\[ \left \{f(t) (a x(t)+b y(t))+x'(t)=g(t),f(t) (c x(t)+d y(t))+y'(t)=h(t)\right \} \] ✗ Mathematica : cpu = 0.00706872 (sec), leaf count = 0 , could not solve
DSolve[{f[t]*(a*x[t] + b*y[t]) + Derivative[1][x][t] == g[t], f[t]*(c*x[t] + d*y[t]) + Derivative[1][y][t] == h[t]}, {x[t], y[t]}, t]
✓ Maple : cpu = 1.226 (sec), leaf count = 2011
\[ \left \{ \left \{ x \left ( t \right ) ={1 \left ( {{\rm e}^{{\frac {1}{2\,da-2\,bc}\int \!f \left ( t \right ) \sqrt {-da+bc}\,{\rm d}t \left ( \sqrt {{\frac {-{a}^{2}+2\,da-4\,bc-{d}^{2}}{da-bc}}} \left ( da-bc \right ) + \left ( a+d \right ) \sqrt {-da+bc} \right ) }}}{\it \_C2}\,\sqrt {-{\frac {{a}^{2}-2\,da+4\,bc+{d}^{2}}{da-bc}}}\sqrt {-da+bc}+{{\rm e}^{-{\frac {1}{2\,da-2\,bc}\int \!f \left ( t \right ) \sqrt {-da+bc}\,{\rm d}t \left ( \sqrt {{\frac {-{a}^{2}+2\,da-4\,bc-{d}^{2}}{da-bc}}} \left ( da-bc \right ) - \left ( a+d \right ) \sqrt {-da+bc} \right ) }}}{\it \_C1}\,\sqrt {-{\frac {{a}^{2}-2\,da+4\,bc+{d}^{2}}{da-bc}}}\sqrt {-da+bc}-\int \!-{\frac {dg \left ( t \right ) \left ( f \left ( t \right ) \right ) ^{2}-bh \left ( t \right ) \left ( f \left ( t \right ) \right ) ^{2}+ \left ( {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) \right ) f \left ( t \right ) - \left ( {\frac {\rm d}{{\rm d}t}}f \left ( t \right ) \right ) g \left ( t \right ) }{ \left ( f \left ( t \right ) \right ) ^{2}}{{\rm e}^{-{\frac {\int \!f \left ( t \right ) \,{\rm d}t}{2} \left ( \sqrt {-{\frac {{a}^{2}-2\,da+4\,bc+{d}^{2}}{da-bc}}}\sqrt {-da+bc}-a-d \right ) }}}}\,{\rm d}t{{\rm e}^{\int \!f \left ( t \right ) \,{\rm d}t\sqrt {-{\frac {{a}^{2}-2\,da+4\,bc+{d}^{2}}{da-bc}}}\sqrt {-da+bc}+{\frac {\int \!f \left ( t \right ) \,{\rm d}t}{2} \left ( \sqrt {{\frac {-{a}^{2}+2\,da-4\,bc-{d}^{2}}{da-bc}}} \left ( da-bc \right ) - \left ( a+d \right ) \sqrt {-da+bc} \right ) {\frac {1}{\sqrt {-da+bc}}}}}}+\int \!-{\frac {dg \left ( t \right ) \left ( f \left ( t \right ) \right ) ^{2}-bh \left ( t \right ) \left ( f \left ( t \right ) \right ) ^{2}+ \left ( {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) \right ) f \left ( t \right ) - \left ( {\frac {\rm d}{{\rm d}t}}f \left ( t \right ) \right ) g \left ( t \right ) }{ \left ( f \left ( t \right ) \right ) ^{2}}{{\rm e}^{{\frac {\int \!f \left ( t \right ) \,{\rm d}t}{2} \left ( \sqrt {-{\frac {{a}^{2}-2\,da+4\,bc+{d}^{2}}{da-bc}}}\sqrt {-da+bc}+a+d \right ) }}}}\,{\rm d}t{{\rm e}^{{\frac {\int \!f \left ( t \right ) \,{\rm d}t}{2} \left ( \sqrt {{\frac {-{a}^{2}+2\,da-4\,bc-{d}^{2}}{da-bc}}} \left ( da-bc \right ) - \left ( a+d \right ) \sqrt {-da+bc} \right ) {\frac {1}{\sqrt {-da+bc}}}}}} \right ) {\frac {1}{\sqrt {-{\frac {{a}^{2}-2\,da+4\,bc+{d}^{2}}{da-bc}}}}}{\frac {1}{\sqrt {-da+bc}}}},y \left ( t \right ) = \left ( -{\frac {d}{2\,b}{{\rm e}^{-{\frac {\int \!f \left ( t \right ) \,{\rm d}t}{2} \left ( \sqrt {-{\frac {{a}^{2}-2\,da+4\,bc+{d}^{2}}{da-bc}}}ad-\sqrt {-{\frac {{a}^{2}-2\,da+4\,bc+{d}^{2}}{da-bc}}}bc+\sqrt {-da+bc}a+\sqrt {-da+bc}d \right ) {\frac {1}{\sqrt {-da+bc}}}}}}{\frac {1}{\sqrt {-{\frac {{a}^{2}-2\,da+4\,bc+{d}^{2}}{da-bc}}}}}{\frac {1}{\sqrt {-da+bc}}}}+{\frac {1}{b} \left ( {\frac {a}{2}{{\rm e}^{-{\frac {\int \!f \left ( t \right ) \,{\rm d}t}{2} \left ( \sqrt {-{\frac {{a}^{2}-2\,da+4\,bc+{d}^{2}}{da-bc}}}ad-\sqrt {-{\frac {{a}^{2}-2\,da+4\,bc+{d}^{2}}{da-bc}}}bc+\sqrt {-da+bc}a+\sqrt {-da+bc}d \right ) {\frac {1}{\sqrt {-da+bc}}}}}}{\frac {1}{\sqrt {-{\frac {{a}^{2}-2\,da+4\,bc+{d}^{2}}{da-bc}}}}}{\frac {1}{\sqrt {-da+bc}}}}+{\frac {1}{2}{{\rm e}^{-{\frac {\int \!f \left ( t \right ) \,{\rm d}t}{2} \left ( \sqrt {-{\frac {{a}^{2}-2\,da+4\,bc+{d}^{2}}{da-bc}}}ad-\sqrt {-{\frac {{a}^{2}-2\,da+4\,bc+{d}^{2}}{da-bc}}}bc+\sqrt {-da+bc}a+\sqrt {-da+bc}d \right ) {\frac {1}{\sqrt {-da+bc}}}}}}} \right ) } \right ) \int \!-{\frac {dg \left ( t \right ) \left ( f \left ( t \right ) \right ) ^{2}-bh \left ( t \right ) \left ( f \left ( t \right ) \right ) ^{2}+ \left ( {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) \right ) f \left ( t \right ) - \left ( {\frac {\rm d}{{\rm d}t}}f \left ( t \right ) \right ) g \left ( t \right ) }{ \left ( f \left ( t \right ) \right ) ^{2}}{{\rm e}^{-{\frac {\int \!f \left ( t \right ) \,{\rm d}t}{2} \left ( \sqrt {-{\frac {{a}^{2}-2\,da+4\,bc+{d}^{2}}{da-bc}}}\sqrt {-da+bc}-a-d \right ) }}}}\,{\rm d}t+ \left ( {\frac {d}{2\,b}{\frac {1}{\sqrt {-{\frac {{a}^{2}-2\,da+4\,bc+{d}^{2}}{da-bc}}}}}{\frac {1}{\sqrt {-da+bc}}}}+{\frac {1}{b} \left ( {\frac {1}{2}}-{\frac {a}{2}{\frac {1}{\sqrt {-{\frac {{a}^{2}-2\,da+4\,bc+{d}^{2}}{da-bc}}}}}{\frac {1}{\sqrt {-da+bc}}}} \right ) } \right ) {{\rm e}^{{\frac {\int \!f \left ( t \right ) \,{\rm d}t}{2} \left ( \sqrt {-{\frac {{a}^{2}-2\,da+4\,bc+{d}^{2}}{da-bc}}} \left ( da-bc \right ) - \left ( a+d \right ) \sqrt {-da+bc} \right ) {\frac {1}{\sqrt {-da+bc}}}}}}\int \!