\[ y''(x)-a \sqrt {b y(x)^2+y'(x)^2}=0 \] ✓ Mathematica : cpu = 0.504302 (sec), leaf count = 76
\[\text {Solve}\left [\int _1^{y(x)}\frac {1}{\text {InverseFunction}\left [\int \frac {\text {$\#$1}}{K[1] \left (\frac {\text {$\#$1}^2}{K[1]^2}-a \sqrt {\frac {\text {$\#$1}^2}{K[1]^2}+b}\right )}d\frac {\text {$\#$1}}{K[1]}\& \right ][c_1-\log (K[1])]}dK[1]=x-c_2,y(x)\right ]\] ✓ Maple : cpu = 2.596 (sec), leaf count = 36
\[\left \{y \left (x \right ) = {\mathrm e}^{c_{2}+\int \RootOf \left (c_{1}+x -\left (\int _{}^{\textit {\_Z}}\frac {1}{-\textit {\_f}^{2}+\sqrt {\textit {\_f}^{2}+b}\, a}d \textit {\_f} \right )\right )d x}\right \}\]