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 = 0.334977 (sec), leaf count = 46
\[\text {Solve}\left [\left \{\text {K$\$$259675} y(\text {K$\$$259675})+x=f^{(-1)}\left (\sqrt {\text {K$\$$259675}^2+1} y(\text {K$\$$259675})\right ),y(x)=\frac {e^{c_1}}{\sqrt {\text {K$\$$259675}^2+1}}\right \},\{y(x),\text {K$\$$259675}\}\right ]\]
Maple ✓
cpu = 0.435 (sec), leaf count = 45
\[ \left \{ [x \left ( {\it \_T} \right ) ={1 \left ( {\it RootOf} \left ( -f \left ( {\it \_Z} \right ) +{\it \_C1} \right ) \sqrt {{{\it \_T}}^{2}+1}-{\it \_C1}\,{\it \_T} \right ) {\frac {1}{\sqrt {{{\it \_T}}^{2}+1}}}},y \left ( {\it \_T} \right ) ={{\it \_C1}{\frac {1}{\sqrt {{{\it \_T}}^{2}+1}}}}] \right \} \] 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$259675*y[K$259675] == InverseFunction[f, 1, 1][Sqrt[1 + K$259675^2]*y[K$259675]], y[x] == E^C[1]/Sqrt[1 + K$259675^2]}, {y[x], K$259675}]
Maple raw input
dsolve(y(x)*(1+diff(y(x),x)^2)^(1/2) = f(y(x)*diff(y(x),x)+x), y(x),'implicit')
Maple raw output
[x(_T) = (RootOf(-f(_Z)+_C1)*(_T^2+1)^(1/2)-_C1*_T)/(_T^2+1)^(1/2), y(_T) = _C1/(_T^2+1)^(1/2)]