\[ y''(x)-a y(x) \left (y'(x)^2+1\right )^{3/2}=0 \] ✓ Mathematica : cpu = 0.779438 (sec), leaf count = 350
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {\frac {\text {$\#$1}^2 a-2+2 c_1}{-1+c_1}} \sqrt {\frac {\text {$\#$1}^2 a+2+2 c_1}{1+c_1}} \left (F\left (i \sinh ^{-1}\left (\sqrt {\frac {a}{2 c_1+2}} \text {$\#$1}\right )|\frac {c_1+1}{c_1-1}\right )+(-1+c_1) E\left (i \sinh ^{-1}\left (\sqrt {\frac {a}{2 c_1+2}} \text {$\#$1}\right )|\frac {c_1+1}{c_1-1}\right )\right )}{\sqrt {\frac {a}{2+2 c_1}} \sqrt {\text {$\#$1}^4 a^2+4 \text {$\#$1}^2 a c_1-4+4 c_1{}^2}}\& \right ][x+c_2]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {\sqrt {\frac {\text {$\#$1}^2 a-2+2 c_1}{-1+c_1}} \sqrt {\frac {\text {$\#$1}^2 a+2+2 c_1}{1+c_1}} \left (F\left (i \sinh ^{-1}\left (\sqrt {\frac {a}{2 c_1+2}} \text {$\#$1}\right )|\frac {c_1+1}{c_1-1}\right )+(-1+c_1) E\left (i \sinh ^{-1}\left (\sqrt {\frac {a}{2 c_1+2}} \text {$\#$1}\right )|\frac {c_1+1}{c_1-1}\right )\right )}{\sqrt {\frac {a}{2+2 c_1}} \sqrt {\text {$\#$1}^4 a^2+4 \text {$\#$1}^2 a c_1-4+4 c_1{}^2}}\& \right ][x+c_2]\right \}\right \}\] ✓ Maple : cpu = 1.878 (sec), leaf count = 84
\[\int _{}^{y \relax (x )}\frac {\left (\textit {\_a}^{2}+2 c_{1}\right ) a}{\sqrt {4-a^{2} \left (\textit {\_a}^{2}+2 c_{1}\right )^{2}}}d \textit {\_a} -x -c_{2} = 0\]