\[ y''(x) \left (a x^2+2 b x+c+y(x)^2\right )^2+d y(x)=0 \] ✓ Mathematica : cpu = 19.8382 (sec), leaf count = 260
DSolve[d*y[x] + (c + 2*b*x + a*x^2 + y[x]^2)^2*Derivative[2][y][x] == 0,y[x],x]
\[\left \{\text {Solve}\left [a \tan ^{-1}\left (\frac {a x+b}{\sqrt {a c-b^2}}\right )+\sqrt {a c-b^2} \int _1^{\frac {y(x)}{\sqrt {c+x (2 b+a x)}}}\frac {a \left (K[2]^2+1\right )}{\sqrt {\left (K[2]^2+1\right ) \left (d+\left (K[2]^2+1\right ) \left (c_1 a^2+\left (b^2-a c\right ) K[2]^2\right )\right )}}dK[2]=c_2 \sqrt {a c-b^2},y(x)\right ],\text {Solve}\left [a \tan ^{-1}\left (\frac {a x+b}{\sqrt {a c-b^2}}\right )-\sqrt {a c-b^2} \int _1^{\frac {y(x)}{\sqrt {c+x (2 b+a x)}}}\frac {a \left (K[3]^2+1\right )}{\sqrt {\left (K[3]^2+1\right ) \left (d+\left (K[3]^2+1\right ) \left (c_1 a^2+\left (b^2-a c\right ) K[3]^2\right )\right )}}dK[3]=c_2 \sqrt {a c-b^2},y(x)\right ]\right \}\] ✓ Maple : cpu = 0.419 (sec), leaf count = 336
dsolve((c+2*b*x+a*x^2+y(x)^2)^2*diff(diff(y(x),x),x)+d*y(x)=0,y(x))
\[y \left (x \right ) = \RootOf \left (-a \arctan \left (\frac {a x +b}{\sqrt {c a -b^{2}}}\right )+\left (\int _{}^{\textit {\_Z}}\frac {\sqrt {\left (\textit {\_f}^{2}+1\right ) \left (-\textit {\_f}^{4} a c +\textit {\_f}^{4} b^{2}+\textit {\_f}^{2} a^{2} c_{1}-c \,\textit {\_f}^{2} a +b^{2} \textit {\_f}^{2}+a^{2} c_{1}+d \right )}\, a}{-\textit {\_f}^{4} a c +\textit {\_f}^{4} b^{2}+\textit {\_f}^{2} a^{2} c_{1}-c \,\textit {\_f}^{2} a +b^{2} \textit {\_f}^{2}+a^{2} c_{1}+d}d \textit {\_f} \right ) \sqrt {c a -b^{2}}+c_{2} \sqrt {c a -b^{2}}\right ) \sqrt {a \,x^{2}+2 b x +c}\]