ODE
\[ f(x) y'(x)+g(y(x)) y'(x)^2+y''(x)=0 \] ODE Classification
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✓
cpu = 1.97088 (sec), leaf count = 57
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}} e^{-\int _1^{K[4]} -g(K[1]) \, dK[1]} \, dK[4]\& \right ]\left [\int _1^x c_1 \left (-e^{-\int _1^{K[5]} f(K[2]) \, dK[2]}\right ) \, dK[5]+c_2\right ]\right \}\right \}\]
Maple ✓
cpu = 0.029 (sec), leaf count = 29
\[ \left \{ \int ^{y \left ( x \right ) }\!{{\rm e}^{\int \!g \left ( {\it \_b} \right ) \,{\rm d}{\it \_b}}}{d{\it \_b}}-{\it \_C1}\,\int \!{{\rm e}^{-\int \!f \left ( x \right ) \,{\rm d}x}}\,{\rm d}x-{\it \_C2}=0 \right \} \] Mathematica raw input
DSolve[f[x]*y'[x] + g[y[x]]*y'[x]^2 + y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[Integrate[E^(-Integrate[-g[K[1]], {K[1], 1, K[4]}]), {K[4], 1, #1}] & ][C[2] + Integrate[-(C[1]/E^Integrate[f[K[2]], {K[2], 1, K[5]}]), {K[5], 1, x}]]}}
Maple raw input
dsolve(diff(diff(y(x),x),x)+g(y(x))*diff(y(x),x)^2+f(x)*diff(y(x),x) = 0, y(x),'implicit')
Maple raw output
Intat(exp(Int(g(_b),_b)),_b = y(x))-_C1*Int(exp(-Int(f(x),x)),x)-_C2 = 0