\[ \boxed { 4\, \left ( {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) \right ) y \left ( x \right ) -3\, \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{2}+a \left ( y \left ( x \right ) \right ) ^{3}+b \left ( y \left ( x \right ) \right ) ^{2}+cy \left ( x \right ) =0} \]
Mathematica: cpu = 3.890994 (sec), leaf count = 2281 \[ \left \{\text {Solve}\left [-\frac {4 F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,4\right ]\right ) \left (\sqrt {y(x)}-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,1\right ]\right )}{\left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,1\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,4\right ]\right ) \left (\sqrt {y(x)}-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]\right )}}\right )|\frac {\left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,3\right ]\right ) \left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,1\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,4\right ]\right )}{\left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,1\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,3\right ]\right ) \left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,4\right ]\right )}\right ) \left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,1\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,4\right ]\right ) \left (\sqrt {y(x)}-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]\right ){}^2 \sqrt {\frac {\left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,1\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]\right ) \left (\sqrt {y(x)}-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,3\right ]\right )}{\left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,1\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,3\right ]\right ) \left (\sqrt {y(x)}-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]\right )}} \sqrt {\frac {\left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,1\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]\right ) \left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,4\right ]\right ) \left (\sqrt {y(x)}-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,1\right ]\right ) \left (\sqrt {y(x)}-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,4\right ]\right )}{\left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,1\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,4\right ]\right ){}^2 \left (\sqrt {y(x)}-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]\right ){}^2}} \sqrt {y(x)}}{\left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,1\right ]\right ) \left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,4\right ]\right ) \sqrt {-\frac {1}{3} a y(x)^3-b y(x)^2+c_1 y(x)^{3/2}+c y(x)}}=x+c_2,y(x)\right ],\text {Solve}\left [\frac {4 F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,4\right ]\right ) \left (\sqrt {y(x)}-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,1\right ]\right )}{\left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,1\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,4\right ]\right ) \left (\sqrt {y(x)}-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]\right )}}\right )|\frac {\left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,3\right ]\right ) \left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,1\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,4\right ]\right )}{\left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,1\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,3\right ]\right ) \left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,4\right ]\right )}\right ) \left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,1\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,4\right ]\right ) \left (\sqrt {y(x)}-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]\right ){}^2 \sqrt {\frac {\left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,1\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]\right ) \left (\sqrt {y(x)}-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,3\right ]\right )}{\left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,1\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,3\right ]\right ) \left (\sqrt {y(x)}-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]\right )}} \sqrt {\frac {\left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,1\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]\right ) \left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,4\right ]\right ) \left (\sqrt {y(x)}-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,1\right ]\right ) \left (\sqrt {y(x)}-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,4\right ]\right )}{\left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,1\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,4\right ]\right ){}^2 \left (\sqrt {y(x)}-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]\right ){}^2}} \sqrt {y(x)}}{\left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,1\right ]\right ) \left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,4\right ]\right ) \sqrt {-\frac {1}{3} a y(x)^3-b y(x)^2+c_1 y(x)^{3/2}+c y(x)}}=x+c_2,y(x)\right ]\right \} \]
Maple: cpu = 1.716 (sec), leaf count = 87 \[ \left \{ \int ^{y \left ( x \right ) }\!-3\,{\frac {1}{\sqrt {9\,{\it \_C1}\,{{\it \_a}}^{3/2}-3\,a{{\it \_a}}^{3}-9\,b{{\it \_a}}^{2}+9\,c{ \it \_a}}}}{d{\it \_a}}-x-{\it \_C2}=0,\int ^{y \left ( x \right ) }\!3 \,{\frac {1}{\sqrt {9\,{\it \_C1}\,{{\it \_a}}^{3/2}-3\,a{{\it \_a}}^{ 3}-9\,b{{\it \_a}}^{2}+9\,c{\it \_a}}}}{d{\it \_a}}-x-{\it \_C2}=0,y \left ( x \right ) =0 \right \} \]