\[ -f(x) ((y(x)-1) y(x) (y(x)-x))^{3/2}+2 (1-y(x)) \left (x^2-2 x y(x)+y(x)\right ) y(x) y'(x)+(1-x) x \left (3 y(x)^2-2 x y(x)-2 y(x)+x\right ) y'(x)^2-2 (1-x) x (1-y(x)) (x-y(x)) y(x) y''(x)-(1-y(x))^2 y(x)^2=0 \] ✗ Mathematica : cpu = 14.1791 (sec), leaf count = 0 , could not solve
DSolve[-((1 - y[x])^2*y[x]^2) - f[x]*((-1 + y[x])*y[x]*(-x + y[x]))^(3/2) + 2*(1 - y[x])*y[x]*(x^2 + y[x] - 2*x*y[x])*Derivative[1][y][x] + (1 - x)*x*(x - 2*y[x] - 2*x*y[x] + 3*y[x]^2)*Derivative[1][y][x]^2 - 2*(1 - x)*x*(1 - y[x])*(x - y[x])*y[x]*Derivative[2][y][x] == 0, y[x], x]
✓ Maple : cpu = 7.063 (sec), leaf count = 733
\[ \left \{ -{\frac {{\it \_C1}}{2}{\it eval} \left ( \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 \!{\int \!{\frac {1}{y \left ( y-1 \right ) \left ( -y+x \right ) ^{2}}\sqrt {-y \left ( y-1 \right ) \left ( -y+x \right ) }}\,{\rm d}y{{\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, \left \{ y=y \left ( x \right ) \right \} \right ) }+{\it \_C1}\,{\it eval} \left ( \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 \!{\frac {1}{y \left ( y-1 \right ) \left ( -y+x \right ) ^{2}}\sqrt {-y \left ( y-1 \right ) \left ( -y+x \right ) }}\,{\rm d}y{{\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, \left \{ y=y \left ( x \right ) \right \} \right ) +{\frac {3\,{\it \_C1}}{4}{\it eval} \left ( \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 \!{\frac {1}{ \left ( -y+x \right ) ^{2}}{\frac {1}{\sqrt {-x{y}^{2}+{y}^{3}+xy-{y}^{2}}}}}\,{\rm d}y{{\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, \left \{ y=y \left ( x \right ) \right \} \right ) }-{\frac {3\,{\it \_C1}}{4}{\it eval} \left ( \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 \!{\frac {1}{ \left ( -y+x \right ) ^{2}}{\frac {1}{\sqrt {-x{y}^{2}+{y}^{3}+xy-{y}^{2}}}}}\,{\rm d}y{{\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, \left \{ y=y \left ( x \right ) \right \} \right ) }+{\frac {{\it \_C1}\, \left ( y \left ( x \right ) \right ) ^{2} \left ( -1+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 \!{{{\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 ( -y \left ( x \right ) \left ( -1+y \left ( x \right ) \right ) \left ( {\it \_f}-y \left ( x \right ) \right ) \right ) ^{-{\frac {3}{2}}}}\,{\rm d}{\it \_f}}{d{\it \_f}}}+{\frac {{\it \_C1}\,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 \!{{{\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}-{\it \_C1}\,{{\rm e}^{\int \!{\frac {1}{2\,x \left ( x-1 \right ) } \left ( \left ( x-1 \right ) {\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}}+{\frac {{\it \_C1}}{4}{\it eval} \left ( \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 \!{\int \!{\frac {1}{\sqrt {-x{y}^{2}+{y}^{3}+xy-{y}^{2}}}}\,{\rm d}y{{\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, \left \{ y=y \left ( x \right ) \right \} \right ) }+\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 \} \]