\[ \boxed { \left ( {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) \right ) y \left ( x \right ) - \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{2}-1-2\,ay \left ( x \right ) \left ( \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{2}+1 \right ) ^{3/2}=0} \]
Mathematica: cpu = 1.926745 (sec), leaf count = 797 \[ \left \{\left \{y(x)\to \text {InverseFunction}\left [-\frac {\left (\left (4 c_1 a^2+\sqrt {8 c_1 a^2+1}+1\right ) E\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {a^2}{-4 c_1 a^2+\sqrt {8 c_1 a^2+1}-1}} \text {$\#$1}\right )|\frac {4 c_1 a^2-\sqrt {8 c_1 a^2+1}+1}{4 c_1 a^2+\sqrt {8 c_1 a^2+1}+1}\right )-\left (\sqrt {8 c_1 a^2+1}+1\right ) F\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {a^2}{-4 c_1 a^2+\sqrt {8 c_1 a^2+1}-1}} \text {$\#$1}\right )|\frac {4 c_1 a^2-\sqrt {8 c_1 a^2+1}+1}{4 c_1 a^2+\sqrt {8 c_1 a^2+1}+1}\right )\right ) \sqrt {1-\frac {2 a^2 \text {$\#$1}^2}{4 c_1 a^2-\sqrt {8 c_1 a^2+1}+1}} \sqrt {1-\frac {2 a^2 \text {$\#$1}^2}{4 c_1 a^2+\sqrt {8 c_1 a^2+1}+1}}}{2 a \sqrt {\frac {a^2}{-4 c_1 a^2+\sqrt {8 c_1 a^2+1}-1}} \sqrt {2 a^2 \text {$\#$1}^4-2 \left (4 c_1 a^2+1\right ) \text {$\#$1}^2+8 a^2 c_1^2}}\& \right ]\left [x+c_2\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {\left (\left (4 c_1 a^2+\sqrt {8 c_1 a^2+1}+1\right ) E\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {a^2}{-4 c_1 a^2+\sqrt {8 c_1 a^2+1}-1}} \text {$\#$1}\right )|\frac {4 c_1 a^2-\sqrt {8 c_1 a^2+1}+1}{4 c_1 a^2+\sqrt {8 c_1 a^2+1}+1}\right )-\left (\sqrt {8 c_1 a^2+1}+1\right ) F\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {a^2}{-4 c_1 a^2+\sqrt {8 c_1 a^2+1}-1}} \text {$\#$1}\right )|\frac {4 c_1 a^2-\sqrt {8 c_1 a^2+1}+1}{4 c_1 a^2+\sqrt {8 c_1 a^2+1}+1}\right )\right ) \sqrt {1-\frac {2 a^2 \text {$\#$1}^2}{4 c_1 a^2-\sqrt {8 c_1 a^2+1}+1}} \sqrt {1-\frac {2 a^2 \text {$\#$1}^2}{4 c_1 a^2+\sqrt {8 c_1 a^2+1}+1}}}{2 a \sqrt {\frac {a^2}{-4 c_1 a^2+\sqrt {8 c_1 a^2+1}-1}} \sqrt {2 a^2 \text {$\#$1}^4-2 \left (4 c_1 a^2+1\right ) \text {$\#$1}^2+8 a^2 c_1^2}}\& \right ]\left [x+c_2\right ]\right \}\right \} \]
Maple: cpu = 3.339 (sec), leaf count = 98 \[ \left \{ \int ^{y \left ( x \right ) }\!{({{\it \_a}}^{2}a+{\it \_C1}){ \frac {1}{\sqrt {-{{\it \_a}}^{4}{a}^{2}-2\,{\it \_C1}\,{{\it \_a}}^{2 }a-{{\it \_C1}}^{2}+{{\it \_a}}^{2}}}}}{d{\it \_a}}-x-{\it \_C2}=0, \int ^{y \left ( x \right ) }\!-{({{\it \_a}}^{2}a+{\it \_C1}){\frac {1} {\sqrt {-{{\it \_a}}^{4}{a}^{2}-2\,{\it \_C1}\,{{\it \_a}}^{2}a-{{\it \_C1}}^{2}+{{\it \_a}}^{2}}}}}{d{\it \_a}}-x-{\it \_C2}=0 \right \} \]