\[ \boxed { \left \{ {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) = \left ( ay \left ( t \right ) +b \right ) x \left ( t \right ) ,{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) = \left ( cx \left ( t \right ) +d \right ) y \left ( t \right ) \right \} } \]
Mathematica: cpu = 0.508065 (sec), leaf count = 201 \[ \left \{\left \{y(t)\to \frac {b W\left (\frac {a \text {InverseFunction}\left [\int _1^{\text {$\#$1}} \frac {1}{K[1] \left (W\left (\frac {a e^{\frac {c K[1]}{b}+\frac {c_1}{b}} K[1]^{\frac {d}{b}}}{b}\right )+1\right )} \, dK[1]\& \right ]\left [b t+c_2\right ]{}^{\frac {d}{b}} \exp \left (\frac {c \text {InverseFunction}\left [\int _1^{\text {$\#$1}} \frac {1}{K[1] \left (W\left (\frac {a e^{\frac {c K[1]}{b}+\frac {c_1}{b}} K[1]^{\frac {d}{b}}}{b}\right )+1\right )} \, dK[1]\& \right ]\left [b t+c_2\right ]}{b}+\frac {c_1}{b}\right )}{b}\right )}{a},x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}} \frac {1}{K[1] \left (W\left (\frac {a e^{\frac {c K[1]}{b}+\frac {c_1}{b}} K[1]^{\frac {d}{b}}}{b}\right )+1\right )} \, dK[1]\& \right ]\left [b t+c_2\right ]\right \}\right \} \]
Maple: cpu = 0.250 (sec), leaf count = 92 \[ \left \{ [ \left \{ x \left ( t \right ) =0 \right \} , \left \{ y \left ( t \right ) ={\it \_C1}\,{{\rm e}^{dt}} \right \} ],[ \left \{ x \left ( t \right ) ={\it RootOf} \left ( -\int ^{{\it \_Z}}\!{\frac {1}{b{\it \_a }} \left ( {\it lambertW} \left ( {\frac {{{\rm e}^{-1}}}{b}{{\rm e}^{{ \frac {c{\it \_a}}{b}}}}{{\it \_a}}^{{\frac {d}{b}}}{{\rm e}^{{\frac { {\it \_C1}}{b}}}}} \right ) +1 \right ) ^{-1}}{d{\it \_a}}+t+{\it \_C2} \right ) \right \} , \left \{ y \left ( t \right ) ={\frac {-bx \left ( t \right ) +{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) }{ax \left ( t \right ) }} \right \} ] \right \} \]