{\it \_f}} \right ) \left ( {\it EllipticK} \left ( \sqrt {{\it \_f}} \right ) \right ) ^{-1}}\,{\rm d}{ \it \_f}}}}{\frac {1}{\sqrt {{\it \_f}}}} \left ( -{\it \_f}\, \left ( y \left ( x \right ) \right ) ^{2}+ \left ( y \left ( x \right ) \right ) ^{ 3}+{\it \_f}\,y \left ( x \right ) - \left ( y \left ( x \right ) \right ) ^{2} \right ) ^{-{\frac {3}{2}}}}\,{\rm d}{\it \_f}}{d{\it \_f}}}-{ {\rm e}^{\int \!{\frac {1}{2\,x \left ( x-1 \right ) } \left ( {\it EllipticK} \left ( \sqrt {x} \right ) x-{\it EllipticK} \left ( \sqrt {x} \right ) +{\it EllipticE} \left ( \sqrt {x} \right ) \right ) \left ( { \it EllipticK} \left ( \sqrt {x} \right ) \right ) ^{-1}}\,{\rm d}x}} \int ^{y \left ( x \right ) }\!{\frac {1}{\sqrt {-{\it \_a}\, \left ( { \it \_a}-1 \right ) \left ( -{\it \_a}+x \right ) }}}{d{\it \_a}} \right ) {\it \_C1}+\int \!{\frac {1}{x-1}{{\rm e}^{\int \!{\frac {1}{ x \left ( x-1 \right ) }{\it EllipticE} \left ( \sqrt {x} \right ) \left ( {\it EllipticK} \left ( \sqrt {x} \right ) \right ) ^{-1}} \,{\rm d}x}}}\,{\rm d}x-{\it \_C2}=0 \right \} \]