\[ y''(x)-a y(x) \left (y'(x)^2+1\right )^{3/2}=0 \] ✓ Mathematica : cpu = 0.871877 (sec), leaf count = 350
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {\frac {\text {$\#$1}^2 a+2 c_1-2}{c_1-1}} \sqrt {\frac {\text {$\#$1}^2 a+2 c_1+2}{c_1+1}} \left (F\left (i \sinh ^{-1}\left (\sqrt {\frac {a}{2 c_1+2}} \text {$\#$1}\right )|\frac {c_1+1}{c_1-1}\right )+\left (c_1-1\right ) E\left (i \sinh ^{-1}\left (\sqrt {\frac {a}{2 c_1+2}} \text {$\#$1}\right )|\frac {c_1+1}{c_1-1}\right )\right )}{\sqrt {\frac {a}{2 c_1+2}} \sqrt {\text {$\#$1}^4 a^2+4 \text {$\#$1}^2 a c_1+4 c_1^2-4}}\& \right ]\left [c_2+x\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {\sqrt {\frac {\text {$\#$1}^2 a+2 c_1-2}{c_1-1}} \sqrt {\frac {\text {$\#$1}^2 a+2 c_1+2}{c_1+1}} \left (F\left (i \sinh ^{-1}\left (\sqrt {\frac {a}{2 c_1+2}} \text {$\#$1}\right )|\frac {c_1+1}{c_1-1}\right )+\left (c_1-1\right ) E\left (i \sinh ^{-1}\left (\sqrt {\frac {a}{2 c_1+2}} \text {$\#$1}\right )|\frac {c_1+1}{c_1-1}\right )\right )}{\sqrt {\frac {a}{2 c_1+2}} \sqrt {\text {$\#$1}^4 a^2+4 \text {$\#$1}^2 a c_1+4 c_1^2-4}}\& \right ]\left [c_2+x\right ]\right \}\right \}\]
✓ Maple : cpu = 0.275 (sec), leaf count = 84
\[ \left \{ \int ^{y \left ( x \right ) }\!{a \left ( {{\it \_a}}^{2}+2\,{\it \_C1} \right ) {\frac {1}{\sqrt {4-{a}^{2} \left ( {{\it \_a}}^{2}+2\,{\it \_C1} \right ) ^{2}}}}}{d{\it \_a}}-x-{\it \_C2}=0,\int ^{y \left ( x \right ) }\!-{a \left ( {{\it \_a}}^{2}+2\,{\it \_C1} \right ) {\frac {1}{\sqrt {4-{a}^{2} \left ( {{\it \_a}}^{2}+2\,{\it \_C1} \right ) ^{2}}}}}{d{\it \_a}}-x-{\it \_C2}=0 \right \} \]