\[ \boxed { \left \{ at{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) =bc \left ( y \left ( t \right ) -z \left ( t \right ) \right ) ,bt{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) =ca \left ( z \left ( t \right ) -x \left ( t \right ) \right ) ,ct{\frac {\rm d}{{\rm d}t}}z \left ( t \right ) =ab \left ( x \left ( t \right ) -y \left ( t \right ) \right ) \right \} } \]
Mathematica: cpu = 0.037505 (sec), leaf count = 64 \[ \text {DSolve}\left [\left \{a t x'(t)=b c (y(t)-z(t)),b t y'(t)=a c (z(t)-x(t)),c t z'(t)=a b (x(t)-y(t))\right \},\{x(t),y(t),z(t)\},t\right ] \]
Maple: cpu = 0.110 (sec), leaf count = 322 \[ \left \{ \left \{ x \left ( t \right ) ={\it \_C1}+{\it \_C2}\,\sin \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}\ln \left ( t \right ) \right ) +{\it \_C3}\,\cos \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}\ln \left ( t \right ) \right ) ,y \left ( t \right ) ={\frac {1}{b \left ( {b}^{2}+{c} ^{2} \right ) } \left ( \cos \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}\ln \left ( t \right ) \right ) \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}{\it \_C2}\, ac-\cos \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}\ln \left ( t \right ) \right ) {\it \_C3}\,{a}^{2}b-\sin \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{ 2}}\ln \left ( t \right ) \right ) \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}{\it \_C3}\,ac-\sin \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}\ln \left ( t \right ) \right ) {\it \_C2}\,{a}^{2}b+{\it \_C1}\,{b}^{3}+{\it \_C1} \,b{c}^{2} \right ) },z \left ( t \right ) =-{\frac {1}{ \left ( {b}^{2}+{ c}^{2} \right ) c} \left ( \cos \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}} \ln \left ( t \right ) \right ) \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}{\it \_C2}\,ab+\cos \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}\ln \left ( t \right ) \right ) {\it \_C3}\,{a}^{2}c-\sin \left ( \sqrt {{a}^{2}+{b}^ {2}+{c}^{2}}\ln \left ( t \right ) \right ) \sqrt {{a}^{2}+{b}^{2}+{c}^ {2}}{\it \_C3}\,ab+\sin \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}\ln \left ( t \right ) \right ) {\it \_C2}\,{a}^{2}c-{\it \_C1}\,{b}^{2}c-{ \it \_C1}\,{c}^{3} \right ) } \right \} \right \} \]