\[ \boxed { \left \{ {\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}x \left ( t \right ) =-{\frac {C \left ( y \left ( t \right ) \right ) f \left ( \sqrt { \left ( {\frac {\rm d}{{\rm d}t}}y \left ( t \right ) \right ) ^{2}} \right ) {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) }{\sqrt { \left ( {\frac {\rm d}{{\rm d}t}}y \left ( t \right ) \right ) ^{2}}}},{\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}y \left ( t \right ) =-{\frac {C \left ( y \left ( t \right ) \right ) f \left ( \sqrt { \left ( {\frac {\rm d}{{\rm d}t}}y \left ( t \right ) \right ) ^{2}} \right ) {\frac {\rm d}{{\rm d}t}}y \left ( t \right ) }{\sqrt { \left ( {\frac {\rm d}{{\rm d}t}}y \left ( t \right ) \right ) ^{2}}}}-g \right \} } \]
Mathematica: cpu = 0.008001 (sec), leaf count = 110 \[ \text {DSolve}\left [\left \{x''(t)=-\frac {c y(t) x'(t) f\left (\sqrt {x'(t)^2+y'(t)^2}\right )}{\sqrt {x'(t)^2+y'(t)^2}},y''(t)=-\frac {c y(t) y'(t) f\left (\sqrt {x'(t)^2+y'(t)^2}\right )}{\sqrt {x'(t)^2+y'(t)^2}}-g\right \},\{x(t),y(t)\},t\right ] \]
Maple: cpu = 2.247 (sec), leaf count = 116 \[ \left \{ [ \left \{ y \left ( t \right ) ={\it ODESolStruc} \left ( {\it \_a},[ \left \{ \left ( {\frac {\rm d}{{\rm d}{\it \_a}}}{\it \_b} \left ( {\it \_a} \right ) \right ) {\it \_b} \left ( {\it \_a} \right ) +{1 \left ( C \left ( {\it \_a} \right ) f \left ( \sqrt { \left ( {\it \_b } \left ( {\it \_a} \right ) \right ) ^{2}} \right ) {\it \_b} \left ( { \it \_a} \right ) +g\sqrt { \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{2}} \right ) {\frac {1}{\sqrt { \left ( {\it \_b} \left ( { \it \_a} \right ) \right ) ^{2}}}}}=0 \right \} , \left \{ {\it \_a}=y \left ( t \right ) ,{\it \_b} \left ( {\it \_a} \right ) ={\frac {\rm d}{ {\rm d}t}}y \left ( t \right ) \right \} , \left \{ t=\int \! \left ( { \it \_b} \left ( {\it \_a} \right ) \right ) ^{-1}\,{\rm d}{\it \_a}+{ \it \_C3},y \left ( t \right ) ={\it \_a} \right \} ] \right ) \right \} , \left \{ x \left ( t \right ) ={\it \_C1}+\int \!{{\rm e}^{\int \!-{ \frac {C \left ( y \left ( t \right ) \right ) }{ \left ( {\frac {\rm d}{ {\rm d}t}}y \left ( t \right ) \right ) ^{2}}\sqrt { \left ( {\frac {\rm d}{{\rm d}t}}y \left ( t \right ) \right ) ^{2}}f \left ( \sqrt { \left ( {\frac {\rm d}{{\rm d}t}}y \left ( t \right ) \right ) ^{2}} \right ) }\,{\rm d}t}}\,{\rm d}t{\it \_C2} \right \} ] \right \} \]