\[ y''(x)-2 a x \left (y'(x)^2+1\right )^{3/2}=0 \] ✓ Mathematica : cpu = 0.447012 (sec), leaf count = 308
DSolve[-2*a*x*(1 + Derivative[1][y][x]^2)^(3/2) + Derivative[2][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to c_2-\frac {\sqrt {\frac {a x^2-1+c_1}{-1+c_1}} \sqrt {\frac {a x^2+1+c_1}{1+c_1}} \left (F\left (i \sinh ^{-1}\left (x \sqrt {\frac {a}{c_1+1}}\right )|\frac {c_1+1}{c_1-1}\right )+(-1+c_1) E\left (i \sinh ^{-1}\left (x \sqrt {\frac {a}{c_1+1}}\right )|\frac {c_1+1}{c_1-1}\right )\right )}{\sqrt {\frac {a}{1+c_1}} \sqrt {a^2 x^4+2 a c_1 x^2-1+c_1{}^2}}\right \},\left \{y(x)\to \frac {\sqrt {\frac {a x^2-1+c_1}{-1+c_1}} \sqrt {\frac {a x^2+1+c_1}{1+c_1}} \left (F\left (i \sinh ^{-1}\left (x \sqrt {\frac {a}{c_1+1}}\right )|\frac {c_1+1}{c_1-1}\right )+(-1+c_1) E\left (i \sinh ^{-1}\left (x \sqrt {\frac {a}{c_1+1}}\right )|\frac {c_1+1}{c_1-1}\right )\right )}{\sqrt {\frac {a}{1+c_1}} \sqrt {a^2 x^4+2 a c_1 x^2-1+c_1{}^2}}+c_2\right \}\right \}\] ✓ Maple : cpu = 1.818 (sec), leaf count = 38
dsolve(diff(diff(y(x),x),x)-2*a*x*(diff(y(x),x)^2+1)^(3/2)=0,y(x))
\[y \left (x \right ) = \int \sqrt {-\frac {1}{-1+\left (x^{2}+2 c_{1}\right )^{2} a^{2}}}\, a \left (x^{2}+2 c_{1}\right )d x +c_{2}\]