\[ \boxed { \left ( c+2\,bx+a{x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2} \right ) ^{2}{\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) +dy \left ( x \right ) =0} \]
Mathematica: cpu = 43.116975 (sec), leaf count = 33 \[ \text {DSolve}\left [y''(x) \left (a x^2+2 b x+c+y(x)^2\right )^2+d y(x)=0,y(x),x\right ] \]
Maple: cpu = 0.437 (sec), leaf count = 382 \[ \left \{ y \left ( x \right ) ={\it RootOf} \left ( -\int ^{{\it \_Z}}\!{ \frac {a}{-{{\it \_f}}^{4}ac+{{\it \_f}}^{4}{b}^{2}+{\it \_C1}\,{{\it \_f}}^{2}{a}^{2}-c{{\it \_f}}^{2}a+{b}^{2}{{\it \_f}}^{2}+{\it \_C1}\, {a}^{2}+d}\sqrt {-{{\it \_f}}^{6}ac+{{\it \_f}}^{6}{b}^{2}+{\it \_C1} \,{{\it \_f}}^{4}{a}^{2}-2\,{{\it \_f}}^{4}ac+2\,{{\it \_f}}^{4}{b}^{2 }+2\,{\it \_C1}\,{{\it \_f}}^{2}{a}^{2}-c{{\it \_f}}^{2}a+{b}^{2}{{ \it \_f}}^{2}+{\it \_C1}\,{a}^{2}+{{\it \_f}}^{2}d+d}}{d{\it \_f}} \sqrt {ac-{b}^{2}}+{\it \_C2}\,\sqrt {ac-{b}^{2}}-a\arctan \left ( {(ax +b){\frac {1}{\sqrt {ac-{b}^{2}}}}} \right ) \right ) \sqrt {a{x}^{2}+2 \,bx+c},y \left ( x \right ) ={\it RootOf} \left ( \int ^{{\it \_Z}}\!{ \frac {a}{-{{\it \_f}}^{4}ac+{{\it \_f}}^{4}{b}^{2}+{\it \_C1}\,{{\it \_f}}^{2}{a}^{2}-c{{\it \_f}}^{2}a+{b}^{2}{{\it \_f}}^{2}+{\it \_C1}\, {a}^{2}+d}\sqrt {-{{\it \_f}}^{6}ac+{{\it \_f}}^{6}{b}^{2}+{\it \_C1} \,{{\it \_f}}^{4}{a}^{2}-2\,{{\it \_f}}^{4}ac+2\,{{\it \_f}}^{4}{b}^{2 }+2\,{\it \_C1}\,{{\it \_f}}^{2}{a}^{2}-c{{\it \_f}}^{2}a+{b}^{2}{{ \it \_f}}^{2}+{\it \_C1}\,{a}^{2}+{{\it \_f}}^{2}d+d}}{d{\it \_f}} \sqrt {ac-{b}^{2}}+{\it \_C2}\,\sqrt {ac-{b}^{2}}-a\arctan \left ( {(ax +b){\frac {1}{\sqrt {ac-{b}^{2}}}}} \right ) \right ) \sqrt {a{x}^{2}+2 \,bx+c} \right \} \]