-{\frac {dg \left ( t \right ) \left ( f \left ( t \right ) \right ) ^{2}-bh \left ( t \right ) \left ( f \left ( t \right ) \right ) ^{2}+ \left ( {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) \right ) f \left ( t \right ) - \left ( {\frac {\rm d}{{\rm d}t}}f \left ( t \right ) \right ) g \left ( t \right ) }{ \left ( f \left ( t \right ) \right ) ^{2}}{{\rm e}^{{\frac {\int \!f \left ( t \right ) \,{\rm d}t}{2} \left ( \sqrt {-{\frac {{a}^{2}-2\,da+4\,bc+{d}^{2}}{da-bc}}}\sqrt {-da+bc}+a+d \right ) }}}}\,{\rm d}t+ \left ( -{\frac {{d}^{2}}{2\,b}{\frac {1}{\sqrt {-{\frac {{a}^{2}-2\,da+4\,bc+{d}^{2}}{da-bc}}}}}{\frac {1}{\sqrt {-da+bc}}}}+{\frac {d}{b} \left ( {\frac {1}{2}}+{a{\frac {1}{\sqrt {-{\frac {{a}^{2}-2\,da+4\,bc+{d}^{2}}{da-bc}}}}}{\frac {1}{\sqrt {-da+bc}}}} \right ) }-2\,{\frac {c}{\sqrt {-da+bc}}{\frac {1}{\sqrt {-{\frac {{a}^{2}-2\,da+4\,bc+{d}^{2}}{da-bc}}}}}}+{\frac {1}{b} \left ( -{\frac {{a}^{2}}{2}{\frac {1}{\sqrt {-{\frac {{a}^{2}-2\,da+4\,bc+{d}^{2}}{da-bc}}}}}{\frac {1}{\sqrt {-da+bc}}}}-{\frac {a}{2}} \right ) } \right ) {\it \_C2}\,{{\rm e}^{{\frac {1}{2\,da-2\,bc}\int \!f \left ( t \right ) \sqrt {-da+bc}\,{\rm d}t \left ( \sqrt {-{\frac {{a}^{2}-2\,da+4\,bc+{d}^{2}}{da-bc}}} \left ( da-bc \right ) + \left ( a+d \right ) \sqrt {-da+bc} \right ) }}}+ \left ( {\frac {{d}^{2}}{2\,b}{\frac {1}{\sqrt {-{\frac {{a}^{2}-2\,da+4\,bc+{d}^{2}}{da-bc}}}}}{\frac {1}{\sqrt {-da+bc}}}}+{\frac {d}{b} \left ( {\frac {1}{2}}-{a{\frac {1}{\sqrt {-{\frac {{a}^{2}-2\,da+4\,bc+{d}^{2}}{da-bc}}}}}{\frac {1}{\sqrt {-da+bc}}}} \right ) }+2\,{\frac {c}{\sqrt {-da+bc}}{\frac {1}{\sqrt {-{\frac {{a}^{2}-2\,da+4\,bc+{d}^{2}}{da-bc}}}}}}+{\frac {1}{b} \left ( {\frac {{a}^{2}}{2}{\frac {1}{\sqrt {-{\frac {{a}^{2}-2\,da+4\,bc+{d}^{2}}{da-bc}}}}}{\frac {1}{\sqrt {-da+bc}}}}-{\frac {a}{2}} \right ) } \right ) {\it \_C1}\,{{\rm e}^{-{\frac {1}{2\,da-2\,bc}\int \!f \left ( t \right ) \sqrt {-da+bc}\,{\rm d}t \left ( \sqrt {-{\frac {{a}^{2}-2\,da+4\,bc+{d}^{2}}{da-bc}}} \left ( da-bc \right ) - \left ( a+d \right ) \sqrt {-da+bc} \right ) }}}+{\frac {g \left ( t \right ) }{bf \left ( t \right ) }} \right \} \right \} \]