\[ y''(x) \left (a x^2+b x+c\right )^{3/2}-f\left (\frac {y(x)}{\sqrt {a x^2+b x+c}}\right )=0 \] ✓ Mathematica : cpu = 61.0355 (sec), leaf count = 251
\[\left \{\text {Solve}\left [2 a \tan ^{-1}\left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )+2 \sqrt {4 a c-b^2} \int _1^{\frac {y(x)}{\sqrt {c+x (b+a x)}}}\frac {a}{\sqrt {4 c_1 a^2+\left (b^2-4 a c\right ) K[2]^2+8 \int _1^{K[2]}f(K[1])dK[1]}}dK[2]=c_2 \sqrt {4 a c-b^2},y(x)\right ],\text {Solve}\left [2 a \tan ^{-1}\left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )-2 \sqrt {4 a c-b^2} \int _1^{\frac {y(x)}{\sqrt {c+x (b+a x)}}}\frac {a}{\sqrt {4 c_1 a^2+\left (b^2-4 a c\right ) K[4]^2+8 \int _1^{K[4]}f(K[3])dK[3]}}dK[4]=c_2 \sqrt {4 a c-b^2},y(x)\right ]\right \}\] ✓ Maple : cpu = 0.948 (sec), leaf count = 254
\[\left \{y \left (x \right ) = \sqrt {a \,x^{2}+b x +c}\, \RootOf \left (-2 a \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )+c_{2} \sqrt {4 a c -b^{2}}-2 \sqrt {4 a c -b^{2}}\, \left (\int _{}^{\textit {\_Z}}\frac {a}{\sqrt {-4 \textit {\_g}^{2} a c +\textit {\_g}^{2} b^{2}+4 c_{1} a^{2}+8 \left (\int F \left (\textit {\_g} \right )d \textit {\_g} \right )}}d \textit {\_g} \right )\right ), y \left (x \right ) = \sqrt {a \,x^{2}+b x +c}\, \RootOf \left (-2 a \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )+c_{2} \sqrt {4 a c -b^{2}}+2 \sqrt {4 a c -b^{2}}\, \left (\int _{}^{\textit {\_Z}}\frac {a}{\sqrt {-4 \textit {\_g}^{2} a c +\textit {\_g}^{2} b^{2}+4 c_{1} a^{2}+8 \left (\int F \left (\textit {\_g} \right )d \textit {\_g} \right )}}d \textit {\_g} \right )\right ), y \left (x \right ) = \RootOf \left (4 \textit {\_Z} a c -\textit {\_Z} \,b^{2}-4 \sqrt {a \,x^{2}+b x +c}\, F \left (\frac {\textit {\_Z}}{\sqrt {a \,x^{2}+b x +c}}\right )\right )\right \}\]