\[ \boxed { \left ( \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{2}+1 \right ) {\frac {{\rm d}^{3}}{{\rm d}{x}^{3}}}y \left ( x \right ) - \left ( 3\,{\frac {\rm d}{{\rm d}x}}y \left ( x \right ) +a \right ) \left ( {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) \right ) ^{2}=0} \]
Mathematica: cpu = 594.052935 (sec), leaf count = 37 \[ \text {DSolve}\left [y''(x)^2 \left (-a-3 y'(x)\right )+y^{(3)}(x) \left (y'(x)^2+1\right )=0,y(x),x\right ] \]
Maple: cpu = 0.843 (sec), leaf count = 789 \[ \left \{ y \left ( x \right ) =\int \!{\frac {\sin \left ( {\it RootOf} \left ( {{\rm e}^{2\,{\it \_Z}\,a}}{{\it \_C1}}^{2}{{\it \_C2}}^{2}{a} ^{4}+2\,{{\rm e}^{2\,{\it \_Z}\,a}}{{\it \_C1}}^{2}{\it \_C2}\,{a}^{4} x+{{\rm e}^{2\,{\it \_Z}\,a}}{{\it \_C1}}^{2}{a}^{4}{x}^{2}+2\,{ {\rm e}^{2\,{\it \_Z}\,a}}{{\it \_C1}}^{2}{{\it \_C2}}^{2}{a}^{2}+4\,{ {\rm e}^{2\,{\it \_Z}\,a}}{{\it \_C1}}^{2}{\it \_C2}\,{a}^{2}x+2\,{ {\rm e}^{2\,{\it \_Z}\,a}}{{\it \_C1}}^{2}{a}^{2}{x}^{2}-2\,\cos \left ( {\it \_Z} \right ) {{\rm e}^{{\it \_Z}\,a}}{\it \_C1}\,{\it \_C2}\,{a}^{3}-2\,\cos \left ( {\it \_Z} \right ) {{\rm e}^{{\it \_Z}\,a }}{\it \_C1}\,{a}^{3}x+{{\rm e}^{2\,{\it \_Z}\,a}}{{\it \_C1}}^{2}{{ \it \_C2}}^{2}+2\,{{\rm e}^{2\,{\it \_Z}\,a}}{{\it \_C1}}^{2}{\it \_C2 }\,x+{{\rm e}^{2\,{\it \_Z}\,a}}{{\it \_C1}}^{2}{x}^{2}-2\,\cos \left ( {\it \_Z} \right ) {{\rm e}^{{\it \_Z}\,a}}{\it \_C1}\,{\it \_C2}\,a-2\,\cos \left ( {\it \_Z} \right ) {{\rm e}^{{\it \_Z}\,a}}{ \it \_C1}\,ax+ \left ( \cos \left ( {\it \_Z} \right ) \right ) ^{2}{a}^{ 2}+ \left ( \cos \left ( {\it \_Z} \right ) \right ) ^{2}-1 \right ) \right ) }{\cos \left ( {\it RootOf} \left ( {{\rm e}^{2\,{\it \_Z}\,a}} {{\it \_C1}}^{2}{{\it \_C2}}^{2}{a}^{4}+2\,{{\rm e}^{2\,{\it \_Z}\,a}} {{\it \_C1}}^{2}{\it \_C2}\,{a}^{4}x+{{\rm e}^{2\,{\it \_Z}\,a}}{{\it \_C1}}^{2}{a}^{4}{x}^{2}+2\,{{\rm e}^{2\,{\it \_Z}\,a}}{{\it \_C1}}^{2 }{{\it \_C2}}^{2}{a}^{2}+4\,{{\rm e}^{2\,{\it \_Z}\,a}}{{\it \_C1}}^{2 }{\it \_C2}\,{a}^{2}x+2\,{{\rm e}^{2\,{\it \_Z}\,a}}{{\it \_C1}}^{2}{a }^{2}{x}^{2}-2\,\cos \left ( {\it \_Z} \right ) {{\rm e}^{{\it \_Z}\,a}} {\it \_C1}\,{\it \_C2}\,{a}^{3}-2\,\cos \left ( {\it \_Z} \right ) { {\rm e}^{{\it \_Z}\,a}}{\it \_C1}\,{a}^{3}x+{{\rm e}^{2\,{\it \_Z}\,a} }{{\it \_C1}}^{2}{{\it \_C2}}^{2}+2\,{{\rm e}^{2\,{\it \_Z}\,a}}{{\it \_C1}}^{2}{\it \_C2}\,x+{{\rm e}^{2\,{\it \_Z}\,a}}{{\it \_C1}}^{2}{x} ^{2}-2\,\cos \left ( {\it \_Z} \right ) {{\rm e}^{{\it \_Z}\,a}}{\it \_C1}\,{\it \_C2}\,a-2\,\cos \left ( {\it \_Z} \right ) {{\rm e}^{{\it \_Z}\,a}}{\it \_C1}\,ax+ \left ( \cos \left ( {\it \_Z} \right ) \right ) ^{2}{a}^{2}+ \left ( \cos \left ( {\it \_Z} \right ) \right ) ^{ 2}-1 \right ) \right ) }}\,{\rm d}x+{\it \_C3},y \left ( x \right ) = \int \!