\[ \boxed { \left \{ {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) =-3\,x \left ( t \right ) +48\,y \left ( t \right ) -28\,z \left ( t \right ) ,{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) =-4\,x \left ( t \right ) +40\,y \left ( t \right ) -22\,z \left ( t \right ) ,{\frac {\rm d}{{\rm d}t}}z \left ( t \right ) =-6\,x \left ( t \right ) +57\,y \left ( t \right ) -31\,z \left ( t \right ) \right \} } \]
Mathematica: cpu = 0.009501 (sec), leaf count = 179 \[ \left \{\left \{x(t)\to c_1 \left (-e^t\right ) \left (2 e^{2 t}-3\right )+6 c_2 e^t \left (2 e^t+3 e^{2 t}-5\right )-2 c_3 e^t \left (4 e^t+5 e^{2 t}-9\right ),y(t)\to -2 c_1 e^t \left (e^{2 t}-1\right )+c_2 e^t \left (3 e^t+18 e^{2 t}-20\right )-2 c_3 e^t \left (e^t+5 e^{2 t}-6\right ),z(t)\to -3 c_1 e^t \left (e^{2 t}-1\right )+3 c_2 e^t \left (e^t+9 e^{2 t}-10\right )-c_3 e^t \left (2 e^t+15 e^{2 t}-18\right )\right \}\right \} \]
Maple: cpu = 0.046 (sec), leaf count = 66 \[ \left \{ \left \{ x \left ( t \right ) ={\it \_C1}\,{{\rm e}^{3\,t}}+{ \it \_C2}\,{{\rm e}^{2\,t}}+{\it \_C3}\,{{\rm e}^{t}},y \left ( t \right ) ={\it \_C1}\,{{\rm e}^{3\,t}}+{\frac {{\it \_C2}\,{{\rm e}^{2 \,t}}}{4}}+{\frac {2\,{\it \_C3}\,{{\rm e}^{t}}}{3}},z \left ( t \right ) ={\frac {3\,{\it \_C1}\,{{\rm e}^{3\,t}}}{2}}+{\frac {{\it \_C2}\,{{\rm e}^{2\,t}}}{4}}+{\it \_C3}\,{{\rm e}^{t}} \right \} \right \} \]