\[ -2 a y(x) \left (y'(x)^2+1\right )^{3/2}+y(x) y''(x)-y'(x)^2-1=0 \] ✓ Mathematica : cpu = 2.02454 (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 = 0.454 (sec), leaf count = 98
\[ \left \{ \int ^{y \left ( x \right ) }\!{(a{{\it \_a}}^{2}+{\it \_C1}){\frac {1}{\sqrt {-{{\it \_a}}^{4}{a}^{2}-2\,{\it \_C1}\,{{\it \_a}}^{2}a-{{\it \_C1}}^{2}+{{\it \_a}}^{2}}}}}{d{\it \_a}}-x-{\it \_C2}=0,\int ^{y \left ( x \right ) }\!-{(a{{\it \_a}}^{2}+{\it \_C1}){\frac {1}{\sqrt {-{{\it \_a}}^{4}{a}^{2}-2\,{\it \_C1}\,{{\it \_a}}^{2}a-{{\it \_C1}}^{2}+{{\it \_a}}^{2}}}}}{d{\it \_a}}-x-{\it \_C2}=0 \right \} \]