\[ a \sqrt [3]{y'(x)^3+1}+b x y'(x)-y(x)=0 \] ✓ Mathematica : cpu = 0.125057 (sec), leaf count = 84
DSolve[-y[x] + b*x*Derivative[1][y][x] + a*(1 + Derivative[1][y][x]^3)^(1/3) == 0,y[x],x]
\[\text {Solve}\left [\left \{x=K[1]^{\frac {b}{1-b}} \left (\frac {a \int \frac {K[1]^{\frac {2 b-1}{b-1}}}{\left (K[1]^3+1\right )^{2/3}}dK[1]}{1-b}+c_1\right ),y(x)=a \sqrt [3]{K[1]^3+1}+b x K[1]\right \},\{K[1],y(x)\}\right ]\] ✓ Maple : cpu = 0.184 (sec), leaf count = 3306
dsolve(a*(diff(y(x),x)^3+1)^(1/3)+b*x*diff(y(x),x)-y(x)=0,y(x))
\[\left (-\left (\int _{}^{\frac {2 b^{2} x^{2} y \left (x \right ) \left (-4 b^{6} x^{6}-8 a^{3} b^{3} x^{3}-4 b^{3} x^{3} y \left (x \right )^{3}+4 \sqrt {b^{6} x^{6}+2 a^{3} b^{3} x^{3}+2 b^{3} x^{3} y \left (x \right )^{3}+a^{6}-2 y \left (x \right )^{3} a^{3}+y \left (x \right )^{6}}\, b^{3} x^{3}-4 a^{6}+4 y \left (x \right )^{3} a^{3}+4 a^{3} \sqrt {b^{6} x^{6}+2 a^{3} b^{3} x^{3}+2 b^{3} x^{3} y \left (x \right )^{3}+a^{6}-2 y \left (x \right )^{3} a^{3}+y \left (x \right )^{6}}\right )^{\frac {1}{3}}-4 y \left (x \right )^{2} a^{2} b x +a \left (-4 b^{6} x^{6}-8 a^{3} b^{3} x^{3}-4 b^{3} x^{3} y \left (x \right )^{3}+4 \sqrt {b^{6} x^{6}+2 a^{3} b^{3} x^{3}+2 b^{3} x^{3} y \left (x \right )^{3}+a^{6}-2 y \left (x \right )^{3} a^{3}+y \left (x \right )^{6}}\, b^{3} x^{3}-4 a^{6}+4 y \left (x \right )^{3} a^{3}+4 a^{3} \sqrt {b^{6} x^{6}+2 a^{3} b^{3} x^{3}+2 b^{3} x^{3} y \left (x \right )^{3}+a^{6}-2 y \left (x \right )^{3} a^{3}+y \left (x \right )^{6}}\right )^{\frac {2}{3}}}{\left (-4 b^{6} x^{6}-8 a^{3} b^{3} x^{3}-4 b^{3} x^{3} y \left (x \right )^{3}+4 \sqrt {b^{6} x^{6}+2 a^{3} b^{3} x^{3}+2 b^{3} x^{3} y \left (x \right )^{3}+a^{6}-2 y \left (x \right )^{3} a^{3}+y \left (x \right )^{6}}\, b^{3} x^{3}-4 a^{6}+4 y \left (x \right )^{3} a^{3}+4 a^{3} \sqrt {b^{6} x^{6}+2 a^{3} b^{3} x^{3}+2 b^{3} x^{3} y \left (x \right )^{3}+a^{6}-2 y \left (x \right )^{3} a^{3}+y \left (x \right )^{6}}\right )^{\frac {1}{3}} \left (2 b^{3} x^{3}+2 a^{3}\right )}}-\frac {a \,\textit {\_a}^{1+\frac {b}{b -1}}}{\left (b -1\right ) \left (\textit {\_a}^{3}+1\right )^{\frac {2}{3}}}d \textit {\_a} \right )-c_{1}\right ) \left (\frac {2 b^{2} x^{2} y \left (x \right ) \left (-4 b^{6} x^{6}-8 a^{3} b^{3} x^{3}-4 b^{3} x^{3} y \left (x \right )^{3}+4 \sqrt {b^{6} x^{6}+2 a^{3} b^{3} x^{3}+2 b^{3} x^{3} y \left (x \right )^{3}+a^{6}-2 y \left (x \right )^{3} a^{3}+y \left (x \right )^{6}}\, b^{3} x^{3}-4 a^{6}+4 y \left (x \right )^{3} a^{3}+4 a^{3} \sqrt {b^{6} x^{6}+2 a^{3} b^{3} x^{3}+2 b^{3} x^{3} y \left (x \right )^{3}+a^{6}-2 y \left (x \right )^{3} a^{3}+y \left (x \right )^{6}}\right )^{\frac {1}{3}}-4 y \left (x \right )^{2} a^{2} b x +a \left (-4 b^{6} x^{6}-8 a^{3} b^{3} x^{3}-4 b^{3} x^{3} y \left (x \right )^{3}+4 \sqrt {b^{6} x^{6}+2 a^{3} b^{3} x^{3}+2 b^{3} x^{3} y \left (x \right )^{3}+a^{6}-2 y \left (x \right )^{3} a^{3}+y \left (x \right )^{6}}\, b^{3} x^{3}-4 a^{6}+4 y \left (x \right )^{3} a^{3}+4 a^{3} \sqrt {b^{6} x^{6}+2 a^{3} b^{3} x^{3}+2 b^{3} x^{3} y \left (x \right )^{3}+a^{6}-2 y \left (x \right )^{3} a^{3}+y \left (x \right )^{6}}\right )^{\frac {2}{3}}}{\left (-4 b^{6} x^{6}-8 a^{3} b^{3} x^{3}-4 b^{3} x^{3} y \left (x \right )^{3}+4 \sqrt {b^{6} x^{6}+2 a^{3} b^{3} x^{3}+2 b^{3} x^{3} y \left (x \right )^{3}+a^{6}-2 y \left (x \right )^{3} a^{3}+y \left (x \right )^{6}}\, b^{3} x^{3}-4 a^{6}+4 y \left (x \right )^{3} a^{3}+4 a^{3} \sqrt {b^{6} x^{6}+2 a^{3} b^{3} x^{3}+2 b^{3} x^{3} y \left (x \right )^{3}+a^{6}-2 y \left (x \right )^{3} a^{3}+y \left (x \right )^{6}}\right )^{\frac {1}{3}} \left (2 b^{3} x^{3}+2 a^{3}\right )}\right )^{-\frac {b}{b -1}}+x = 0\]