\[ \left \{f(t) (a x(t)+b y(t))+x'(t)=g(t),f(t) (c x(t)+d y(t))+y'(t)=h(t)\right \} \] ✓ Mathematica : cpu = 1.4733 (sec), leaf count = 3181
\[\left \{\left \{x(t)\to \frac {\left (-a+d+\sqrt {a^2-2 d a+d^2+4 b c}\right ) e^{\int _1^t-\frac {1}{2} \left (a+d-\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[1])dK[1]} c_1}{2 \sqrt {a^2-2 d a+d^2+4 b c}}+\frac {\left (a-d+\sqrt {a^2-2 d a+d^2+4 b c}\right ) e^{\int _1^t-\frac {1}{2} \left (a+d+\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[2])dK[2]} c_2}{2 \sqrt {a^2-2 d a+d^2+4 b c}}+\frac {\left (-a+d+\sqrt {a^2-2 d a+d^2+4 b c}\right ) e^{\int _1^t-\frac {1}{2} \left (a+d-\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[1])dK[1]} \int _1^t\frac {\sqrt {a^2-2 d a+d^2+4 b c} \left (2 c e^{\int _1^{K[5]}-\frac {1}{2} \left (a+d+\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[4])dK[4]} g(K[5])-\left (a-d+\sqrt {a^2-2 d a+d^2+4 b c}\right ) e^{\int _1^{K[5]}-\frac {1}{2} \left (a+d+\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[2])dK[2]} h(K[5])\right )}{c \left (a \left (e^{\int _1^{K[5]}-\frac {1}{2} \left (a+d+\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[2])dK[2]+\int _1^{K[5]}-\frac {1}{2} \left (a+d-\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[3])dK[3]}-e^{\int _1^{K[5]}-\frac {1}{2} \left (a+d-\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[1])dK[1]+\int _1^{K[5]}-\frac {1}{2} \left (a+d+\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[4])dK[4]}\right )+d \left (-e^{\int _1^{K[5]}-\frac {1}{2} \left (a+d+\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[2])dK[2]+\int _1^{K[5]}-\frac {1}{2} \left (a+d-\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[3])dK[3]}+e^{\int _1^{K[5]}-\frac {1}{2} \left (a+d-\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[1])dK[1]+\int _1^{K[5]}-\frac {1}{2} \left (a+d+\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[4])dK[4]}\right )+\sqrt {a^2-2 d a+d^2+4 b c} \left (e^{\int _1^{K[5]}-\frac {1}{2} \left (a+d+\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[2])dK[2]+\int _1^{K[5]}-\frac {1}{2} \left (a+d-\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[3])dK[3]}+e^{\int _1^{K[5]}-\frac {1}{2} \left (a+d-\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[1])dK[1]+\int _1^{K[5]}-\frac {1}{2} \left (a+d+\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[4])dK[4]}\right )\right )}dK[5]+\left (a-d+\sqrt {a^2-2 d a+d^2+4 b c}\right ) e^{\int _1^t-\frac {1}{2} \left (a+d+\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[2])dK[2]} \int _1^t\frac {\sqrt {a^2-2 d a+d^2+4 b c} \left (2 c e^{\int _1^{K[6]}-\frac {1}{2} \left (a+d-\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[3])dK[3]} g(K[6])+\left (-a+d+\sqrt {a^2-2 d a+d^2+4 b c}\right ) e^{\int _1^{K[6]}-\frac {1}{2} \left (a+d-\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[1])dK[1]} h(K[6])\right )}{c \left (a \left (e^{\int _1^{K[6]}-\frac {1}{2} \left (a+d+\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[2])dK[2]+\int _1^{K[6]}-\frac {1}{2} \left (a+d-\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[3])dK[3]}-e^{\int _1^{K[6]}-\frac {1}{2} \left (a+d-\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[1])dK[1]+\int _1^{K[6]}-\frac {1}{2} \left (a+d+\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[4])dK[4]}\right )+d \left (-e^{\int _1^{K[6]}-\frac {1}{2} \left (a+d+\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[2])dK[2]+\int _1^{K[6]}-\frac {1}{2} \left (a+d-\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[3])dK[3]}+e^{\int _1^{K[6]}-\frac {1}{2} \left (a+d-\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[1])dK[1]+\int _1^{K[6]}-\frac {1}{2} \left (a+d+\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[4])dK[4]}\right )+\sqrt {a^2-2 d a+d^2+4 b c} \left (e^{\int _1^{K[6]}-\frac {1}{2} \left (a+d+\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[2])dK[2]+\int _1^{K[6]}-\frac {1}{2} \left (a+d-\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[3])dK[3]}+e^{\int _1^{K[6]}-\frac {1}{2} \left (a+d-\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[1])dK[1]+\int _1^{K[6]}-\frac {1}{2} \left (a+d+\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[4])dK[4]}\right )\right )}dK[6]}{2 \sqrt {a^2-2 d a+d^2+4 b c}},y(t)\to -\frac {c e^{\int _1^t-\frac {1}{2} \left (a+d-\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[3])dK[3]} c_1}{\sqrt {a^2-2 d a+d^2+4 b c}}+\frac {c e^{\int _1^t-\frac {1}{2} \left (a+d+\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[4])dK[4]} c_2}{\sqrt {a^2-2 d a+d^2+4 b c}}+\frac {c e^{\int _1^t-\frac {1}{2} \left (a+d+\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[4])dK[4]} \int _1^t\frac {\sqrt {a^2-2 d a+d^2+4 b c} \left (2 c e^{\int _1^{K[6]}-\frac {1}{2} \left (a+d-\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[3])dK[3]} g(K[6])+\left (-a+d+\sqrt {a^2-2 d a+d^2+4 b c}\right ) e^{\int _1^{K[6]}-\frac {1}{2} \left (a+d-\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[1])dK[1]} h(K[6])\right )}{c \left (a \left (e^{\int _1^{K[6]}-\frac {1}{2} \left (a+d+\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[2])dK[2]+\int _1^{K[6]}-\frac {1}{2} \left (a+d-\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[3])dK[3]}-e^{\int _1^{K[6]}-\frac {1}{2} \left (a+d-\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[1])dK[1]+\int _1^{K[6]}-\frac {1}{2} \left (a+d+\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[4])dK[4]}\right )+d \left (-e^{\int _1^{K[6]}-\frac {1}{2} \left (a+d+\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[2])dK[2]+\int _1^{K[6]}-\frac {1}{2} \left (a+d-\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[3])dK[3]}+e^{\int _1^{K[6]}-\frac {1}{2} \left (a+d-\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[1])dK[1]+\int _1^{K[6]}-\frac {1}{2} \left (a+d+\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[4])dK[4]}\right )+\sqrt {a^2-2 d a+d^2+4 b c} \left (e^{\int _1^{K[6]}-\frac {1}{2} \left (a+d+\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[2])dK[2]+\int _1^{K[6]}-\frac {1}{2} \left (a+d-\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[3])dK[3]}+e^{\int _1^{K[6]}-\frac {1}{2} \left (a+d-\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[1])dK[1]+\int _1^{K[6]}-\frac {1}{2} \left (a+d+\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[4])dK[4]}\right )\right )}dK[6]-c e^{\int _1^t-\frac {1}{2} \left (a+d-\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[3])dK[3]} \int _1^t\frac {\sqrt {a^2-2 d a+d^2+4 b c} \left (2 c e^{\int _1^{K[5]}-\frac {1}{2} \left (a+d+\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[4])dK[4]} g(K[5])-\left (a-d+\sqrt {a^2-2 d a+d^2+4 b c}\right ) e^{\int _1^{K[5]}-\frac {1}{2} \left (a+d+\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[2])dK[2]} h(K[5])\right )}{c \left (a \left (e^{\int _1^{K[5]}-\frac {1}{2} \left (a+d+\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[2])dK[2]+\int _1^{K[5]}-\frac {1}{2} \left (a+d-\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[3])dK[3]}-e^{\int _1^{K[5]}-\frac {1}{2} \left (a+d-\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[1])dK[1]+\int _1^{K[5]}-\frac {1}{2} \left (a+d+\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[4])dK[4]}\right )+d \left (-e^{\int _1^{K[5]}-\frac {1}{2} \left (a+d+\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[2])dK[2]+\int _1^{K[5]}-\frac {1}{2} \left (a+d-\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[3])dK[3]}+e^{\int _1^{K[5]}-\frac {1}{2} \left (a+d-\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[1])dK[1]+\int _1^{K[5]}-\frac {1}{2} \left (a+d+\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[4])dK[4]}\right )+\sqrt {a^2-2 d a+d^2+4 b c} \left (e^{\int _1^{K[5]}-\frac {1}{2} \left (a+d+\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[2])dK[2]+\int _1^{K[5]}-\frac {1}{2} \left (a+d-\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[3])dK[3]}+e^{\int _1^{K[5]}-\frac {1}{2} \left (a+d-\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[1])dK[1]+\int _1^{K[5]}-\frac {1}{2} \left (a+d+\sqrt {a^2-2 d a+d^2+4 b c}\right ) f(K[4])dK[4]}\right )\right )}dK[5]}{\sqrt {a^2-2 d a+d^2+4 b c}}\right \}\right \}\] ✓ Maple : cpu = 2.962 (sec), leaf count = 2606
\[\text {Expression too large to display}\]