\[ \left \{x'(t)=x(t) (a y(t)+b),y'(t)=y(t) (c x(t)+d)\right \} \] ✓ Mathematica : cpu = 0.239057 (sec), leaf count = 204
\[\left \{\left \{y(t)\to \frac {b W\left (\frac {a \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (W\left (\frac {a e^{\frac {c_1}{b}+\frac {c K[1]}{b}} K[1]^{\frac {d}{b}}}{b}\right )+1\right )}dK[1]\& \right ][b t+c_2]{}^{\frac {d}{b}} \exp \left (\frac {c \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (W\left (\frac {a e^{\frac {c_1}{b}+\frac {c K[1]}{b}} K[1]^{\frac {d}{b}}}{b}\right )+1\right )}dK[1]\& \right ][b t+c_2]}{b}+\frac {c_1}{b}\right )}{b}\right )}{a},x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (W\left (\frac {a e^{\frac {c_1}{b}+\frac {c K[1]}{b}} K[1]^{\frac {d}{b}}}{b}\right )+1\right )}dK[1]\& \right ][b t+c_2]\right \}\right \}\] ✓ Maple : cpu = 1.461 (sec), leaf count = 92
\[ \left \{ [ \left \{ x \left ( t \right ) =0 \right \} , \left \{ y \left ( t \right ) ={\it \_C1}\,{{\rm e}^{dt}} \right \} ],[ \left \{ x \left ( t \right ) ={\it RootOf} \left ( -\int ^{{\it \_Z}}\!{\frac {1}{{\it \_a}\,b} \left ( {\it lambertW} \left ( {\frac {{{\rm e}^{-1}}}{b}{{\rm e}^{{\frac {c{\it \_a}}{b}}}}{{\it \_a}}^{{\frac {d}{b}}}{{\rm e}^{{\frac {{\it \_C1}}{b}}}}} \right ) +1 \right ) ^{-1}}{d{\it \_a}}+t+{\it \_C2} \right ) \right \} , \left \{ y \left ( t \right ) ={\frac {-bx \left ( t \right ) +{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) }{ax \left ( t \right ) }} \right \} ] \right \} \]