\[ \left \{x'(t)=x(t) (a y(t)+b),y'(t)=y(t) (c x(t)+d)\right \} \] ✓ Mathematica : cpu = 0.28613 (sec), leaf count = 204
DSolve[{Derivative[1][x][t] == x[t]*(b + a*y[t]), Derivative[1][y][t] == (d + c*x[t])*y[t]},{x[t], y[t]},t]
\[\left \{\left \{y(t)\to \frac {b W\left (\frac {a \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (W\left (\frac {a e^{\frac {c_1}{b}+\frac {c K[1]}{b}} K[1]^{\frac {d}{b}}}{b}\right )+1\right )}dK[1]\& \right ][b t+c_2]{}^{\frac {d}{b}} \exp \left (\frac {c \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (W\left (\frac {a e^{\frac {c_1}{b}+\frac {c K[1]}{b}} K[1]^{\frac {d}{b}}}{b}\right )+1\right )}dK[1]\& \right ][b t+c_2]}{b}+\frac {c_1}{b}\right )}{b}\right )}{a},x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (W\left (\frac {a e^{\frac {c_1}{b}+\frac {c K[1]}{b}} K[1]^{\frac {d}{b}}}{b}\right )+1\right )}dK[1]\& \right ][b t+c_2]\right \}\right \}\] ✓ Maple : cpu = 0.46 (sec), leaf count = 92
dsolve({diff(x(t),t) = (a*y(t)+b)*x(t), diff(y(t),t) = (c*x(t)+d)*y(t)})
\[[\{x \left (t \right ) = 0\}, \{y \left (t \right ) = c_{1} {\mathrm e}^{d t}\}]\]