##### 4.24.22 $$f\left (y(x) y'(x)+x\right )=y(x)^2 \left (y'(x)^2+1\right )$$

ODE
$f\left (y(x) y'(x)+x\right )=y(x)^2 \left (y'(x)^2+1\right )$ ODE Classiﬁcation

[x=_G(y,y')]

Book solution method
Change of variable

Mathematica
cpu = 0.294762 (sec), leaf count = 48

$\text {Solve}\left [\left \{K[1] y(K[1])+x=f^{(-1)}\left (\left (K[1]^2+1\right ) y(K[1])^2\right ),y(x)=\frac {c_1}{\sqrt {K[1]^2+1}}\right \},\{y(x),K[1]\}\right ]$

Maple
cpu = 2.267 (sec), leaf count = 38

$[y \left (x \right )-\RootOf \left (-\textit {\_C1} +\textit {\_Z}^{2}+x^{2}-2 x \RootOf \left (-f \left (\textit {\_Z} \right )+\textit {\_C1} \right )+\RootOf \left (-f \left (\textit {\_Z} \right )+\textit {\_C1} \right )^{2}\right ) = 0]$ Mathematica raw input

DSolve[f[x + y[x]*y'[x]] == y[x]^2*(1 + y'[x]^2),y[x],x]

Mathematica raw output

Solve[{x + K[1]*y[K[1]] == InverseFunction[f, 1, 1][(1 + K[1]^2)*y[K[1]]^2], y[x
] == C[1]/Sqrt[1 + K[1]^2]}, {y[x], K[1]}]

Maple raw input

dsolve(f(y(x)*diff(y(x),x)+x) = (1+diff(y(x),x)^2)*y(x)^2, y(x))

Maple raw output

[y(x)-RootOf(-_C1+_Z^2+x^2-2*x*RootOf(-f(_Z)+_C1)+RootOf(-f(_Z)+_C1)^2) = 0]