\[ \boxed { \left \{ {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) +3\,x \left ( t \right ) -y \left ( t \right ) ={{\rm e}^{2\,t}},{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) +x \left ( t \right ) +5\,y \left ( t \right ) ={{\rm e}^{t}} \right \} } \]
Mathematica: cpu = 0.046006 (sec), leaf count = 162 \[ \left \{\left \{x(t)\to c_1 e^{-4 t} (t+1)+c_2 e^{-4 t} t-e^t (t+1) \left (\frac {t}{5}+\frac {1}{36} e^t (6 t-7)-\frac {1}{25}\right )+e^t t \left (\frac {t}{5}+\frac {1}{36} e^t (6 t-1)+\frac {4}{25}\right ),y(t)\to -c_1 e^{-4 t} t-c_2 e^{-4 t} (t-1)+e^t t \left (\frac {t}{5}+\frac {1}{36} e^t (6 t-7)-\frac {1}{25}\right )-e^t (t-1) \left (\frac {t}{5}+\frac {1}{36} e^t (6 t-1)+\frac {4}{25}\right )\right \}\right \} \]
Maple: cpu = 0.047 (sec), leaf count = 64 \[ \left \{ \left \{ x \left ( t \right ) ={{\rm e}^{-4\,t}}{\it \_C2}+{ {\rm e}^{-4\,t}}t{\it \_C1}+{\frac {7\,{{\rm e}^{2\,t}}}{36}}+{\frac { {{\rm e}^{t}}}{25}},y \left ( t \right ) =-{\frac {{{\rm e}^{2\,t}}}{36} }-{{\rm e}^{-4\,t}}{\it \_C2}-{{\rm e}^{-4\,t}}t{\it \_C1}+{{\rm e}^{- 4\,t}}{\it \_C1}+{\frac {4\,{{\rm e}^{t}}}{25}} \right \} \right \} \]