\[ y''(x) \left (a x^2+b x+c\right )^{3/2}-f\left (\frac {y(x)}{\sqrt {a x^2+b x+c}}\right )=0 \] ✓ Mathematica : cpu = 60.8995 (sec), leaf count = 251
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]
\[\left \{\text {Solve}\left [2 a \tan ^{-1}\left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )+2 \sqrt {4 a c-b^2} \int _1^{\frac {y(x)}{\sqrt {c+x (b+a x)}}}\frac {a}{\sqrt {4 c_1 a^2+\left (b^2-4 a c\right ) K[2]^2+8 \int _1^{K[2]}f(K[1])dK[1]}}dK[2]=c_2 \sqrt {4 a c-b^2},y(x)\right ],\text {Solve}\left [2 a \tan ^{-1}\left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )-2 \sqrt {4 a c-b^2} \int _1^{\frac {y(x)}{\sqrt {c+x (b+a x)}}}\frac {a}{\sqrt {4 c_1 a^2+\left (b^2-4 a c\right ) K[4]^2+8 \int _1^{K[4]}f(K[3])dK[3]}}dK[4]=c_2 \sqrt {4 a c-b^2},y(x)\right ]\right \}\] ✓ Maple : cpu = 0.804 (sec), leaf count = 254
dsolve((a*x^2+b*x+c)^(3/2)*diff(diff(y(x),x),x)-F(y(x)/(a*x^2+b*x+c)^(1/2))=0,y(x))
\[y \left (x \right ) = \RootOf \left (4 \textit {\_Z} a c -\textit {\_Z} \,b^{2}-4 F \left (\frac {\textit {\_Z}}{\sqrt {a \,x^{2}+b x +c}}\right ) \sqrt {a \,x^{2}+b x +c}\right )\]