#### 2.313   ODE No. 313

$y'(x) \left (3 a x y(x)^2+2 a y(x)^3-b x^3+c x^2\right )-a y(x)^3+2 b x^3+3 b x^2 y(x)+c y(x)^2=0$ Mathematica : cpu = 0.319681 (sec), leaf count = 537

$\left \{\left \{y(x)\to \frac {\sqrt [3]{2} (3 a c x+3 a c_1)}{3 a \sqrt [3]{\sqrt {\left (27 a^2 b x^3+27 a^2 c_1 x\right ){}^2+4 (3 a c x+3 a c_1){}^3}+27 a^2 b x^3+27 a^2 c_1 x}}-\frac {\sqrt [3]{\sqrt {\left (27 a^2 b x^3+27 a^2 c_1 x\right ){}^2+4 (3 a c x+3 a c_1){}^3}+27 a^2 b x^3+27 a^2 c_1 x}}{3 \sqrt [3]{2} a}\right \},\left \{y(x)\to \frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{\sqrt {\left (27 a^2 b x^3+27 a^2 c_1 x\right ){}^2+4 (3 a c x+3 a c_1){}^3}+27 a^2 b x^3+27 a^2 c_1 x}}{6 \sqrt [3]{2} a}-\frac {\left (1+i \sqrt {3}\right ) (3 a c x+3 a c_1)}{3\ 2^{2/3} a \sqrt [3]{\sqrt {\left (27 a^2 b x^3+27 a^2 c_1 x\right ){}^2+4 (3 a c x+3 a c_1){}^3}+27 a^2 b x^3+27 a^2 c_1 x}}\right \},\left \{y(x)\to \frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {\left (27 a^2 b x^3+27 a^2 c_1 x\right ){}^2+4 (3 a c x+3 a c_1){}^3}+27 a^2 b x^3+27 a^2 c_1 x}}{6 \sqrt [3]{2} a}-\frac {\left (1-i \sqrt {3}\right ) (3 a c x+3 a c_1)}{3\ 2^{2/3} a \sqrt [3]{\sqrt {\left (27 a^2 b x^3+27 a^2 c_1 x\right ){}^2+4 (3 a c x+3 a c_1){}^3}+27 a^2 b x^3+27 a^2 c_1 x}}\right \}\right \}$ Maple : cpu = 0.155 (sec), leaf count = 748

$\left \{ y \left ( x \right ) ={\frac {1}{6\,a} \left ( \left ( -12\,cx+12\,{\it \_C1} \right ) a+ \left ( \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 ) ^{{\frac {2}{3}}} \right ) {\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} \left ( \left ( \left ( 12\,icx-12\,i{\it \_C1} \right ) a+i \left ( \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 ) ^{{\frac {2}{3}}} \right ) \sqrt {3}+ \left ( -12\,cx+12\,{\it \_C1} \right ) a+ \left ( \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 ) ^{{\frac {2}{3}}} \right ) {\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} \left ( \left ( \left ( 12\,icx-12\,i{\it \_C1} \right ) a+i \left ( \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 ) ^{{\frac {2}{3}}} \right ) \sqrt {3}+ \left ( 12\,cx-12\,{\it \_C1} \right ) a- \left ( \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 ) ^{{\frac {2}{3}}} \right ) {\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 \}$