\[ y^{(3)}(x) y''(x)-a \sqrt {b^2 y''(x)^2+1}=0 \] ✓ Mathematica : cpu = 0.555936 (sec), leaf count = 426
\[\left \{\left \{y(x)\to \frac {\frac {\left (a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1{}^2-1\right ){}^{3/2}}{3 a b^2}+\frac {\sqrt {a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1{}^2-1}}{a b^2}-\frac {c_1 \log \left (\sqrt {a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1{}^2-1}+a b^2 x+b^2 c_1\right )}{a}-x \log \left (b^2 \left (\sqrt {a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1{}^2-1}+a b^2 x+b^2 c_1\right )\right )}{2 a b^3}+c_3 x+c_2\right \},\left \{y(x)\to \frac {-\frac {\left (a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1{}^2-1\right ){}^{3/2}}{3 a b^2}-\frac {\sqrt {a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1{}^2-1}}{a b^2}+\frac {c_1 \log \left (\sqrt {a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1{}^2-1}+a b^2 x+b^2 c_1\right )}{a}+x \log \left (b^2 \left (\sqrt {a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1{}^2-1}+a b^2 x+b^2 c_1\right )\right )}{2 a b^3}+c_3 x+c_2\right \}\right \}\] ✓ Maple : cpu = 0.523 (sec), leaf count = 197
\[ \left \{ y \left ( x \right ) ={\it \_C2}\,x+\int \!{\frac {1}{2\,b} \left ( -{\ln \left ( \sqrt { \left ( -1+{b}^{2} \left ( x+{\it \_C1} \right ) a \right ) \left ( 1+{b}^{2} \left ( x+{\it \_C1} \right ) a \right ) }+{ \left ( x+{\it \_C1} \right ) {b}^{4}{a}^{2}{\frac {1}{\sqrt {{a}^{2}{b}^{4}}}}} \right ) {\frac {1}{\sqrt {{a}^{2}{b}^{4}}}}}+\sqrt { \left ( -1+{b}^{2} \left ( x+{\it \_C1} \right ) a \right ) \left ( 1+{b}^{2} \left ( x+{\it \_C1} \right ) a \right ) } \left ( x+{\it \_C1} \right ) \right ) }\,{\rm d}x+{\it \_C3},y \left ( x \right ) ={\it \_C2}\,x+\int \!{\frac {1}{2\,b} \left ( {\ln \left ( \sqrt { \left ( -1+{b}^{2} \left ( x+{\it \_C1} \right ) a \right ) \left ( 1+{b}^{2} \left ( x+{\it \_C1} \right ) a \right ) }+{ \left ( x+{\it \_C1} \right ) {b}^{4}{a}^{2}{\frac {1}{\sqrt {{a}^{2}{b}^{4}}}}} \right ) {\frac {1}{\sqrt {{a}^{2}{b}^{4}}}}}-\sqrt { \left ( -1+{b}^{2} \left ( x+{\it \_C1} \right ) a \right ) \left ( 1+{b}^{2} \left ( x+{\it \_C1} \right ) a \right ) } \left ( x+{\it \_C1} \right ) \right ) }\,{\rm d}x+{\it \_C3} \right \} \]