\[ -a x y'(x)+x^3+y'(x)^3=0 \] ✓ Mathematica : cpu = 145.263 (sec), leaf count = 392
DSolve[x^3 - a*x*Derivative[1][y][x] + Derivative[1][y][x]^3 == 0,y[x],x]
\[\left \{\left \{y(x)\to \int _1^x\left (\frac {\sqrt [3]{\frac {2}{3}} a K[1]}{\sqrt [3]{\sqrt {3} \sqrt {27 K[1]^6-4 a^3 K[1]^3}-9 K[1]^3}}+\frac {\sqrt [3]{\sqrt {3} \sqrt {27 K[1]^6-4 a^3 K[1]^3}-9 K[1]^3}}{\sqrt [3]{2} 3^{2/3}}\right )dK[1]+c_1\right \},\left \{y(x)\to \int _1^x\left (-\frac {\left (1+i \sqrt {3}\right ) a K[2]}{2^{2/3} \sqrt [3]{3} \sqrt [3]{\sqrt {3} \sqrt {27 K[2]^6-4 a^3 K[2]^3}-9 K[2]^3}}-\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{\sqrt {3} \sqrt {27 K[2]^6-4 a^3 K[2]^3}-9 K[2]^3}}{2 \sqrt [3]{2} 3^{2/3}}\right )dK[2]+c_1\right \},\left \{y(x)\to \int _1^x\left (-\frac {\left (1-i \sqrt {3}\right ) a K[3]}{2^{2/3} \sqrt [3]{3} \sqrt [3]{\sqrt {3} \sqrt {27 K[3]^6-4 a^3 K[3]^3}-9 K[3]^3}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {3} \sqrt {27 K[3]^6-4 a^3 K[3]^3}-9 K[3]^3}}{2 \sqrt [3]{2} 3^{2/3}}\right )dK[3]+c_1\right \}\right \}\] ✓ Maple : cpu = 0.067 (sec), leaf count = 231
dsolve(diff(y(x),x)^3-a*x*diff(y(x),x)+x^3=0,y(x))
\[y \left (x \right ) = \int \frac {i \left (\left (\frac {i}{12}+\frac {\sqrt {3}}{12}\right ) \left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{\frac {2}{3}}+\left (i-\sqrt {3}\right ) a x \right )}{\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{\frac {1}{3}}}d x +c_{1}\]