\[ -2 a y(x) \left (y'(x)^2+1\right )^{3/2}+y(x) y''(x)-y'(x)^2-1=0 \] ✓ Mathematica : cpu = 1.27014 (sec), leaf count = 697
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {1-\frac {2 \text {$\#$1}^2 a^2}{-2 a c_1+\sqrt {1-4 a c_1}+1}} \sqrt {1+\frac {2 \text {$\#$1}^2 a^2}{2 a c_1+\sqrt {1-4 a c_1}-1}} \left (\left (-2 a c_1+\sqrt {1-4 a c_1}+1\right ) E\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {a^2}{2 a c_1+\sqrt {1-4 a c_1}-1}} \text {$\#$1}\right )|-\frac {2 a c_1+\sqrt {1-4 a c_1}-1}{-2 a c_1+\sqrt {1-4 a c_1}+1}\right )-\left (1+\sqrt {1-4 a c_1}\right ) F\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {a^2}{2 a c_1+\sqrt {1-4 a c_1}-1}} \text {$\#$1}\right )|-\frac {2 a c_1+\sqrt {1-4 a c_1}-1}{-2 a c_1+\sqrt {1-4 a c_1}+1}\right )\right )}{2 \sqrt {2} a \sqrt {\frac {a^2}{2 a c_1+\sqrt {1-4 a c_1}-1}} \sqrt {\text {$\#$1}^4 a^2+\text {$\#$1}^2 (-1+2 a c_1)+c_1{}^2}}\& \right ][x+c_2]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {\sqrt {1-\frac {2 \text {$\#$1}^2 a^2}{-2 a c_1+\sqrt {1-4 a c_1}+1}} \sqrt {1+\frac {2 \text {$\#$1}^2 a^2}{2 a c_1+\sqrt {1-4 a c_1}-1}} \left (\left (-2 a c_1+\sqrt {1-4 a c_1}+1\right ) E\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {a^2}{2 a c_1+\sqrt {1-4 a c_1}-1}} \text {$\#$1}\right )|-\frac {2 a c_1+\sqrt {1-4 a c_1}-1}{-2 a c_1+\sqrt {1-4 a c_1}+1}\right )-\left (1+\sqrt {1-4 a c_1}\right ) F\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {a^2}{2 a c_1+\sqrt {1-4 a c_1}-1}} \text {$\#$1}\right )|-\frac {2 a c_1+\sqrt {1-4 a c_1}-1}{-2 a c_1+\sqrt {1-4 a c_1}+1}\right )\right )}{2 \sqrt {2} a \sqrt {\frac {a^2}{2 a c_1+\sqrt {1-4 a c_1}-1}} \sqrt {\text {$\#$1}^4 a^2+\text {$\#$1}^2 (-1+2 a c_1)+c_1{}^2}}\& \right ][x+c_2]\right \}\right \}\] ✓ Maple : cpu = 2.223 (sec), leaf count = 98
\[\int _{}^{y \relax (x )}\frac {\textit {\_a}^{2} a +c_{1}}{\sqrt {-a^{2} \textit {\_a}^{4}-2 \textit {\_a}^{2} a c_{1}+\textit {\_a}^{2}-c_{1}^{2}}}d \textit {\_a} -x -c_{2} = 0\]