\[ \boxed { \left ( {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) \right ) y \left ( x \right ) +a \left ( \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{2}+1 \right ) =0} \]
Mathematica: cpu = 0.645082 (sec), leaf count = 172 \[ \left \{\left \{y(x)\to \text {InverseFunction}\left [-\frac {\text {$\#$1} \sqrt {1-e^{2 c_1} \text {$\#$1}^{-2 a}} \, _2F_1\left (\frac {1}{2},-\frac {1}{2 a};1-\frac {1}{2 a};e^{2 c_1} \text {$\#$1}^{-2 a}\right )}{\sqrt {e^{2 c_1} \text {$\#$1}^{-2 a}-1}}\& \right ]\left [c_2+x\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {\text {$\#$1} \sqrt {1-e^{2 c_1} \text {$\#$1}^{-2 a}} \, _2F_1\left (\frac {1}{2},-\frac {1}{2 a};1-\frac {1}{2 a};e^{2 c_1} \text {$\#$1}^{-2 a}\right )}{\sqrt {e^{2 c_1} \text {$\#$1}^{-2 a}-1}}\& \right ]\left [c_2+x\right ]\right \}\right \} \]
Maple: cpu = 1.622 (sec), leaf count = 68 \[ \left \{ \int ^{y \left ( x \right ) }\!{\frac {1}{{{\it \_a}}^{-a}}{ \frac {1}{\sqrt {-{{\it \_a}}^{2\,a}+{\it \_C1}}}}}{d{\it \_a}}-x-{ \it \_C2}=0,\int ^{y \left ( x \right ) }\!-{\frac {1}{{{\it \_a}}^{-a}} {\frac {1}{\sqrt {-{{\it \_a}}^{2\,a}+{\it \_C1}}}}}{d{\it \_a}}-x-{ \it \_C2}=0 \right \} \]