\[ \left \{a x'(t)=b c (y(t)-z(t)),b y'(t)=a c (z(t)-x(t)),c z'(t)=a b (x(t)-y(t))\right \} \] ✓ Mathematica : cpu = 0.0898444 (sec), leaf count = 741
\[\left \{\left \{x(t)\to \frac {e^{-i t \sqrt {a^2+b^2+c^2}} \left (a \left (b^2 \left (c_1 \left (1+e^{2 i t \sqrt {a^2+b^2+c^2}}\right )-c_2 \left (-1+e^{i t \sqrt {a^2+b^2+c^2}}\right )^2\right )+c^2 \left (c_1 \left (1+e^{2 i t \sqrt {a^2+b^2+c^2}}\right )-c_3 \left (-1+e^{i t \sqrt {a^2+b^2+c^2}}\right )^2\right )\right )-i b c \left (c_2-c_3\right ) \sqrt {a^2+b^2+c^2} \left (-1+e^{2 i t \sqrt {a^2+b^2+c^2}}\right )+2 a^3 c_1 e^{i t \sqrt {a^2+b^2+c^2}}\right )}{2 a \left (a^2+b^2+c^2\right )},y(t)\to \frac {e^{-i t \sqrt {a^2+b^2+c^2}} \left (a^2 (-b) \left (c_1 \left (-1+e^{i t \sqrt {a^2+b^2+c^2}}\right )^2-c_2 \left (1+e^{2 i t \sqrt {a^2+b^2+c^2}}\right )\right )+i a c \left (c_1-c_3\right ) \sqrt {a^2+b^2+c^2} \left (-1+e^{2 i t \sqrt {a^2+b^2+c^2}}\right )+b \left (2 b^2 c_2 e^{i t \sqrt {a^2+b^2+c^2}}+c^2 \left (c_2 \left (1+e^{2 i t \sqrt {a^2+b^2+c^2}}\right )-c_3 \left (-1+e^{i t \sqrt {a^2+b^2+c^2}}\right )^2\right )\right )\right )}{2 b \left (a^2+b^2+c^2\right )},z(t)\to \frac {e^{-i t \sqrt {a^2+b^2+c^2}} \left (a^2 (-c) \left (c_1 \left (-1+e^{i t \sqrt {a^2+b^2+c^2}}\right )^2-c_3 \left (1+e^{2 i t \sqrt {a^2+b^2+c^2}}\right )\right )-i a b \left (c_1-c_2\right ) \sqrt {a^2+b^2+c^2} \left (-1+e^{2 i t \sqrt {a^2+b^2+c^2}}\right )+c \left (b^2 \left (c_3 \left (1+e^{2 i t \sqrt {a^2+b^2+c^2}}\right )-c_2 \left (-1+e^{i t \sqrt {a^2+b^2+c^2}}\right )^2\right )+2 c^2 c_3 e^{i t \sqrt {a^2+b^2+c^2}}\right )\right )}{2 c \left (a^2+b^2+c^2\right )}\right \}\right \}\]
✓ Maple : cpu = 0.155 (sec), leaf count = 299
\[ \left \{ \left \{ x \left ( t \right ) ={\it \_C1}+{\it \_C2}\,\sin \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}t \right ) +{\it \_C3}\,\cos \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}t \right ) ,y \left ( t \right ) ={\frac {1}{b \left ( {b}^{2}+{c}^{2} \right ) } \left ( {\it \_C1}\,{b}^{3}+ \left ( \left ( -{\it \_C2}\,\sin \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}t \right ) -{\it \_C3}\,\cos \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}t \right ) \right ) {a}^{2}+{c}^{2}{\it \_C1} \right ) b-\sqrt {{a}^{2}+{b}^{2}+{c}^{2}}ac \left ( \sin \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}t \right ) {\it \_C3}-\cos \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}t \right ) {\it \_C2} \right ) \right ) },z \left ( t \right ) ={\frac {1}{ \left ( {b}^{2}+{c}^{2} \right ) c} \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}\sin \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}t \right ) {\it \_C3}\,ab-\sqrt {{a}^{2}+{b}^{2}+{c}^{2}}\cos \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}t \right ) {\it \_C2}\,ab-\sin \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}t \right ) {\it \_C2}\,{a}^{2}c-\cos \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}t \right ) {\it \_C3}\,{a}^{2}c+{\it \_C1}\,{b}^{2}c+{\it \_C1}\,{c}^{3} \right ) } \right \} \right \} \]