\[ \boxed { \left \{ {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) =2\,x \left ( t \right ) ,{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) =3\,x \left ( t \right ) -2\,y \left ( t \right ) ,{\frac {\rm d}{{\rm d}t}}z \left ( t \right ) =2\,y \left ( t \right ) +3\,z \left ( t \right ) \right \} } \]
Mathematica: cpu = 0.010501 (sec), leaf count = 112 \[ \left \{\left \{x(t)\to c_1 e^{2 t},y(t)\to \frac {3}{4} c_1 e^{-2 t} \left (e^{4 t}-1\right )+c_2 e^{-2 t},z(t)\to \frac {3}{10} c_1 e^{-2 t} \left (2 e^t+3 e^{2 t}+4 e^{3 t}+1\right ) \left (e^t-1\right )^2+\frac {2}{5} c_2 e^{-2 t} \left (e^{5 t}-1\right )+c_3 e^{3 t}\right \}\right \} \]
Maple: cpu = 0.078 (sec), leaf count = 52 \[ \left \{ \left \{ x \left ( t \right ) ={\it \_C3}\,{{\rm e}^{2\,t}},y \left ( t \right ) ={\frac {3\,{\it \_C3}\,{{\rm e}^{2\,t}}}{4}}+{ {\rm e}^{-2\,t}}{\it \_C2},z \left ( t \right ) ={\it \_C1}\,{{\rm e}^{3 \,t}}-{\frac {3\,{\it \_C3}\,{{\rm e}^{2\,t}}}{2}}-{\frac {2\,{{\rm e} ^{-2\,t}}{\it \_C2}}{5}} \right \} \right \} \]