\[ \boxed { \left ( 2\,a \left ( y \left ( x \right ) \right ) ^{3}+3\,ax \left ( y \left ( x \right ) \right ) ^{2}-b{x}^{3}+c{x}^{2} \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) -a \left ( y \left ( x \right ) \right ) ^{3}+c \left ( y \left ( x \right ) \right ) ^{2}+3\,b{x}^{2}y \left ( x \right ) +2\,b{x}^{3}=0} \]
Mathematica: cpu = 0.069009 (sec), leaf count = 537 \[ \left \{\left \{y(x)\to \frac {\sqrt [3]{2} \left (3 a c x+3 a c_1\right )}{3 a \sqrt [3]{\sqrt {\left (27 a^2 b x^3+27 a^2 c_1 x\right ){}^2+4 \left (3 a c x+3 a c_1\right ){}^3}+27 a^2 b x^3+27 a^2 c_1 x}}-\frac {\sqrt [3]{\sqrt {\left (27 a^2 b x^3+27 a^2 c_1 x\right ){}^2+4 \left (3 a c x+3 a c_1\right ){}^3}+27 a^2 b x^3+27 a^2 c_1 x}}{3 \sqrt [3]{2} a}\right \},\left \{y(x)\to \frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{\sqrt {\left (27 a^2 b x^3+27 a^2 c_1 x\right ){}^2+4 \left (3 a c x+3 a c_1\right ){}^3}+27 a^2 b x^3+27 a^2 c_1 x}}{6 \sqrt [3]{2} a}-\frac {\left (1+i \sqrt {3}\right ) \left (3 a c x+3 a c_1\right )}{3\ 2^{2/3} a \sqrt [3]{\sqrt {\left (27 a^2 b x^3+27 a^2 c_1 x\right ){}^2+4 \left (3 a c x+3 a c_1\right ){}^3}+27 a^2 b x^3+27 a^2 c_1 x}}\right \},\left \{y(x)\to \frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {\left (27 a^2 b x^3+27 a^2 c_1 x\right ){}^2+4 \left (3 a c x+3 a c_1\right ){}^3}+27 a^2 b x^3+27 a^2 c_1 x}}{6 \sqrt [3]{2} a}-\frac {\left (1-i \sqrt {3}\right ) \left (3 a c x+3 a c_1\right )}{3\ 2^{2/3} a \sqrt [3]{\sqrt {\left (27 a^2 b x^3+27 a^2 c_1 x\right ){}^2+4 \left (3 a c x+3 a c_1\right ){}^3}+27 a^2 b x^3+27 a^2 c_1 x}}\right \}\right \} \]
Maple: cpu = 0.156 (sec), leaf count = 912 \[ \left \{ y \left ( x \right ) ={\frac {1}{6\,a}\sqrt [3]{ \left ( -108\,b {x}^{3}+108\,{\it \_C1}\,x+12\,\sqrt {-{\frac {-81\,a{b}^{2}{x}^{6}+ 162\,{\it \_C1}\,ab{x}^{4}-12\,{c}^{3}{x}^{3}-81\,{{\it \_C1}}^{2}a{x} ^{2}+36\,{\it \_C1}\,{c}^{2}{x}^{2}-36\,{{\it \_C1}}^{2}cx+12\,{{\it \_C1}}^{3}}{a}}} \right ) {a}^{2}}}+2\,{(-cx+{\it \_C1}){\frac {1}{ \sqrt [3]{ \left ( -108\,b{x}^{3}+108\,{\it \_C1}\,x+12\,\sqrt {-{ \frac {-81\,a{b}^{2}{x}^{6}+162\,{\it \_C1}\,ab{x}^{4}-12\,{c}^{3}{x}^ {3}-81\,{{\it \_C1}}^{2}a{x}^{2}+36\,{\it \_C1}\,{c}^{2}{x}^{2}-36\,{{ \it \_C1}}^{2}cx+12\,{{\it \_C1}}^{3}}{a}}} \right ) {a}^{2}}}}},y \left ( x \right ) =-{\frac {1}{12\,a}\sqrt [3]{ \left ( -108\,b{x}^{3}+ 108\,{\it \_C1}\,x+12\,\sqrt {-{\frac {-81\,a{b}^{2}{x}^{6}+162\,{\it \_C1}\,ab{x}^{4}-12\,{c}^{3}{x}^{3}-81\,{{\it \_C1}}^{2}a{x}^{2}+36\,{ \it \_C1}\,{c}^{2}{x}^{2}-36\,{{\it \_C1}}^{2}cx+12\,{{\it \_C1}}^{3} }{a}}} \right ) {a}^{2}}}-{(-cx+{\it \_C1}){\frac {1}{\sqrt [3]{ \left ( -108\,b{x}^{3}+108\,{\it \_C1}\,x+12\,\sqrt {-{\frac {-81\,a{b }^{2}{x}^{6}+162\,{\it \_C1}\,ab{x}^{4}-12\,{c}^{3}{x}^{3}-81\,{{\it \_C1}}^{2}a{x}^{2}+36\,{\it \_C1}\,{c}^{2}{x}^{2}-36\,{{\it \_C1}}^{2} cx+12\,{{\it \_C1}}^{3}}{a}}} \right ) {a}^{2}}}}}-{\frac {i}{2}}\sqrt {3} \left ( {\frac {1}{6\,a}\sqrt [3]{ \left ( -108\,b{x}^{3}+108\,{\it \_C1}\,x+12\,\sqrt {-{\frac {-81\,a{b}^{2}{x}^{6}+162\,{\it \_C1}\,ab{ x}^{4}-12\,{c}^{3}{x}^{3}-81\,{{\it \_C1}}^{2}a{x}^{2}+36\,{\it \_C1} \,{c}^{2}{x}^{2}-36\,{{\it \_C1}}^{2}cx+12\,{{\it \_C1}}^{3}}{a}}} \right ) {a}^{2}}}-2\,{(-cx+{\it \_C1}){\frac {1}{\sqrt [3]{ \left ( - 108\,b{x}^{3}+108\,{\it \_C1}\,x+12\,\sqrt {-{\frac {-81\,a{b}^{2}{x}^ {6}+162\,{\it \_C1}\,ab{x}^{4}-12\,{c}^{3}{x}^{3}-81\,{{\it \_C1}}^{2} a{x}^{2}+36\,{\it \_C1}\,{c}^{2}{x}^{2}-36\,{{\it \_C1}}^{2}cx+12\,{{ \it \_C1}}^{3}}{a}}} \right ) {a}^{2}}}}} \right ) ,y \left ( x \right ) = -{\frac {1}{12\,a}\sqrt [3]{ \left ( -108\,b{x}^{3}+108\,{\it \_C1}\,x+ 12\,\sqrt {-{\frac {-81\,a{b}^{2}{x}^{6}+162\,{\it \_C1}\,ab{x}^{4}-12 \,{c}^{3}{x}^{3}-81\,{{\it \_C1}}^{2}a{x}^{2}+36\,{\it \_C1}\,{c}^{2}{ x}^{2}-36\,{{\it \_C1}}^{2}cx+12\,{{\it \_C1}}^{3}}{a}}} \right ) {a}^{ 2}}}-{(-cx+{\it \_C1}){\frac {1}{\sqrt [3]{ \left ( -108\,b{x}^{3}+108 \,{\it \_C1}\,x+12\,\sqrt {-{\frac {-81\,a{b}^{2}{x}^{6}+162\,{\it \_C1}\,ab{x}^{4}-12\,{c}^{3}{x}^{3}-81\,{{\it \_C1}}^{2}a{x}^{2}+36\,{ \it \_C1}\,{c}^{2}{x}^{2}-36\,{{\it \_C1}}^{2}cx+12\,{{\it \_C1}}^{3} }{a}}} \right ) {a}^{2}}}}}+{\frac {i}{2}}\sqrt {3} \left ( {\frac {1}{6 \,a}\sqrt [3]{ \left ( -108\,b{x}^{3}+108\,{\it \_C1}\,x+12\,\sqrt {-{ \frac {-81\,a{b}^{2}{x}^{6}+162\,{\it \_C1}\,ab{x}^{4}-12\,{c}^{3}{x}^ {3}-81\,{{\it \_C1}}^{2}a{x}^{2}+36\,{\it \_C1}\,{c}^{2}{x}^{2}-36\,{{ \it \_C1}}^{2}cx+12\,{{\it \_C1}}^{3}}{a}}} \right ) {a}^{2}}}-2\,{(-cx +{\it \_C1}){\frac {1}{\sqrt [3]{ \left ( -108\,b{x}^{3}+108\,{\it \_C1 }\,x+12\,\sqrt {-{\frac {-81\,a{b}^{2}{x}^{6}+162\,{\it \_C1}\,ab{x}^{ 4}-12\,{c}^{3}{x}^{3}-81\,{{\it \_C1}}^{2}a{x}^{2}+36\,{\it \_C1}\,{c} ^{2}{x}^{2}-36\,{{\it \_C1}}^{2}cx+12\,{{\it \_C1}}^{3}}{a}}} \right ) {a}^{2}}}}} \right ) \right \} \]