ODE
\[ y'(x)^3 (a x+b y(x))+y''(x)=0 \] ODE Classification
[[_2nd_order, _with_exponential_symmetries], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]
Book solution method
TO DO
Mathematica ✓
cpu = 1.87669 (sec), leaf count = 154
\[\left \{\text {Solve}\left [\frac {b \log \left (a^2 x+\sqrt {a} \sqrt {a^3 x^2+2 a^2 b x y(x)+b^2 \left (-2 a c_1+a y(x)^2+1\right )}+a b y(x)\right )}{\sqrt {a}}+a c_2+b y(x)=0,y(x)\right ],\text {Solve}\left [a c_2+b y(x)=\frac {b \log \left (a^2 x+\sqrt {a} \sqrt {a^3 x^2+2 a^2 b x y(x)+b^2 \left (-2 a c_1+a y(x)^2+1\right )}+a b y(x)\right )}{\sqrt {a}},y(x)\right ]\right \}\]
Maple ✓
cpu = 0.123 (sec), leaf count = 44
\[ \left \{ -2\,\sqrt {a}{{\rm e}^{\sqrt {a}y \left ( x \right ) }}x-2\,{\frac {{{\rm e}^{\sqrt {a}y \left ( x \right ) }}by \left ( x \right ) }{\sqrt {a}}}-{{\rm e}^{2\,\sqrt {a}y \left ( x \right ) }}{\it \_C2}+{\it \_C1}=0 \right \} \] Mathematica raw input
DSolve[(a*x + b*y[x])*y'[x]^3 + y''[x] == 0,y[x],x]
Mathematica raw output
{Solve[a*C[2] + (b*Log[a^2*x + a*b*y[x] + Sqrt[a]*Sqrt[a^3*x^2 + 2*a^2*b*x*y[x] + b^2*(1 - 2*a*C[1] + a*y[x]^2)]])/Sqrt[a] + b*y[x] == 0, y[x]], Solve[a*C[2] + b*y[x] == (b*Log[a^2*x + a*b*y[x] + Sqrt[a]*Sqrt[a^3*x^2 + 2*a^2*b*x*y[x] + b^2*(1 - 2*a*C[1] + a*y[x]^2)]])/Sqrt[a], y[x]]}
Maple raw input
dsolve(diff(diff(y(x),x),x)+(a*x+b*y(x))*diff(y(x),x)^3 = 0, y(x),'implicit')
Maple raw output
-2*a^(1/2)*exp(a^(1/2)*y(x))*x-2/a^(1/2)*exp(a^(1/2)*y(x))*b*y(x)-exp(2*a^(1/2)*y(x))*_C2+_C1 = 0