\[ -f(x) ((y(x)-1) y(x) (y(x)-x))^{3/2}+2 (1-y(x)) \left (x^2-2 x y(x)+y(x)\right ) y(x) y'(x)-2 (1-x) x (1-y(x)) (x-y(x)) y(x) y''(x)+(1-x) x \left (3 y(x)^2-2 x y(x)-2 y(x)+x\right ) y'(x)^2-(1-y(x))^2 y(x)^2=0 \] ✗ Mathematica : cpu = 14.0072 (sec), leaf count = 0
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]
, 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 = 2.529 (sec), leaf count = 733
dsolve(-2*x*y(x)*(1-x)*(1-y(x))*(x-y(x))*diff(diff(y(x),x),x)+x*(1-x)*(x-2*x*y(x)-2*y(x)+3*y(x)^2)*diff(y(x),x)^2+2*y(x)*(1-y(x))*(x^2+y(x)-2*x*y(x))*diff(y(x),x)-y(x)^2*(1-y(x))^2-f*(y(x)*(-1+y(x))*(y(x)-x))^(3/2)=0,y(x))
\[-\frac {c_{1} \left (\int \frac {{\mathrm e}^{\int \frac {\EllipticE \left (\sqrt {x}\right )}{x \left (x -1\right ) \EllipticK \left (\sqrt {x}\right )}d x} \left (\int \frac {\left (\int \frac {\sqrt {-y \left (y -1\right ) \left (x -y \right )}}{y \left (y -1\right ) \left (x -y \right )^{2}}d y \right ) {\mathrm e}^{-\frac {\left (\int \frac {\EllipticE \left (\sqrt {x}\right )}{x \left (x -1\right ) \EllipticK \left (\sqrt {x}\right )}d x \right )}{2}}}{\sqrt {x}}d x \right )}{x -1}d x {\raisebox {-0.36em}{$\Big |$}}{\mstack {}{_{\left \{y \hiderel {=}y \left (x \right )\right \}}}}\right )}{2}+c_{1} \left (\int \frac {{\mathrm e}^{\int \frac {\EllipticE \left (\sqrt {x}\right )}{x \left (x -1\right ) \EllipticK \left (\sqrt {x}\right )}d x} \left (\int \sqrt {x}\, \left (\int \frac {\sqrt {-y \left (y -1\right ) \left (x -y \right )}}{y \left (y -1\right ) \left (x -y \right )^{2}}d y \right ) {\mathrm e}^{-\frac {\left (\int \frac {\EllipticE \left (\sqrt {x}\right )}{x \left (x -1\right ) \EllipticK \left (\sqrt {x}\right )}d x \right )}{2}}d x \right )}{x -1}d x {\raisebox {-0.36em}{$\Big |$}}{\mstack {}{_{\left \{y \hiderel {=}y \left (x \right )\right \}}}}\right )+\frac {3 c_{1} \left (\int \frac {{\mathrm e}^{\int \frac {\EllipticE \left (\sqrt {x}\right )}{x \left (x -1\right ) \EllipticK \left (\sqrt {x}\right )}d x} \left (\int x^{\frac {3}{2}} \left (\int \frac {1}{\left (x -y \right )^{2} \sqrt {-x \,y^{2}+y^{3}+y x -y^{2}}}d y \right ) {\mathrm e}^{-\frac {\left (\int \frac {\EllipticE \left (\sqrt {x}\right )}{x \left (x -1\right ) \EllipticK \left (\sqrt {x}\right )}d x \right )}{2}}d x \right )}{x -1}d x {\raisebox {-0.36em}{$\Big |$}}{\mstack {}{_{\left \{y \hiderel {=}y \left (x \right )\right \}}}}\right )}{4}-\frac {3 c_{1} \left (\int \frac {{\mathrm e}^{\int \frac {\EllipticE \left (\sqrt {x}\right )}{x \left (x -1\right ) \EllipticK \left (\sqrt {x}\right )}d x} \left (\int \sqrt {x}\, \left (\int \frac {1}{\left (x -y \right )^{2} \sqrt {-x \,y^{2}+y^{3}+y x -y^{2}}}d y \right ) {\mathrm e}^{-\frac {\left (\int \frac {\EllipticE \left (\sqrt {x}\right )}{x \left (x -1\right ) \EllipticK \left (\sqrt {x}\right )}d x \right )}{2}}d x \right )}{x -1}d x {\raisebox {-0.36em}{$\Big |$}}{\mstack {}{_{\left \{y \hiderel {=}y \left (x \right )\right \}}}}\right )}{4}+\frac {c_{1} y \left (x \right )^{2} \left (-1+y \left (x \right )\right )^{2} \left (\int _{}^{x}\frac {{\mathrm e}^{\int \frac {\EllipticE \left (\sqrt {\textit {\_f}}\right )}{\textit {\_f} \left (\textit {\_f} -1\right ) \EllipticK \left (\sqrt {\textit {\_f}}\right )}d \textit {\_f}} \left (\int \frac {{\mathrm e}^{-\frac {\left (\int \frac {\EllipticE \left (\sqrt {\textit {\_f}}\right )}{\textit {\_f} \left (\textit {\_f} -1\right ) \EllipticK \left (\sqrt {\textit {\_f}}\right )}d \textit {\_f} \right )}{2}}}{\sqrt {\textit {\_f}}\, \left (-y \left (x \right ) \left (-1+y \left (x \right )\right ) \left (\textit {\_f} -y \left (x \right )\right )\right )^{\frac {3}{2}}}d \textit {\_f} \right )}{\textit {\_f} -1}d \textit {\_f} \right )}{2}+\frac {c_{1} f \left (\int \frac {{\mathrm e}^{\int \frac {\EllipticE \left (\sqrt {x}\right )}{x \left (x -1\right ) \EllipticK \left (\sqrt {x}\right )}d x} \left (\int \frac {{\mathrm e}^{-\frac {\left (\int \frac {\EllipticE \left (\sqrt {x}\right )}{x \left (x -1\right ) \EllipticK \left (\sqrt {x}\right )}d x \right )}{2}}}{\sqrt {x}}d x \right )}{x -1}d x \right )}{2}-c_{1} {\mathrm e}^{\int \frac {\left (x -1\right ) \EllipticK \left (\sqrt {x}\right )+\EllipticE \left (\sqrt {x}\right )}{2 x \left (x -1\right ) \EllipticK \left (\sqrt {x}\right )}d x} \left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {-\textit {\_a} \left (\textit {\_a} -1\right ) \left (x -\textit {\_a} \right )}}d \textit {\_a} \right )+\frac {c_{1} \left (\int \frac {{\mathrm e}^{\int \frac {\EllipticE \left (\sqrt {x}\right )}{x \left (x -1\right ) \EllipticK \left (\sqrt {x}\right )}d x} \left (\int \frac {\left (\int \frac {1}{\sqrt {-x \,y^{2}+y^{3}+y x -y^{2}}}d y \right ) {\mathrm e}^{-\frac {\left (\int \frac {\EllipticE \left (\sqrt {x}\right )}{x \left (x -1\right ) \EllipticK \left (\sqrt {x}\right )}d x \right )}{2}}}{\sqrt {x}}d x \right )}{x -1}d x {\raisebox {-0.36em}{$\Big |$}}{\mstack {}{_{\left \{y \hiderel {=}y \left (x \right )\right \}}}}\right )}{4}+\int \frac {{\mathrm e}^{\int \frac {\EllipticE \left (\sqrt {x}\right )}{x \left (x -1\right ) \EllipticK \left (\sqrt {x}\right )}d x}}{x -1}d x -c_{2} = 0\]