\[ f\left (x^2+y(x)^2\right ) \left (y'(x)^2+1\right )-\left (x y'(x)-y(x)\right )^2=0 \] ✓ Mathematica : cpu = 2.19266 (sec), leaf count = 1922
\[\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 = 1.44 (sec), leaf count = 117
\[\left \{y \left (x \right ) = \frac {x}{\tan \left (\RootOf \left (2 c_{1}-2 \textit {\_Z} -\left (\int _{}^{\frac {\left (\tan ^{2}\left (\textit {\_Z} \right )+1\right ) x^{2}}{\tan \left (\textit {\_Z} \right )^{2}}}\frac {\sqrt {-\left (-\textit {\_a} +f \left (\textit {\_a} \right )\right ) f \left (\textit {\_a} \right )}}{\left (-\textit {\_a} +f \left (\textit {\_a} \right )\right ) \textit {\_a}}d \textit {\_a} \right )\right )\right )}, y \left (x \right ) = \frac {x}{\tan \left (\RootOf \left (2 c_{1}-2 \textit {\_Z} +\int _{}^{\frac {\left (\tan ^{2}\left (\textit {\_Z} \right )+1\right ) x^{2}}{\tan \left (\textit {\_Z} \right )^{2}}}\frac {\sqrt {-\left (-\textit {\_a} +f \left (\textit {\_a} \right )\right ) f \left (\textit {\_a} \right )}}{\left (-\textit {\_a} +f \left (\textit {\_a} \right )\right ) \textit {\_a}}d \textit {\_a} \right )\right )}\right \}\]