\[ y''(x)^2 \left (-a-3 y'(x)\right )+y^{(3)}(x) \left (y'(x)^2+1\right )=0 \] ✓ Mathematica : cpu = 0.409356 (sec), leaf count = 187
DSolve[(-a - 3*Derivative[1][y][x])*Derivative[2][y][x]^2 + (1 + Derivative[1][y][x]^2)*Derivative[3][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to c_3-\frac {\left (1-i \text {InverseFunction}\left [\frac {(\text {$\#$1}-a) e^{-a \tan ^{-1}(\text {$\#$1})}}{\sqrt {\text {$\#$1}^2+1} \left (a^2+1\right ) c_1}\& \right ][x+c_2]\right ){}^{-\frac {1}{2}-\frac {i a}{2}} \left (1+i \text {InverseFunction}\left [\frac {(\text {$\#$1}-a) e^{-a \tan ^{-1}(\text {$\#$1})}}{\sqrt {\text {$\#$1}^2+1} \left (a^2+1\right ) c_1}\& \right ][x+c_2]\right ){}^{\frac {1}{2} i (a+i)} \left (1+a \text {InverseFunction}\left [\frac {(\text {$\#$1}-a) e^{-a \tan ^{-1}(\text {$\#$1})}}{\sqrt {\text {$\#$1}^2+1} \left (a^2+1\right ) c_1}\& \right ][x+c_2]\right )}{\left (a^2+1\right ) c_1}\right \}\right \}\] ✓ Maple : cpu = 1.308 (sec), leaf count = 789
dsolve((diff(y(x),x)^2+1)*diff(diff(diff(y(x),x),x),x)-(3*diff(y(x),x)+a)*diff(diff(y(x),x),x)^2=0,y(x))
\[y \left (x \right ) = \int \frac {\sin \left (\RootOf \left ({\mathrm e}^{2 a \textit {\_Z}} c_{1}^{2} c_{2}^{2} a^{4}+2 \,{\mathrm e}^{2 a \textit {\_Z}} c_{1}^{2} c_{2} a^{4} x +{\mathrm e}^{2 a \textit {\_Z}} c_{1}^{2} a^{4} x^{2}+2 \,{\mathrm e}^{2 a \textit {\_Z}} c_{1}^{2} c_{2}^{2} a^{2}+4 \,{\mathrm e}^{2 a \textit {\_Z}} c_{1}^{2} c_{2} a^{2} x +2 \,{\mathrm e}^{2 a \textit {\_Z}} c_{1}^{2} a^{2} x^{2}-2 \,{\mathrm e}^{a \textit {\_Z}} \cos \left (\textit {\_Z} \right ) c_{1} c_{2} a^{3}-2 \,{\mathrm e}^{a \textit {\_Z}} \cos \left (\textit {\_Z} \right ) c_{1} a^{3} x +{\mathrm e}^{2 a \textit {\_Z}} c_{1}^{2} c_{2}^{2}+2 \,{\mathrm e}^{2 a \textit {\_Z}} c_{1}^{2} c_{2} x +{\mathrm e}^{2 a \textit {\_Z}} c_{1}^{2} x^{2}-2 \,{\mathrm e}^{a \textit {\_Z}} \cos \left (\textit {\_Z} \right ) c_{1} c_{2} a -2 \,{\mathrm e}^{a \textit {\_Z}} \cos \left (\textit {\_Z} \right ) c_{1} a x +\left (\cos ^{2}\left (\textit {\_Z} \right )\right ) a^{2}+\cos ^{2}\left (\textit {\_Z} \right )-1\right )\right )}{\cos \left (\RootOf \left ({\mathrm e}^{2 a \textit {\_Z}} c_{1}^{2} c_{2}^{2} a^{4}+2 \,{\mathrm e}^{2 a \textit {\_Z}} c_{1}^{2} c_{2} a^{4} x +{\mathrm e}^{2 a \textit {\_Z}} c_{1}^{2} a^{4} x^{2}+2 \,{\mathrm e}^{2 a \textit {\_Z}} c_{1}^{2} c_{2}^{2} a^{2}+4 \,{\mathrm e}^{2 a \textit {\_Z}} c_{1}^{2} c_{2} a^{2} x +2 \,{\mathrm e}^{2 a \textit {\_Z}} c_{1}^{2} a^{2} x^{2}-2 \,{\mathrm e}^{a \textit {\_Z}} \cos \left (\textit {\_Z} \right ) c_{1} c_{2} a^{3}-2 \,{\mathrm e}^{a \textit {\_Z}} \cos \left (\textit {\_Z} \right ) c_{1} a^{3} x +{\mathrm e}^{2 a \textit {\_Z}} c_{1}^{2} c_{2}^{2}+2 \,{\mathrm e}^{2 a \textit {\_Z}} c_{1}^{2} c_{2} x +{\mathrm e}^{2 a \textit {\_Z}} c_{1}^{2} x^{2}-2 \,{\mathrm e}^{a \textit {\_Z}} \cos \left (\textit {\_Z} \right ) c_{1} c_{2} a -2 \,{\mathrm e}^{a \textit {\_Z}} \cos \left (\textit {\_Z} \right ) c_{1} a x +\left (\cos ^{2}\left (\textit {\_Z} \right )\right ) a^{2}+\cos ^{2}\left (\textit {\_Z} \right )-1\right )\right )}d x +c_{3}\]