\[ \boxed { \left \{ {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) =ax \left ( t \right ) +gy \left ( t \right ) +\beta \,z \left ( t \right ) ,{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) =gx \left ( t \right ) +by \left ( t \right ) +\alpha \,z \left ( t \right ) ,{\frac {\rm d}{{\rm d}t}}z \left ( t \right ) =\beta \,x \left ( t \right ) +\alpha \,y \left ( t \right ) +cz \left ( t \right ) \right \} } \]
Mathematica: cpu = 0.060008 (sec), leaf count = 1630 \[ \left \{\left \{x(t)\to -c_2 \text {RootSum}\left [\text {$\#$1}^3-a \text {$\#$1}^2-b \text {$\#$1}^2-c \text {$\#$1}^2-\alpha ^2 \text {$\#$1}-\beta ^2 \text {$\#$1}-g^2 \text {$\#$1}+a b \text {$\#$1}+a c \text {$\#$1}+b c \text {$\#$1}+a \alpha ^2+b \beta ^2+c g^2-a b c-2 \alpha \beta g\& ,\frac {\alpha e^{t \text {$\#$1}} \beta -c e^{t \text {$\#$1}} g+e^{t \text {$\#$1}} g \text {$\#$1}}{\alpha ^2+\beta ^2+g^2-3 \text {$\#$1}^2-a b-a c-b c+2 a \text {$\#$1}+2 b \text {$\#$1}+2 c \text {$\#$1}}\& \right ]+c_3 \text {RootSum}\left [\text {$\#$1}^3-a \text {$\#$1}^2-b \text {$\#$1}^2-c \text {$\#$1}^2-\alpha ^2 \text {$\#$1}-\beta ^2 \text {$\#$1}-g^2 \text {$\#$1}+a b \text {$\#$1}+a c \text {$\#$1}+b c \text {$\#$1}+a \alpha ^2+b \beta ^2+c g^2-a b c-2 \alpha \beta g\& ,\frac {-b e^{t \text {$\#$1}} \beta +e^{t \text {$\#$1}} \text {$\#$1} \beta +\alpha e^{t \text {$\#$1}} g}{-\alpha ^2-\beta ^2-g^2+3 \text {$\#$1}^2+a b+a c+b c-2 a \text {$\#$1}-2 b \text {$\#$1}-2 c \text {$\#$1}}\& \right ]+c_1 \text {RootSum}\left [\text {$\#$1}^3-a \text {$\#$1}^2-b \text {$\#$1}^2-c \text {$\#$1}^2-\alpha ^2 \text {$\#$1}-\beta ^2 \text {$\#$1}-g^2 \text {$\#$1}+a b \text {$\#$1}+a c \text {$\#$1}+b c \text {$\#$1}+a \alpha ^2+b \beta ^2+c g^2-a b c-2 \alpha \beta g\& ,\frac {-e^{t \text {$\#$1}} \alpha ^2+b c e^{t \text {$\#$1}}+e^{t \text {$\#$1}} \text {$\#$1}^2-b e^{t \text {$\#$1}} \text {$\#$1}-c e^{t \text {$\#$1}} \text {$\#$1}}{-\alpha ^2-\beta ^2-g^2+3 \text {$\#$1}^2+a b+a c+b c-2 a \text {$\#$1}-2 b \text {$\#$1}-2 c \text {$\#$1}}\& \right ],y(t)\to -c_1 \text {RootSum}\left [\text {$\#$1}^3-a \text {$\#$1}^2-b \text {$\#$1}^2-c \text {$\#$1}^2-\alpha ^2 \text {$\#$1}-\beta ^2 \text {$\#$1}-g^2 \text {$\#$1}+a b \text {$\#$1}+a c \text {$\#$1}+b c \text {$\#$1}+a \alpha ^2+b \beta ^2+c g^2-a b c-2 \alpha \beta g\& ,\frac {\alpha e^{t \text {$\#$1}} \beta -c e^{t \text {$\#$1}} g+e^{t \text {$\#$1}} g \text {$\#$1}}{\alpha ^2+\beta ^2+g^2-3 \text {$\#$1}^2-a b-a c-b c+2 a \text {$\#$1}+2 b \text {$\#$1}+2 c \text {$\#$1}}\& \right ]+c_3 \text {RootSum}\left [\text {$\#$1}^3-a \text {$\#$1}^2-b \text {$\#$1}^2-c \text {$\#$1}^2-\alpha ^2 \text {$\#$1}-\beta ^2 \text {$\#$1}-g^2 \text {$\#$1}+a b \text {$\#$1}+a c \text {$\#$1}+b c \text {$\#$1}+a \alpha ^2+b \beta ^2+c g^2-a b c-2 \alpha \beta g\& ,\frac {-a e^{t \text {$\#$1}} \alpha +e^{t \text {$\#$1}} \text {$\#$1} \alpha +\beta e^{t \text {$\#$1}} g}{-\alpha ^2-\beta ^2-g^2+3 \text {$\#$1}^2+a b+a c+b c-2 a \text {$\#$1}-2 b \text {$\#$1}-2 c \text {$\#$1}}\& \right ]+c_2 \text {RootSum}\left [\text {$\#$1}^3-a \text {$\#$1}^2-b \text {$\#$1}^2-c \text {$\#$1}^2-\alpha ^2 \text {$\#$1}-\beta ^2 \text {$\#$1}-g^2 \text {$\#$1}+a b \text {$\#$1}+a c \text {$\#$1}+b c \text {$\#$1}+a \alpha ^2+b \beta ^2+c g^2-a b c-2 \alpha \beta g\& ,\frac {-e^{t \text {$\#$1}} \beta ^2+a c e^{t \text {$\#$1}}+e^{t \text {$\#$1}} \text {$\#$1}^2-a e^{t \text {$\#$1}} \text {$\#$1}-c e^{t \text {$\#$1}} \text {$\#$1}}{-\alpha ^2-\beta ^2-g^2+3 \text {$\#$1}^2+a b+a c+b c-2 a \text {$\#$1}-2 b \text {$\#$1}-2 c \text {$\#$1}}\& \right ],z(t)\to c_2 \text {RootSum}\left [\text {$\#$1}^3-a \text {$\#$1}^2-b \text {$\#$1}^2-c \text {$\#$1}^2-\alpha ^2 \text {$\#$1}-\beta ^2 \text {$\#$1}-g^2 \text {$\#$1}+a b \text {$\#$1}+a c \text {$\#$1}+b c \text {$\#$1}+a \alpha ^2+b \beta ^2+c g^2-a b c-2 \alpha \beta g\& ,\frac {-a e^{t \text {$\#$1}} \alpha +e^{t \text {$\#$1}} \text {$\#$1} \alpha +\beta e^{t \text {$\#$1}} g}{-\alpha ^2-\beta ^2-g^2+3 \text {$\#$1}^2+a b+a c+b c-2 a \text {$\#$1}-2 b \text {$\#$1}-2 c \text {$\#$1}}\& \right ]+c_1 \text {RootSum}\left [\text {$\#$1}^3-a \text {$\#$1}^2-b \text {$\#$1}^2-c \text {$\#$1}^2-\alpha ^2 \text {$\#$1}-\beta ^2 \text {$\#$1}-g^2 \text {$\#$1}+a b \text {$\#$1}+a c \text {$\#$1}+b c \text {$\#$1}+a \alpha ^2+b \beta ^2+c g^2-a b c-2 \alpha \beta g\& ,\frac {-b e^{t \text {$\#$1}} \beta +e^{t \text {$\#$1}} \text {$\#$1} \beta +\alpha e^{t \text {$\#$1}} g}{-\alpha ^2-\beta ^2-g^2+3 \text {$\#$1}^2+a b+a c+b c-2 a \text {$\#$1}-2 b \text {$\#$1}-2 c \text {$\#$1}}\& \right ]+c_3 \text {RootSum}\left [\text {$\#$1}^3-a \text {$\#$1}^2-b \text {$\#$1}^2-c \text {$\#$1}^2-\alpha ^2 \text {$\#$1}-\beta ^2 \text {$\#$1}-g^2 \text {$\#$1}+a b \text {$\#$1}+a c \text {$\#$1}+b c \text {$\#$1}+a \alpha ^2+b \beta ^2+c g^2-a b c-2 \alpha \beta g\& ,\frac {-e^{t \text {$\#$1}} g^2+a b e^{t \text {$\#$1}}+e^{t \text {$\#$1}} \text {$\#$1}^2-a e^{t \text {$\#$1}} \text {$\#$1}-b e^{t \text {$\#$1}} \text {$\#$1}}{-\alpha ^2-\beta ^2-g^2+3 \text {$\#$1}^2+a b+a c+b c-2 a \text {$\#$1}-2 b \text {$\#$1}-2 c \text {$\#$1}}\& \right ]\right \}\right \} \]
Maple: cpu = 23.774 (sec), leaf count = 33085
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