\[ \boxed { {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) -2\,ax \left ( \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{2}+1 \right ) ^{3/2}=0} \]
Mathematica: cpu = 0.311540 (sec), leaf count = 308 \[ \left \{\left \{y(x)\to c_2-\frac {\sqrt {\frac {a x^2+c_1-1}{c_1-1}} \sqrt {\frac {a x^2+c_1+1}{c_1+1}} \left (F\left (i \sinh ^{-1}\left (x \sqrt {\frac {a}{c_1+1}}\right )|\frac {c_1+1}{c_1-1}\right )+\left (c_1-1\right ) E\left (i \sinh ^{-1}\left (x \sqrt {\frac {a}{c_1+1}}\right )|\frac {c_1+1}{c_1-1}\right )\right )}{\sqrt {\frac {a}{c_1+1}} \sqrt {a^2 x^4+2 a c_1 x^2+c_1^2-1}}\right \},\left \{y(x)\to c_2+\frac {\sqrt {\frac {a x^2+c_1-1}{c_1-1}} \sqrt {\frac {a x^2+c_1+1}{c_1+1}} \left (F\left (i \sinh ^{-1}\left (x \sqrt {\frac {a}{c_1+1}}\right )|\frac {c_1+1}{c_1-1}\right )+\left (c_1-1\right ) E\left (i \sinh ^{-1}\left (x \sqrt {\frac {a}{c_1+1}}\right )|\frac {c_1+1}{c_1-1}\right )\right )}{\sqrt {\frac {a}{c_1+1}} \sqrt {a^2 x^4+2 a c_1 x^2+c_1^2-1}}\right \}\right \} \]
Maple: cpu = 0.125 (sec), leaf count = 49 \[ \left \{ y \left ( x \right ) =\int \!\sqrt {- \left ( {a}^{2}{x}^{4}+4\, {\it \_C1}\,{a}^{2}{x}^{2}+4\,{{\it \_C1}}^{2}{a}^{2}-1 \right ) ^{-1}} a \left ( {x}^{2}+2\,{\it \_C1} \right ) \,{\rm d}x+{\it \_C2} \right \} \]