\[ \left \{f(t) (a x(t)+b y(t))+x'(t)=g(t),f(t) (c x(t)+d y(t))+y'(t)=h(t)\right \} \] ✗ Mathematica : cpu = 0.00660219 (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 = 1.426 (sec), leaf count = 1447
\[ \left \{ \left \{ x \left ( t \right ) ={1 \left ( -\int \!{\frac {g \left ( t \right ) {\frac {\rm d}{{\rm d}t}}f \left ( t \right ) -f \left ( t \right ) \left ( {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) -f \left ( t \right ) \left ( h \left ( t \right ) b-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\,ad-4\,bc-{d}^{2}}{ad-bc}}}\sqrt {-ad+bc}+a+d \right ) }}}}\,{\rm d}t{{\rm e}^{{\frac {\int \!f \left ( t \right ) \,{\rm d}t}{2} \left ( \left ( {(ad-bc){\frac {1}{\sqrt {-ad+bc}}}}+2\,\sqrt {-ad+bc} \right ) \sqrt {{\frac {-{a}^{2}+2\,ad-4\,bc-{d}^{2}}{ad-bc}}}-a-d \right ) }}}+{{\rm e}^{{\frac {\int \!f \left ( t \right ) \,{\rm d}t}{2} \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 ) {\frac {1}{\sqrt {-ad+bc}}}}}}\int \!{\frac {g \left ( t \right ) {\frac {\rm d}{{\rm d}t}}f \left ( t \right ) -f \left ( t \right ) \left ( {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) -f \left ( t \right ) \left ( h \left ( t \right ) b-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\,ad-4\,bc-{d}^{2}}{ad-bc}}}\sqrt {-ad+bc}+a+d \right ) }}}}\,{\rm d}t+\sqrt {-ad+bc}\sqrt {{\frac {-{a}^{2}+2\,ad-4\,bc-{d}^{2}}{ad-bc}}} \left ( {{\rm e}^{{\frac {1}{2\,ad-2\,bc}\int \!f \left ( t \right ) \sqrt {-ad+bc}\,{\rm d}t \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 ) }}}{\it \_C2}+{{\rm e}^{-{\frac {1}{2\,ad-2\,bc}\int \!f \left ( t \right ) \sqrt {-ad+bc}\,{\rm d}t \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 ) }}}{\it \_C1} \right ) \right ) {\frac {1}{\sqrt {{\frac {-{a}^{2}+2\,ad-4\,bc-{d}^{2}}{ad-bc}}}}}{\frac {1}{\sqrt {-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 {{\frac {-{a}^{2}+2\,ad-4\,bc-{d}^{2}}{ad-bc}}} \left ( ad-bc \right ) + \left ( a+d \right ) \sqrt {-ad+bc} \right ) {\frac {1}{\sqrt {-ad+bc}}}}}}f \left ( t \right ) \left ( \sqrt {{\frac {-{a}^{2}+2\,ad-4\,bc-{d}^{2}}{ad-bc}}}\sqrt {-ad+bc}+a-d \right ) \int \!{\frac {g \left ( t \right ) {\frac {\rm d}{{\rm d}t}}f \left ( t \right ) -f \left ( t \right ) \left ( {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) -f \left ( t \right ) \left ( h \left ( t \right ) b-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\,ad-4\,bc-{d}^{2}}{ad-bc}}}\sqrt {-ad+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\,ad-4\,bc-{d}^{2}}{ad-bc}}} \left ( ad-bc \right ) - \left ( a+d \right ) \sqrt {-ad+bc} \right ) {\frac {1}{\sqrt {-ad+bc}}}}}}f \left ( t \right ) \left ( -\sqrt {{\frac {-{a}^{2}+2\,ad-4\,bc-{d}^{2}}{ad-bc}}}\sqrt {-ad+bc}+a-d \right ) \int \!{\frac {g \left ( t \right ) {\frac {\rm d}{{\rm d}t}}f \left ( t \right ) -f \left ( t \right ) \left ( {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) -f \left ( t \right ) \left ( h \left ( t \right ) b-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\,ad-4\,bc-{d}^{2}}{ad-bc}}}\sqrt {-ad+bc}+a+d \right ) }}}}\,{\rm d}t+{\it \_C1}\,f \left ( t \right ) \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 ) {{\rm e}^{-{\frac {1}{2\,ad-2\,bc}\int \!f \left ( t \right ) \sqrt {-ad+bc}\,{\rm d}t \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 ) }}}-f \left ( t \right ) {\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 ) {{\rm e}^{{\frac {1}{2\,ad-2\,bc}\int \!f \left ( t \right ) \sqrt {-ad+bc}\,{\rm d}t \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 ) }}}+2\,g \left ( t \right ) \sqrt {-ad+bc}\sqrt {{\frac {-{a}^{2}+2\,ad-4\,bc-{d}^{2}}{ad-bc}}} \right ) {\frac {1}{\sqrt {{\frac {-{a}^{2}+2\,ad-4\,bc-{d}^{2}}{ad-bc}}}}}{\frac {1}{\sqrt {-ad+bc}}}} \right \} \right \} \]