#### 2.323   ODE No. 323

$x y'(x) \left (a x y(x)^3+c\right )+y(x) \left (b x^3 y(x)+c\right )=0$ Mathematica : cpu = 0.250908 (sec), leaf count = 463

$\left \{\left \{y(x)\to \frac {\sqrt [3]{54 a^2 c x^2+\sqrt {2916 a^4 c^2 x^4+108 a^3 x^3 \left (b x^3-2 c_1 x\right ){}^3}}}{3 \sqrt [3]{2} a x}-\frac {\sqrt [3]{2} \left (b x^3-2 c_1 x\right )}{\sqrt [3]{54 a^2 c x^2+\sqrt {2916 a^4 c^2 x^4+108 a^3 x^3 \left (b x^3-2 c_1 x\right ){}^3}}}\right \},\left \{y(x)\to \frac {\left (1+i \sqrt {3}\right ) \left (b x^3-2 c_1 x\right )}{2^{2/3} \sqrt [3]{54 a^2 c x^2+\sqrt {2916 a^4 c^2 x^4+108 a^3 x^3 \left (b x^3-2 c_1 x\right ){}^3}}}-\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{54 a^2 c x^2+\sqrt {2916 a^4 c^2 x^4+108 a^3 x^3 \left (b x^3-2 c_1 x\right ){}^3}}}{6 \sqrt [3]{2} a x}\right \},\left \{y(x)\to \frac {\left (1-i \sqrt {3}\right ) \left (b x^3-2 c_1 x\right )}{2^{2/3} \sqrt [3]{54 a^2 c x^2+\sqrt {2916 a^4 c^2 x^4+108 a^3 x^3 \left (b x^3-2 c_1 x\right ){}^3}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{54 a^2 c x^2+\sqrt {2916 a^4 c^2 x^4+108 a^3 x^3 \left (b x^3-2 c_1 x\right ){}^3}}}{6 \sqrt [3]{2} a x}\right \}\right \}$ Maple : cpu = 0.094 (sec), leaf count = 630

$\left \{ y \left ( x \right ) =-{\frac {{3}^{{\frac {2}{3}}}}{18\,ax} \left ( \left ( 3\,i \left ( b{x}^{2}-2\,{\it \_C1} \right ) {x}^{2}a+i \left ( \left ( 27\,c+3\,\sqrt {{\frac {3\,{b}^{3}{x}^{8}-18\,{\it \_C1}\,{b}^{2}{x}^{6}+36\,{{\it \_C1}}^{2}b{x}^{4}-24\,{{\it \_C1}}^{3}{x}^{2}+81\,{c}^{2}a}{a}}} \right ) {a}^{2}{x}^{2} \right ) ^{{\frac {2}{3}}} \right ) \sqrt {3}+ \left ( -3\,{x}^{4}b+6\,{\it \_C1}\,{x}^{2} \right ) a+ \left ( \left ( 27\,c+3\,\sqrt {{\frac {3\,{b}^{3}{x}^{8}-18\,{\it \_C1}\,{b}^{2}{x}^{6}+36\,{{\it \_C1}}^{2}b{x}^{4}-24\,{{\it \_C1}}^{3}{x}^{2}+81\,{c}^{2}a}{a}}} \right ) {a}^{2}{x}^{2} \right ) ^{{\frac {2}{3}}} \right ) {\frac {1}{\sqrt [3]{ \left ( 9\,c+\sqrt {{\frac {3\,{b}^{3}{x}^{8}-18\,{\it \_C1}\,{b}^{2}{x}^{6}+36\,{{\it \_C1}}^{2}b{x}^{4}-24\,{{\it \_C1}}^{3}{x}^{2}+81\,{c}^{2}a}{a}}} \right ) {a}^{2}{x}^{2}}}}},y \left ( x \right ) ={\frac {{3}^{{\frac {2}{3}}}}{18\,ax} \left ( \left ( 3\,i \left ( b{x}^{2}-2\,{\it \_C1} \right ) {x}^{2}a+i \left ( \left ( 27\,c+3\,\sqrt {{\frac {3\,{b}^{3}{x}^{8}-18\,{\it \_C1}\,{b}^{2}{x}^{6}+36\,{{\it \_C1}}^{2}b{x}^{4}-24\,{{\it \_C1}}^{3}{x}^{2}+81\,{c}^{2}a}{a}}} \right ) {a}^{2}{x}^{2} \right ) ^{{\frac {2}{3}}} \right ) \sqrt {3}+ \left ( 3\,{x}^{4}b-6\,{\it \_C1}\,{x}^{2} \right ) a- \left ( \left ( 27\,c+3\,\sqrt {{\frac {3\,{b}^{3}{x}^{8}-18\,{\it \_C1}\,{b}^{2}{x}^{6}+36\,{{\it \_C1}}^{2}b{x}^{4}-24\,{{\it \_C1}}^{3}{x}^{2}+81\,{c}^{2}a}{a}}} \right ) {a}^{2}{x}^{2} \right ) ^{{\frac {2}{3}}} \right ) {\frac {1}{\sqrt [3]{ \left ( 9\,c+\sqrt {{\frac {3\,{b}^{3}{x}^{8}-18\,{\it \_C1}\,{b}^{2}{x}^{6}+36\,{{\it \_C1}}^{2}b{x}^{4}-24\,{{\it \_C1}}^{3}{x}^{2}+81\,{c}^{2}a}{a}}} \right ) {a}^{2}{x}^{2}}}}},y \left ( x \right ) ={\frac {{3}^{{\frac {2}{3}}}}{9\,ax} \left ( -3\,ab{x}^{4}+6\,{\it \_C1}\,a{x}^{2}+ \left ( \left ( 27\,c+3\,\sqrt {{\frac {3\,{b}^{3}{x}^{8}-18\,{\it \_C1}\,{b}^{2}{x}^{6}+36\,{{\it \_C1}}^{2}b{x}^{4}-24\,{{\it \_C1}}^{3}{x}^{2}+81\,{c}^{2}a}{a}}} \right ) {a}^{2}{x}^{2} \right ) ^{{\frac {2}{3}}} \right ) {\frac {1}{\sqrt [3]{ \left ( 9\,c+\sqrt {{\frac {3\,{b}^{3}{x}^{8}-18\,{\it \_C1}\,{b}^{2}{x}^{6}+36\,{{\it \_C1}}^{2}b{x}^{4}-24\,{{\it \_C1}}^{3}{x}^{2}+81\,{c}^{2}a}{a}}} \right ) {a}^{2}{x}^{2}}}}} \right \}$