\[ \left \{f(t) (a x(t)+b y(t))+x'(t)=g(t),f(t) (c x(t)+d y(t))+y'(t)=h(t)\right \} \] ✗ Mathematica : cpu = 0.00713852 (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 = 1.018 (sec), leaf count = 1361
\[ \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 {-ad+bc}\sqrt {{\frac {-{a}^{2}+2\,ad-4\,bc-{d}^{2}}{ad-bc}}}+a+d \right ) }}}}\,{\rm d}t{{\rm e}^{-{\frac {\int \!f \left ( t \right ) \,{\rm d}t}{2} \left ( -\sqrt {-ad+bc}\sqrt {{\frac {-{a}^{2}+2\,ad-4\,bc-{d}^{2}}{ad-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 {-ad+bc}\sqrt {{\frac {-{a}^{2}+2\,ad-4\,bc-{d}^{2}}{ad-bc}}}+a+d \right ) }}}}\,{\rm d}t{{\rm e}^{-{\frac {\int \!f \left ( t \right ) \,{\rm d}t}{2} \left ( \sqrt {-ad+bc}\sqrt {{\frac {-{a}^{2}+2\,ad-4\,bc-{d}^{2}}{ad-bc}}}+a+d \right ) }}}+\sqrt {-ad+bc}\sqrt {{\frac {-{a}^{2}+2\,ad-4\,bc-{d}^{2}}{ad-bc}}} \left ( {{\rm e}^{{\frac {1}{2\,ad-2\,bc} \left ( \sqrt {{\frac {-{a}^{2}+2\,ad-4\,bc-{d}^{2}}{ad-bc}}} \left ( ad-bc \right ) + \left ( a+d \right ) \sqrt {-ad+bc} \right ) \int \!f \left ( t \right ) \sqrt {-ad+bc}\,{\rm d}t}}}{\it \_C2}+{{\rm e}^{-{\frac {1}{2\,ad-2\,bc} \left ( \sqrt {{\frac {-{a}^{2}+2\,ad-4\,bc-{d}^{2}}{ad-bc}}} \left ( ad-bc \right ) - \left ( a+d \right ) \sqrt {-ad+bc} \right ) \int \!f \left ( t \right ) \sqrt {-ad+bc}\,{\rm d}t}}}{\it \_C1} \right ) \right ) {\frac {1}{\sqrt {-ad+bc}}}{\frac {1}{\sqrt {{\frac {-{a}^{2}+2\,ad-4\,bc-{d}^{2}}{ad-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 {-ad+bc}\sqrt {{\frac {-{a}^{2}+2\,ad-4\,bc-{d}^{2}}{ad-bc}}}+a+d \right ) }}}f \left ( t \right ) \left ( \sqrt {-ad+bc}\sqrt {{\frac {-{a}^{2}+2\,ad-4\,bc-{d}^{2}}{ad-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 {-ad+bc}\sqrt {{\frac {-{a}^{2}+2\,ad-4\,bc-{d}^{2}}{ad-bc}}}+a+d \right ) }}}}\,{\rm d}t-{{\rm e}^{-{\frac {\int \!f \left ( t \right ) \,{\rm d}t}{2} \left ( \sqrt {-ad+bc}\sqrt {{\frac {-{a}^{2}+2\,ad-4\,bc-{d}^{2}}{ad-bc}}}+a+d \right ) }}}f \left ( t \right ) \left ( -\sqrt {-ad+bc}\sqrt {{\frac {-{a}^{2}+2\,ad-4\,bc-{d}^{2}}{ad-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 {-ad+bc}\sqrt {{\frac {-{a}^{2}+2\,ad-4\,bc-{d}^{2}}{ad-bc}}}+a+d \right ) }}}}\,{\rm d}t+ \left ( -\sqrt {-ad+bc} \left ( a-d \right ) \sqrt {{\frac {-{a}^{2}+2\,ad-4\,bc-{d}^{2}}{ad-bc}}}+{a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) f \left ( t \right ) {\it \_C1}\,{{\rm e}^{-{\frac {1}{2\,ad-2\,bc} \left ( \sqrt {{\frac {-{a}^{2}+2\,ad-4\,bc-{d}^{2}}{ad-bc}}} \left ( ad-bc \right ) - \left ( a+d \right ) \sqrt {-ad+bc} \right ) \int \!f \left ( t \right ) \sqrt {-ad+bc}\,{\rm d}t}}}-{\it \_C2}\, \left ( \sqrt {-ad+bc} \left ( a-d \right ) \sqrt {{\frac {-{a}^{2}+2\,ad-4\,bc-{d}^{2}}{ad-bc}}}+{a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) f \left ( t \right ) {{\rm e}^{{\frac {1}{2\,ad-2\,bc} \left ( \sqrt {{\frac {-{a}^{2}+2\,ad-4\,bc-{d}^{2}}{ad-bc}}} \left ( ad-bc \right ) + \left ( a+d \right ) \sqrt {-ad+bc} \right ) \int \!f \left ( t \right ) \sqrt {-ad+bc}\,{\rm d}t}}}+2\,g \left ( t \right ) \sqrt {-ad+bc}\sqrt {{\frac {-{a}^{2}+2\,ad-4\,bc-{d}^{2}}{ad-bc}}} \right ) {\frac {1}{\sqrt {-ad+bc}}}{\frac {1}{\sqrt {{\frac {-{a}^{2}+2\,ad-4\,bc-{d}^{2}}{ad-bc}}}}}} \right \} \right \} \]