#### 2.523   ODE No. 523

$-a x y'(x)+x^3+y'(x)^3=0$ Mathematica : cpu = 145.347 (sec), leaf count = 392

$\left \{\left \{y(x)\to \int _1^x\left (\frac {\sqrt [3]{\frac {2}{3}} a K[1]}{\sqrt [3]{\sqrt {3} \sqrt {27 K[1]^6-4 a^3 K[1]^3}-9 K[1]^3}}+\frac {\sqrt [3]{\sqrt {3} \sqrt {27 K[1]^6-4 a^3 K[1]^3}-9 K[1]^3}}{\sqrt [3]{2} 3^{2/3}}\right )dK[1]+c_1\right \},\left \{y(x)\to \int _1^x\left (-\frac {\left (1+i \sqrt {3}\right ) a K[2]}{2^{2/3} \sqrt [3]{3} \sqrt [3]{\sqrt {3} \sqrt {27 K[2]^6-4 a^3 K[2]^3}-9 K[2]^3}}-\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{\sqrt {3} \sqrt {27 K[2]^6-4 a^3 K[2]^3}-9 K[2]^3}}{2 \sqrt [3]{2} 3^{2/3}}\right )dK[2]+c_1\right \},\left \{y(x)\to \int _1^x\left (-\frac {\left (1-i \sqrt {3}\right ) a K[3]}{2^{2/3} \sqrt [3]{3} \sqrt [3]{\sqrt {3} \sqrt {27 K[3]^6-4 a^3 K[3]^3}-9 K[3]^3}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {3} \sqrt {27 K[3]^6-4 a^3 K[3]^3}-9 K[3]^3}}{2 \sqrt [3]{2} 3^{2/3}}\right )dK[3]+c_1\right \}\right \}$ Maple : cpu = 0.116 (sec), leaf count = 231

$\left \{ y \left ( x \right ) =\int \!{i \left ( \left ( {\frac {i}{12}}-{\frac {\sqrt {3}}{12}} \right ) \left ( -108\,{x}^{3}+12\,\sqrt {-12\,{a}^{3}{x}^{3}+81\,{x}^{6}} \right ) ^{{\frac {2}{3}}}+a \left ( \sqrt {3}+i \right ) x \right ) {\frac {1}{\sqrt [3]{-108\,{x}^{3}+12\,\sqrt {-12\,{a}^{3}{x}^{3}+81\,{x}^{6}}}}}}\,{\rm d}x+{\it \_C1},y \left ( x \right ) =\int \!{i \left ( \left ( {\frac {i}{12}}+{\frac {\sqrt {3}}{12}} \right ) \left ( -108\,{x}^{3}+12\,\sqrt {-12\,{a}^{3}{x}^{3}+81\,{x}^{6}} \right ) ^{{\frac {2}{3}}}+a \left ( i-\sqrt {3} \right ) x \right ) {\frac {1}{\sqrt [3]{-108\,{x}^{3}+12\,\sqrt {-12\,{a}^{3}{x}^{3}+81\,{x}^{6}}}}}}\,{\rm d}x+{\it \_C1},y \left ( x \right ) =\int \!{\frac {1}{6} \left ( \left ( -108\,{x}^{3}+12\,\sqrt {-12\,{a}^{3}{x}^{3}+81\,{x}^{6}} \right ) ^{{\frac {2}{3}}}+12\,ax \right ) {\frac {1}{\sqrt [3]{-108\,{x}^{3}+12\,\sqrt {-12\,{a}^{3}{x}^{3}+81\,{x}^{6}}}}}}\,{\rm d}x+{\it \_C1} \right \}$