\[ \left \{x'(t)=x(t) (a y(t)+b),y'(t)=y(t) (c x(t)+d)\right \} \] ✓ Mathematica : cpu = 0.324561 (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 = 0.582 (sec), leaf count = 92
\[\left \{[\{x \left (t \right ) = 0\}, \{y \left (t \right ) = c_{1} {\mathrm e}^{d t}\}], \left [\left \{x \left (t \right ) = \RootOf \left (c_{2}+t -\left (\int _{}^{\textit {\_Z}}\frac {1}{\left (\LambertW \left (\frac {{\mathrm e}^{-1} \textit {\_a}^{\frac {d}{b}} {\mathrm e}^{\frac {c_{1}}{b}} {\mathrm e}^{\frac {\textit {\_a} c}{b}}}{b}\right )+1\right ) \textit {\_a} b}d \textit {\_a} \right )\right )\right \}, \left \{y \left (t \right ) = \frac {-b x \left (t \right )+\frac {d}{d t}x \left (t \right )}{a x \left (t \right )}\right \}\right ]\right \}\]