{\frac {\sin \left ( {\it RootOf} \left ( {{\rm e}^{2\,{\it \_Z} \,a}}{{\it \_C1}}^{2}{{\it \_C2}}^{2}{a}^{4}+2\,{{\rm e}^{2\,{\it \_Z} \,a}}{{\it \_C1}}^{2}{\it \_C2}\,{a}^{4}x+{{\rm e}^{2\,{\it \_Z}\,a}}{ {\it \_C1}}^{2}{a}^{4}{x}^{2}+2\,{{\rm e}^{2\,{\it \_Z}\,a}}{{\it \_C1 }}^{2}{{\it \_C2}}^{2}{a}^{2}+4\,{{\rm e}^{2\,{\it \_Z}\,a}}{{\it \_C1 }}^{2}{\it \_C2}\,{a}^{2}x+2\,{{\rm e}^{2\,{\it \_Z}\,a}}{{\it \_C1}}^ {2}{a}^{2}{x}^{2}+2\,\cos \left ( {\it \_Z} \right ) {{\rm e}^{{\it \_Z} \,a}}{\it \_C1}\,{\it \_C2}\,{a}^{3}+2\,\cos \left ( {\it \_Z} \right ) {{\rm e}^{{\it \_Z}\,a}}{\it \_C1}\,{a}^{3}x+{{\rm e}^{2\,{\it \_Z}\,a }}{{\it \_C1}}^{2}{{\it \_C2}}^{2}+2\,{{\rm e}^{2\,{\it \_Z}\,a}}{{ \it \_C1}}^{2}{\it \_C2}\,x+{{\rm e}^{2\,{\it \_Z}\,a}}{{\it \_C1}}^{2 }{x}^{2}+2\,\cos \left ( {\it \_Z} \right ) {{\rm e}^{{\it \_Z}\,a}}{ \it \_C1}\,{\it \_C2}\,a+2\,\cos \left ( {\it \_Z} \right ) {{\rm e}^{{ \it \_Z}\,a}}{\it \_C1}\,ax+ \left ( \cos \left ( {\it \_Z} \right ) \right ) ^{2}{a}^{2}+ \left ( \cos \left ( {\it \_Z} \right ) \right ) ^{ 2}-1 \right ) \right ) }{\cos \left ( {\it RootOf} \left ( {{\rm e}^{2\,{ \it \_Z}\,a}}{{\it \_C1}}^{2}{{\it \_C2}}^{2}{a}^{4}+2\,{{\rm e}^{2\,{ \it \_Z}\,a}}{{\it \_C1}}^{2}{\it \_C2}\,{a}^{4}x+{{\rm e}^{2\,{\it \_Z}\,a}}{{\it \_C1}}^{2}{a}^{4}{x}^{2}+2\,{{\rm e}^{2\,{\it \_Z}\,a}} {{\it \_C1}}^{2}{{\it \_C2}}^{2}{a}^{2}+4\,{{\rm e}^{2\,{\it \_Z}\,a}} {{\it \_C1}}^{2}{\it \_C2}\,{a}^{2}x+2\,{{\rm e}^{2\,{\it \_Z}\,a}}{{ \it \_C1}}^{2}{a}^{2}{x}^{2}+2\,\cos \left ( {\it \_Z} \right ) {{\rm e} ^{{\it \_Z}\,a}}{\it \_C1}\,{\it \_C2}\,{a}^{3}+2\,\cos \left ( {\it \_Z} \right ) {{\rm e}^{{\it \_Z}\,a}}{\it \_C1}\,{a}^{3}x+{{\rm e}^{2 \,{\it \_Z}\,a}}{{\it \_C1}}^{2}{{\it \_C2}}^{2}+2\,{{\rm e}^{2\,{\it \_Z}\,a}}{{\it \_C1}}^{2}{\it \_C2}\,x+{{\rm e}^{2\,{\it \_Z}\,a}}{{ \it \_C1}}^{2}{x}^{2}+2\,\cos \left ( {\it \_Z} \right ) {{\rm e}^{{\it \_Z}\,a}}{\it \_C1}\,{\it \_C2}\,a+2\,\cos \left ( {\it \_Z} \right ) { {\rm e}^{{\it \_Z}\,a}}{\it \_C1}\,ax+ \left ( \cos \left ( {\it \_Z} \right ) \right ) ^{2}{a}^{2}+ \left ( \cos \left ( {\it \_Z} \right ) \right ) ^{2}-1 \right ) \right ) }}\,{\rm d}x+{\it \_C3} \right \} \]