\[ \boxed { \left ( a{x}^{2}+bx+c \right ) ^{3/2}{\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) -F \left ( {\frac {y \left ( x \right ) }{\sqrt {a{x}^{2}+bx+c}}} \right ) =0} \]
Mathematica: cpu = 61.997373 (sec), leaf count = 46 \[ \text {DSolve}\left [y''(x) \left (a x^2+b x+c\right )^{3/2}-f\left (\frac {y(x)}{\sqrt {a x^2+b x+c}}\right )=0,y(x),x\right ] \]
Maple: cpu = 0.795 (sec), leaf count = 254 \[ \left \{ y \left ( x \right ) ={\it RootOf} \left ( -2\,a\arctan \left ( { \frac {2\,ax+b}{\sqrt {4\,ac-{b}^{2}}}} \right ) -2\,\int ^{{\it \_Z}} \!{\frac {a}{\sqrt {4\,{\it \_C1}\,{a}^{2}-4\,c{{\it \_g}}^{2}a+{b}^{2 }{{\it \_g}}^{2}+8\,\int \!F \left ( {\it \_g} \right ) \,{\rm d}{\it \_g}}}}{d{\it \_g}}\sqrt {4\,ac-{b}^{2}}+{\it \_C2}\,\sqrt {4\,ac-{b}^ {2}} \right ) \sqrt {a{x}^{2}+bx+c},y \left ( x \right ) ={\it RootOf} \left ( -2\,a\arctan \left ( {\frac {2\,ax+b}{\sqrt {4\,ac-{b}^{2}}}} \right ) +2\,\int ^{{\it \_Z}}\!{\frac {a}{\sqrt {4\,{\it \_C1}\,{a}^{ 2}-4\,c{{\it \_g}}^{2}a+{b}^{2}{{\it \_g}}^{2}+8\,\int \!F \left ( { \it \_g} \right ) \,{\rm d}{\it \_g}}}}{d{\it \_g}}\sqrt {4\,ac-{b}^{2} }+{\it \_C2}\,\sqrt {4\,ac-{b}^{2}} \right ) \sqrt {a{x}^{2}+bx+c},y \left ( x \right ) ={\it RootOf} \left ( 4\,{\it \_Z}\,ac-{\it \_Z}\,{b} ^{2}-4\,F \left ( {\frac {{\it \_Z}}{\sqrt {a{x}^{2}+bx+c}}} \right ) \sqrt {a{x}^{2}+bx+c} \right ) \right \} \]