ODE
\[ y(x) \sqrt {y'(x)^2+1}=f\left (y(x) y'(x)+x\right ) \] ODE Classification
[`x=_G(y,y')`]
Book solution method
The method of Lagrange
Mathematica ✓
cpu = 0.450167 (sec), leaf count = 52
\[\text {Solve}\left [\left \{K[1] y(K[1])+x=f^{(-1)}\left (\sqrt {K[1]^2+1} y(K[1])\right ),y(x)=\frac {e^{c_1}}{\sqrt {K[1]^2+1}}\right \},\{y(x),K[1]\}\right ]\]
Maple ✓
cpu = 4.037 (sec), leaf count = 40
\[[y \left (x \right )-\RootOf \left (-\textit {\_C1}^{2}+\textit {\_Z}^{2}+x^{2}-2 x \RootOf \left (-f \left (\textit {\_Z} \right )+\textit {\_C1} \right )+\RootOf \left (-f \left (\textit {\_Z} \right )+\textit {\_C1} \right )^{2}\right ) = 0]\] Mathematica raw input
DSolve[y[x]*Sqrt[1 + y'[x]^2] == f[x + y[x]*y'[x]],y[x],x]
Mathematica raw output
Solve[{x + K[1]*y[K[1]] == InverseFunction[f, 1, 1][Sqrt[1 + K[1]^2]*y[K[1]]], y[x] == E^C[1]/Sqrt[1 + K[1]^2]}, {y[x], K[1]}]
Maple raw input
dsolve(y(x)*(1+diff(y(x),x)^2)^(1/2) = f(y(x)*diff(y(x),x)+x), y(x))
Maple raw output
[y(x)-RootOf(-_C1^2+_Z^2+x^2-2*x*RootOf(-f(_Z)+_C1)+RootOf(-f(_Z)+_C1)^2) = 0]