\[ \left (x^2+y(x)^2\right ) f\left (\frac {x}{\sqrt {x^2+y(x)^2}}\right ) \left (y'(x)^2+1\right )-\left (x y'(x)-y(x)\right )^2=0 \] ✓ Mathematica : cpu = 0.523367 (sec), leaf count = 253
DSolve[-(-y[x] + x*Derivative[1][y][x])^2 + f[x/Sqrt[x^2 + y[x]^2]]*(x^2 + y[x]^2)*(1 + Derivative[1][y][x]^2) == 0,y[x],x]
\[\left \{\text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {f\left (\frac {1}{\sqrt {K[1]^2+1}}\right ) K[1]^2+f\left (\frac {1}{\sqrt {K[1]^2+1}}\right )-1}{\sqrt {f\left (\frac {1}{\sqrt {K[1]^2+1}}\right )} (K[1]-i) (K[1]+i) \left (\sqrt {f\left (\frac {1}{\sqrt {K[1]^2+1}}\right )} K[1]+i \sqrt {f\left (\frac {1}{\sqrt {K[1]^2+1}}\right )-1}\right )}dK[1]=-\log (x)+c_1,y(x)\right ],\text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {f\left (\frac {1}{\sqrt {K[2]^2+1}}\right ) K[2]^2+f\left (\frac {1}{\sqrt {K[2]^2+1}}\right )-1}{\sqrt {f\left (\frac {1}{\sqrt {K[2]^2+1}}\right )} (K[2]-i) (K[2]+i) \left (\sqrt {f\left (\frac {1}{\sqrt {K[2]^2+1}}\right )} K[2]-i \sqrt {f\left (\frac {1}{\sqrt {K[2]^2+1}}\right )-1}\right )}dK[2]=-\log (x)+c_1,y(x)\right ]\right \}\] ✓ Maple : cpu = 1.023 (sec), leaf count = 70
dsolve((y(x)^2+x^2)*f(x/(y(x)^2+x^2)^(1/2))*(diff(y(x),x)^2+1)-(x*diff(y(x),x)-y(x))^2=0,y(x))
\[y \left (x \right ) = \RootOf \left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}-\frac {\textit {\_a} f \left (\frac {1}{\sqrt {\textit {\_a}^{2}+1}}\right )+\sqrt {-f \left (\frac {1}{\sqrt {\textit {\_a}^{2}+1}}\right )^{2}+f \left (\frac {1}{\sqrt {\textit {\_a}^{2}+1}}\right )}}{\left (\textit {\_a}^{2}+1\right ) f \left (\frac {1}{\sqrt {\textit {\_a}^{2}+1}}\right )}d \textit {\_a} +c_{1}\right ) x\]