\[ \sqrt {x^2+y(x)^2} y''(x)-a \left (y'(x)^2+1\right )^{3/2}=0 \] ✗ Mathematica : cpu = 300.004 (sec), leaf count = 0 , timed out
$Aborted
✓ Maple : cpu = 3.85 (sec), leaf count = 1862
\[ \left \{ y \left ( x \right ) ={\it RootOf} \left ( -\ln \left ( x \right ) +\int ^{{\it \_Z}}\!{\frac {1}{{{\it \_g}}^{2}+1} \left ( {\it RootOf} \left ( \arctan \left ( {\it \_g} \right ) +\int ^{{\it \_Z}}\!{\frac {1}{ \left ( {{\it \_f}}^{2}{a}^{2}+{a}^{2}-1 \right ) \left ( {{\it \_f}}^{2}+1 \right ) } \left ( 1+\sqrt {{a}^{2} \left ( {{\it \_f}}^{2}+1 \right ) } \right ) }{d{\it \_f}}+{\it \_C1} \right ) -{\it \_g} \right ) }{d{\it \_g}}+{\it \_C2} \right ) x,y \left ( x \right ) ={\it RootOf} \left ( -\ln \left ( x \right ) +\int ^{{\it \_Z}}\!{\it RootOf} \left ( -\sqrt {\pi }\sqrt {2} \left ( {\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} \right ) ^{{\frac {5}{2}}}{{\rm e}^{{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{2\,{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}}{\it \_C1}\,{{\it \_Z}}^{4}{{\it \_f}}^{4}-2\,\sqrt {\pi }\sqrt {2} \left ( {\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} \right ) ^{5/2}{{\rm e}^{1/2\,{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}}{\it \_C1}\,{{\it \_Z}}^{4}{{\it \_f}}^{2}-3\,\sqrt {\pi }\sqrt {2} \left ( {\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} \right ) ^{3/2}{{\rm e}^{1/2\,{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}}{\it \_C1}\,{{\it \_Z}}^{4}{{\it \_f}}^{4}-2\,\sqrt {\pi }\sqrt {2} \left ( {\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} \right ) ^{3/2}{{\rm e}^{1/2\,{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}}{\it \_C1}\,{{\it \_Z}}^{4}{{\it \_f}}^{2}-2\,\sqrt {\pi }\sqrt {2} \left ( {\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} \right ) ^{3/2}{{\rm e}^{1/2\,{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}}{\it \_C1}\,{{\it \_Z}}^{3}{{\it \_f}}^{3}-2\,\sqrt {\pi }\sqrt {2} \left ( {\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} \right ) ^{3/2}{{\rm e}^{1/2\,{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}}{\it \_f}\,{\it \_C1}\,{{\it \_Z}}^{3}+2\,{\it \_C1}\,{{\it \_Z}}^{4}{{\it \_f}}^{8}-4\,{\it \_C1}\,{{\it \_Z}}^{4}{{\it \_f}}^{6}+8\,{\it \_C1}\,{{\it \_Z}}^{3}{{\it \_f}}^{7}-12\,{\it \_C1}\,{{\it \_Z}}^{3}{{\it \_f}}^{5}+12\,{\it \_C1}\,{{\it \_Z}}^{2}{{\it \_f}}^{6}+4\,{\it \_C1}\,{{\it \_Z}}^{4}{{\it \_f}}^{2}-12\,{\it \_C1}\,{{\it \_Z}}^{2}{{\it \_f}}^{4}+8\,{\it \_C1}\,{\it \_Z}\,{{\it \_f}}^{5}+4\,{\it \_C1}\,{{\it \_Z}}^{3}{\it \_f}-4\,{\it \_C1}\,{\it \_Z}\,{{\it \_f}}^{3}+\sqrt {\pi }\sqrt {2} \left ( {\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} \right ) ^{{\frac {5}{2}}}{{\rm e}^{{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{2\,{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}}{\it Erf} \left ( {\frac {\sqrt {2}}{2}\sqrt {{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}} \right ) {\it \_C1}\,{{\it \_Z}}^{4}-\sqrt {\pi }\sqrt {2} \left ( {\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} \right ) ^{{\frac {3}{2}}}{{\rm e}^{{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{2\,{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}}{\it Erf} \left ( {\frac {\sqrt {2}}{2}\sqrt {{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}} \right ) {\it \_C1}\,{{\it \_Z}}^{4}-2\,{\it \_C1}\,{{\it \_Z}}^{4}+2\,{\it \_C1}\,{{\it \_f}}^{4}+4\, \left ( {\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} \right ) ^{3/2}{{\rm e}^{1/2\,{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}}{{\it \_Z}}^{2}{{\it \_f}}^{4}+4\, \left ( {\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} \right ) ^{3/2}{{\rm e}^{1/2\,{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}}{{\it \_Z}}^{2}{{\it \_f}}^{2}+4\, \left ( {\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} \right ) ^{3/2}{{\rm e}^{1/2\,{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}}{{\it \_Z}}^{4}{{\it \_f}}^{6}+8\, \left ( {\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} \right ) ^{3/2}{{\rm e}^{1/2\,{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}}{{\it \_Z}}^{4}{{\it \_f}}^{4}+8\, \left ( {\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} \right ) ^{3/2}{{\rm e}^{1/2\,{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}}{{\it \_Z}}^{3}{{\it \_f}}^{5}+4\, \left ( {\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} \right ) ^{3/2}{{\rm e}^{1/2\,{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}}{{\it \_Z}}^{4}{{\it \_f}}^{2}+8\, \left ( {\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} \right ) ^{3/2}{{\rm e}^{1/2\,{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}}{{\it \_Z}}^{3}{{\it \_f}}^{3}+\sqrt {\pi }\sqrt {2} \left ( {\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} \right ) ^{{\frac {5}{2}}}{{\rm e}^{{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{2\,{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}}{\it Erf} \left ( {\frac {\sqrt {2}}{2}\sqrt {{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}} \right ) {\it \_C1}\,{{\it \_Z}}^{4}{{\it \_f}}^{4}+2\,\sqrt {\pi }\sqrt {2} \left ( {\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} \right ) ^{5/2}{{\rm e}^{1/2\,{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}}{\it Erf} \left ( 1/2\,\sqrt {2}\sqrt {{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}} \right ) {\it \_C1}\,{{\it \_Z}}^{4}{{\it \_f}}^{2}+3\,\sqrt {\pi }\sqrt {2} \left ( {\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} \right ) ^{3/2}{{\rm e}^{1/2\,{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}}{\it Erf} \left ( 1/2\,\sqrt {2}\sqrt {{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}} \right ) {\it \_C1}\,{{\it \_Z}}^{4}{{\it \_f}}^{4}+2\,\sqrt {\pi }\sqrt {2} \left ( {\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} \right ) ^{3/2}{{\rm e}^{1/2\,{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}}{\it Erf} \left ( 1/2\,\sqrt {2}\sqrt {{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}} \right ) {\it \_C1}\,{{\it \_Z}}^{4}{{\it \_f}}^{2}+2\,\sqrt {\pi }\sqrt {2} \left ( {\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} \right ) ^{3/2}{{\rm e}^{1/2\,{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}}{\it Erf} \left ( 1/2\,\sqrt {2}\sqrt {{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}} \right ) {\it \_C1}\,{{\it \_Z}}^{3}{{\it \_f}}^{3}+2\,\sqrt {\pi }\sqrt {2} \left ( {\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} \right ) ^{3/2}{{\rm e}^{1/2\,{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}}{\it Erf} \left ( 1/2\,\sqrt {2}\sqrt {{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}} \right ) {\it \_f}\,{\it \_C1}\,{{\it \_Z}}^{3}-\sqrt {\pi }\sqrt {2} \left ( {\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} \right ) ^{{\frac {5}{2}}}{{\rm e}^{{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{2\,{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}}{\it \_C1}\,{{\it \_Z}}^{4}+\sqrt {\pi }\sqrt {2} \left ( {\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} \right ) ^{{\frac {3}{2}}}{{\rm e}^{{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{2\,{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}}{\it \_C1}\,{{\it \_Z}}^{4} \right ) {d{\it \_f}}+{\it \_C2} \right ) x \right \} \]