\[ f\left (x^2+y(x)^2\right ) \left (y'(x)^2+1\right )-\left (x y'(x)-y(x)\right )^2=0 \] ✓ Mathematica : cpu = 1.88204 (sec), leaf count = 1922
DSolve[-(-y[x] + x*Derivative[1][y][x])^2 + f[x^2 + y[x]^2]*(1 + Derivative[1][y][x]^2) == 0,y[x],x]
\[\left \{\text {Solve}\left [\int _1^x\left (\frac {\sqrt {f\left (K[1]^2+y(x)^2\right ) \left (K[1]^2+y(x)^2-f\left (K[1]^2+y(x)^2\right )\right )} K[1]}{f\left (K[1]^2+y(x)^2\right ) \left (K[1]^2+y(x)^2\right )}-\frac {\sqrt {f\left (K[1]^2+y(x)^2\right ) \left (K[1]^2+y(x)^2-f\left (K[1]^2+y(x)^2\right )\right )} K[1]}{f\left (K[1]^2+y(x)^2\right ) \left (K[1]^2+y(x)^2-f\left (K[1]^2+y(x)^2\right )\right )}+\frac {y(x)}{K[1]^2+y(x)^2}\right )dK[1]+\int _1^{y(x)}\left (-\frac {x}{x^2+K[2]^2}-\int _1^x\left (-\frac {2 K[2]^2}{\left (K[1]^2+K[2]^2\right )^2}-\frac {2 K[1] \sqrt {f\left (K[1]^2+K[2]^2\right ) \left (K[1]^2+K[2]^2-f\left (K[1]^2+K[2]^2\right )\right )} f'\left (K[1]^2+K[2]^2\right ) K[2]}{f\left (K[1]^2+K[2]^2\right )^2 \left (K[1]^2+K[2]^2\right )}+\frac {2 K[1] \sqrt {f\left (K[1]^2+K[2]^2\right ) \left (K[1]^2+K[2]^2-f\left (K[1]^2+K[2]^2\right )\right )} f'\left (K[1]^2+K[2]^2\right ) K[2]}{f\left (K[1]^2+K[2]^2\right )^2 \left (K[1]^2+K[2]^2-f\left (K[1]^2+K[2]^2\right )\right )}-\frac {2 K[1] \sqrt {f\left (K[1]^2+K[2]^2\right ) \left (K[1]^2+K[2]^2-f\left (K[1]^2+K[2]^2\right )\right )} K[2]}{f\left (K[1]^2+K[2]^2\right ) \left (K[1]^2+K[2]^2\right )^2}+\frac {K[1] \sqrt {f\left (K[1]^2+K[2]^2\right ) \left (K[1]^2+K[2]^2-f\left (K[1]^2+K[2]^2\right )\right )} \left (2 K[2]-2 K[2] f'\left (K[1]^2+K[2]^2\right )\right )}{f\left (K[1]^2+K[2]^2\right ) \left (K[1]^2+K[2]^2-f\left (K[1]^2+K[2]^2\right )\right )^2}+\frac {K[1] \left (2 K[2] \left (K[1]^2+K[2]^2-f\left (K[1]^2+K[2]^2\right )\right ) f'\left (K[1]^2+K[2]^2\right )+f\left (K[1]^2+K[2]^2\right ) \left (2 K[2]-2 K[2] f'\left (K[1]^2+K[2]^2\right )\right )\right )}{2 f\left (K[1]^2+K[2]^2\right ) \left (K[1]^2+K[2]^2\right ) \sqrt {f\left (K[1]^2+K[2]^2\right ) \left (K[1]^2+K[2]^2-f\left (K[1]^2+K[2]^2\right )\right )}}-\frac {K[1] \left (2 K[2] \left (K[1]^2+K[2]^2-f\left (K[1]^2+K[2]^2\right )\right ) f'\left (K[1]^2+K[2]^2\right )+f\left (K[1]^2+K[2]^2\right ) \left (2 K[2]-2 K[2] f'\left (K[1]^2+K[2]^2\right )\right )\right )}{2 f\left (K[1]^2+K[2]^2\right ) \left (K[1]^2+K[2]^2-f\left (K[1]^2+K[2]^2\right )\right ) \sqrt {f\left (K[1]^2+K[2]^2\right ) \left (K[1]^2+K[2]^2-f\left (K[1]^2+K[2]^2\right )\right )}}+\frac {1}{K[1]^2+K[2]^2}\right )dK[1]+\frac {K[2] \sqrt {f\left (x^2+K[2]^2\right ) \left (x^2+K[2]^2-f\left (x^2+K[2]^2\right )\right )}}{f\left (x^2+K[2]^2\right ) \left (x^2+K[2]^2\right )}-\frac {K[2] \sqrt {f\left (x^2+K[2]^2\right ) \left (x^2+K[2]^2-f\left (x^2+K[2]^2\right )\right )}}{f\left (x^2+K[2]^2\right ) \left (x^2+K[2]^2-f\left (x^2+K[2]^2\right )\right )}\right )dK[2]=c_1,y(x)\right ],\text {Solve}\left [\int _1^x\left (-\frac {\sqrt {f\left (K[3]^2+y(x)^2\right ) \left (K[3]^2+y(x)^2-f\left (K[3]^2+y(x)^2\right )\right )} K[3]}{f\left (K[3]^2+y(x)^2\right ) \left (K[3]^2+y(x)^2\right )}+\frac {\sqrt {f\left (K[3]^2+y(x)^2\right ) \left (K[3]^2+y(x)^2-f\left (K[3]^2+y(x)^2\right )\right )} K[3]}{f\left (K[3]^2+y(x)^2\right ) \left (K[3]^2+y(x)^2-f\left (K[3]^2+y(x)^2\right )\right )}+\frac {y(x)}{K[3]^2+y(x)^2}\right )dK[3]+\int _1^{y(x)}\left (-\frac {x}{x^2+K[4]^2}-\int _1^x\left (-\frac {2 K[4]^2}{\left (K[3]^2+K[4]^2\right )^2}+\frac {2 K[3] \sqrt {f\left (K[3]^2+K[4]^2\right ) \left (K[3]^2+K[4]^2-f\left (K[3]^2+K[4]^2\right )\right )} f'\left (K[3]^2+K[4]^2\right ) K[4]}{f\left (K[3]^2+K[4]^2\right )^2 \left (K[3]^2+K[4]^2\right )}-\frac {2 K[3] \sqrt {f\left (K[3]^2+K[4]^2\right ) \left (K[3]^2+K[4]^2-f\left (K[3]^2+K[4]^2\right )\right )} f'\left (K[3]^2+K[4]^2\right ) K[4]}{f\left (K[3]^2+K[4]^2\right )^2 \left (K[3]^2+K[4]^2-f\left (K[3]^2+K[4]^2\right )\right )}+\frac {2 K[3] \sqrt {f\left (K[3]^2+K[4]^2\right ) \left (K[3]^2+K[4]^2-f\left (K[3]^2+K[4]^2\right )\right )} K[4]}{f\left (K[3]^2+K[4]^2\right ) \left (K[3]^2+K[4]^2\right )^2}-\frac {K[3] \sqrt {f\left (K[3]^2+K[4]^2\right ) \left (K[3]^2+K[4]^2-f\left (K[3]^2+K[4]^2\right )\right )} \left (2 K[4]-2 K[4] f'\left (K[3]^2+K[4]^2\right )\right )}{f\left (K[3]^2+K[4]^2\right ) \left (K[3]^2+K[4]^2-f\left (K[3]^2+K[4]^2\right )\right )^2}-\frac {K[3] \left (2 K[4] \left (K[3]^2+K[4]^2-f\left (K[3]^2+K[4]^2\right )\right ) f'\left (K[3]^2+K[4]^2\right )+f\left (K[3]^2+K[4]^2\right ) \left (2 K[4]-2 K[4] f'\left (K[3]^2+K[4]^2\right )\right )\right )}{2 f\left (K[3]^2+K[4]^2\right ) \left (K[3]^2+K[4]^2\right ) \sqrt {f\left (K[3]^2+K[4]^2\right ) \left (K[3]^2+K[4]^2-f\left (K[3]^2+K[4]^2\right )\right )}}+\frac {K[3] \left (2 K[4] \left (K[3]^2+K[4]^2-f\left (K[3]^2+K[4]^2\right )\right ) f'\left (K[3]^2+K[4]^2\right )+f\left (K[3]^2+K[4]^2\right ) \left (2 K[4]-2 K[4] f'\left (K[3]^2+K[4]^2\right )\right )\right )}{2 f\left (K[3]^2+K[4]^2\right ) \left (K[3]^2+K[4]^2-f\left (K[3]^2+K[4]^2\right )\right ) \sqrt {f\left (K[3]^2+K[4]^2\right ) \left (K[3]^2+K[4]^2-f\left (K[3]^2+K[4]^2\right )\right )}}+\frac {1}{K[3]^2+K[4]^2}\right )dK[3]-\frac {K[4] \sqrt {f\left (x^2+K[4]^2\right ) \left (x^2+K[4]^2-f\left (x^2+K[4]^2\right )\right )}}{f\left (x^2+K[4]^2\right ) \left (x^2+K[4]^2\right )}+\frac {K[4] \sqrt {f\left (x^2+K[4]^2\right ) \left (x^2+K[4]^2-f\left (x^2+K[4]^2\right )\right )}}{f\left (x^2+K[4]^2\right ) \left (x^2+K[4]^2-f\left (x^2+K[4]^2\right )\right )}\right )dK[4]=c_1,y(x)\right ]\right \}\] ✓ Maple : cpu = 1.548 (sec), leaf count = 117
dsolve(f(y(x)^2+x^2)*(diff(y(x),x)^2+1)-(x*diff(y(x),x)-y(x))^2=0,y(x))
\[y \left (x \right ) = \frac {x}{\tan \left (\RootOf \left (-2 \textit {\_Z} -\left (\int _{}^{\frac {x^{2} \left (\tan ^{2}\left (\textit {\_Z} \right )+1\right )}{\tan \left (\textit {\_Z} \right )^{2}}}\frac {\sqrt {-f \left (\textit {\_a} \right ) \left (f \left (\textit {\_a} \right )-\textit {\_a} \right )}}{\textit {\_a} \left (f \left (\textit {\_a} \right )-\textit {\_a} \right )}d \textit {\_a} \right )+2 c_{1}\right )\right )}\]