\[ y''(x) \left (a x^2+b x+c\right )^{3/2}-f\left (\frac {y(x)}{\sqrt {a x^2+b x+c}}\right )=0 \] ✗ Mathematica : cpu = 60.6351 (sec), leaf count = 0 , could not solve
DSolve[-f[y[x]/Sqrt[c + b*x + a*x^2]] + (c + b*x + a*x^2)^(3/2)*Derivative[2][y][x] == 0, y[x], x]
✓ Maple : cpu = 1.861 (sec), leaf count = 254
\[ \left \{ y \left ( x \right ) ={\it RootOf} \left ( -2\,\int ^{{\it \_Z}}\!{\frac {a}{\sqrt {4\,{\it \_C1}\,{a}^{2}-4\,c{{\it \_g}}^{2}a+{b}^{2}{{\it \_g}}^{2}+8\,\int \!F \left ( {\it \_g} \right ) \,{\rm d}{\it \_g}}}}{d{\it \_g}}\sqrt {4\,ac-{b}^{2}}+{\it \_C2}\,\sqrt {4\,ac-{b}^{2}}-2\,\arctan \left ( {\frac {2\,ax+b}{\sqrt {4\,ac-{b}^{2}}}} \right ) a \right ) \sqrt {a{x}^{2}+bx+c},y \left ( x \right ) ={\it RootOf} \left ( 2\,\int ^{{\it \_Z}}\!{\frac {a}{\sqrt {4\,{\it \_C1}\,{a}^{2}-4\,c{{\it \_g}}^{2}a+{b}^{2}{{\it \_g}}^{2}+8\,\int \!F \left ( {\it \_g} \right ) \,{\rm d}{\it \_g}}}}{d{\it \_g}}\sqrt {4\,ac-{b}^{2}}+{\it \_C2}\,\sqrt {4\,ac-{b}^{2}}-2\,\arctan \left ( {\frac {2\,ax+b}{\sqrt {4\,ac-{b}^{2}}}} \right ) a \right ) \sqrt {a{x}^{2}+bx+c},y \left ( x \right ) ={\it RootOf} \left ( 4\,{\it \_Z}\,ac-{\it \_Z}\,{b}^{2}-4\,F \left ( {\frac {{\it \_Z}}{\sqrt {a{x}^{2}+bx+c}}} \right ) \sqrt {a{x}^{2}+bx+c} \right ) \right \} \]