\[ \left \{f(t) (a x(t)+b y(t))+x'(t)=g(t),f(t) (c x(t)+d y(t))+y'(t)=h(t)\right \} \] ✗ Mathematica : cpu = 0.00797435 (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 = 1.264 (sec), leaf count = 1447
\[ \left \{ \left \{ x \left ( t \right ) ={1 \left ( -\int \!{\frac { \left ( {\frac {\rm d}{{\rm d}t}}f \left ( t \right ) \right ) g \left ( t \right ) -f \left ( t \right ) \left ( {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) -f \left ( t \right ) \left ( bh \left ( t \right ) -g \left ( t \right ) d \right ) \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 ( \left ( {(da-bc){\frac {1}{\sqrt {-da+bc}}}}+2\,\sqrt {-da+bc} \right ) \sqrt {{\frac {-{a}^{2}+2\,da-4\,bc-{d}^{2}}{da-bc}}}-a-d \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 { \left ( {\frac {\rm d}{{\rm d}t}}f \left ( t \right ) \right ) g \left ( t \right ) -f \left ( t \right ) \left ( {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) -f \left ( t \right ) \left ( bh \left ( t \right ) -g \left ( t \right ) d \right ) \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+\sqrt {-da+bc}\sqrt {{\frac {-{a}^{2}+2\,da-4\,bc-{d}^{2}}{da-bc}}} \left ( {{\rm e}^{{\frac {1}{2\,da-2\,bc} \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 ) \int \!f \left ( t \right ) \sqrt {-da+bc}\,{\rm d}t}}}{\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 ) }}}{\it \_C1} \right ) \right ) {\frac {1}{\sqrt {{\frac {-{a}^{2}+2\,da-4\,bc-{d}^{2}}{da-bc}}}}}{\frac {1}{\sqrt {-da+bc}}}},y \left ( t \right ) =-{\frac {1}{2\,bf \left ( t \right ) } \left ( -{{\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}}}}}}f \left ( t \right ) \left ( \sqrt {{\frac {-{a}^{2}+2\,da-4\,bc-{d}^{2}}{da-bc}}}\sqrt {-da+bc}+a-d \right ) \int \!{\frac { \left ( {\frac {\rm d}{{\rm d}t}}f \left ( t \right ) \right ) g \left ( t \right ) -f \left ( t \right ) \left ( {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) -f \left ( t \right ) \left ( bh \left ( t \right ) -g \left ( t \right ) d \right ) \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}}}}}}f \left ( t \right ) \left ( -\sqrt {{\frac {-{a}^{2}+2\,da-4\,bc-{d}^{2}}{da-bc}}}\sqrt {-da+bc}+a-d \right ) \int \!{\frac { \left ( {\frac {\rm d}{{\rm d}t}}f \left ( t \right ) \right ) g \left ( t \right ) -f \left ( t \right ) \left ( {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) -f \left ( t \right ) \left ( bh \left ( t \right ) -g \left ( t \right ) d \right ) \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-f \left ( t \right ) {\it \_C1}\, \left ( -\sqrt {-da+bc} \left ( a-d \right ) \sqrt {{\frac {-{a}^{2}+2\,da-4\,bc-{d}^{2}}{da-bc}}}+{a}^{2}-2\,da+4\,bc+{d}^{2} \right ) {{\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 ) }}}+f \left ( t \right ) {\it \_C2}\, \left ( \sqrt {-da+bc} \left ( a-d \right ) \sqrt {{\frac {-{a}^{2}+2\,da-4\,bc-{d}^{2}}{da-bc}}}+{a}^{2}-2\,da+4\,bc+{d}^{2} \right ) {{\rm e}^{{\frac {1}{2\,da-2\,bc} \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 ) \int \!f \left ( t \right ) \sqrt {-da+bc}\,{\rm d}t}}}-2\,g \left ( t \right ) \sqrt {{\frac {-{a}^{2}+2\,da-4\,bc-{d}^{2}}{da-bc}}}\sqrt {-da+bc} \right ) {\frac {1}{\sqrt {{\frac {-{a}^{2}+2\,da-4\,bc-{d}^{2}}{da-bc}}}}}{\frac {1}{\sqrt {-da+bc}}}} \right \} \right \} \]