\[ \left \{a x'(t)=b c (y(t)-z(t)),b y'(t)=a c (z(t)-x(t)),c z'(t)=a b (x(t)-y(t))\right \} \] ✓ Mathematica : cpu = 0.0668711 (sec), leaf count = 1304
DSolve[{a*Derivative[1][x][t] == b*c*(y[t] - z[t]), b*Derivative[1][y][t] == a*c*(-x[t] + z[t]), c*Derivative[1][z][t] == a*b*(x[t] - y[t])},{x[t], y[t], z[t]},t]
\[\left \{\left \{x(t)\to \frac {e^{-i \sqrt {a^2+b^2+c^2} t} \left (2 e^{i \sqrt {a^2+b^2+c^2} t} a^2+b^2 e^{2 i \sqrt {a^2+b^2+c^2} t}+c^2 e^{2 i \sqrt {a^2+b^2+c^2} t}+b^2+c^2\right ) c_1}{2 \left (a^2+b^2+c^2\right )}-\frac {b e^{-i \sqrt {a^2+b^2+c^2} t} \left (-1+e^{i \sqrt {a^2+b^2+c^2} t}\right ) \left (a e^{i \sqrt {a^2+b^2+c^2} t} b-a b+i c \sqrt {a^2+b^2+c^2} e^{i \sqrt {a^2+b^2+c^2} t}+i c \sqrt {a^2+b^2+c^2}\right ) c_2}{2 a \left (a^2+b^2+c^2\right )}-\frac {c e^{-i \sqrt {a^2+b^2+c^2} t} \left (-1+e^{i \sqrt {a^2+b^2+c^2} t}\right ) \left (-i \sqrt {a^2+b^2+c^2} e^{i \sqrt {a^2+b^2+c^2} t} b-i \sqrt {a^2+b^2+c^2} b+a c e^{i \sqrt {a^2+b^2+c^2} t}-a c\right ) c_3}{2 a \left (a^2+b^2+c^2\right )},y(t)\to -\frac {a e^{-i \sqrt {a^2+b^2+c^2} t} \left (-1+e^{i \sqrt {a^2+b^2+c^2} t}\right ) \left (a e^{i \sqrt {a^2+b^2+c^2} t} b-a b-i c \sqrt {a^2+b^2+c^2} e^{i \sqrt {a^2+b^2+c^2} t}-i c \sqrt {a^2+b^2+c^2}\right ) c_1}{2 b \left (a^2+b^2+c^2\right )}+\frac {e^{-i \sqrt {a^2+b^2+c^2} t} \left (e^{2 i \sqrt {a^2+b^2+c^2} t} a^2+a^2+2 b^2 e^{i \sqrt {a^2+b^2+c^2} t}+c^2 e^{2 i \sqrt {a^2+b^2+c^2} t}+c^2\right ) c_2}{2 \left (a^2+b^2+c^2\right )}-\frac {c e^{-i \sqrt {a^2+b^2+c^2} t} \left (-1+e^{i \sqrt {a^2+b^2+c^2} t}\right ) \left (i \sqrt {a^2+b^2+c^2} e^{i \sqrt {a^2+b^2+c^2} t} a+i \sqrt {a^2+b^2+c^2} a+b c e^{i \sqrt {a^2+b^2+c^2} t}-b c\right ) c_3}{2 b \left (a^2+b^2+c^2\right )},z(t)\to -\frac {a e^{-i \sqrt {a^2+b^2+c^2} t} \left (-1+e^{i \sqrt {a^2+b^2+c^2} t}\right ) \left (i \sqrt {a^2+b^2+c^2} e^{i \sqrt {a^2+b^2+c^2} t} b+i \sqrt {a^2+b^2+c^2} b+a c e^{i \sqrt {a^2+b^2+c^2} t}-a c\right ) c_1}{2 c \left (a^2+b^2+c^2\right )}-\frac {b e^{-i \sqrt {a^2+b^2+c^2} t} \left (-1+e^{i \sqrt {a^2+b^2+c^2} t}\right ) \left (-i \sqrt {a^2+b^2+c^2} e^{i \sqrt {a^2+b^2+c^2} t} a-i \sqrt {a^2+b^2+c^2} a+b c e^{i \sqrt {a^2+b^2+c^2} t}-b c\right ) c_2}{2 c \left (a^2+b^2+c^2\right )}+\frac {e^{-i \sqrt {a^2+b^2+c^2} t} \left (e^{2 i \sqrt {a^2+b^2+c^2} t} a^2+a^2+2 c^2 e^{i \sqrt {a^2+b^2+c^2} t}+b^2 e^{2 i \sqrt {a^2+b^2+c^2} t}+b^2\right ) c_3}{2 \left (a^2+b^2+c^2\right )}\right \}\right \}\] ✓ Maple : cpu = 0.143 (sec), leaf count = 299
dsolve({a*diff(x(t),t) = b*c*(y(t)-z(t)), b*diff(y(t),t) = c*a*(z(t)-x(t)), c*diff(z(t),t) = a*b*(x(t)-y(t))})
\[\left \{x \left (t \right ) = c_{1}+c_{2} \sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right )+c_{3} \cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ), y \left (t \right ) = \frac {c_{1} b^{3}+\left (\left (-c_{2} \sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right )-c_{3} \cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right )\right ) a^{2}+c^{2} c_{1}\right ) b -\left (\sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_{3}-\cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_{2}\right ) \sqrt {a^{2}+b^{2}+c^{2}}\, c a}{b \left (b^{2}+c^{2}\right )}, z \left (t \right ) = \frac {\sqrt {a^{2}+b^{2}+c^{2}}\, \sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_{3} a b -\sqrt {a^{2}+b^{2}+c^{2}}\, \cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_{2} a b -\sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_{2} a^{2} c -\cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_{3} a^{2} c +c_{1} b^{2} c +c_{1} c^{3}}{\left (b^{2}+c^{2}\right ) c}\right \}\]