\[ y^{(3)}(x) y''(x)-a \sqrt {b^2 y''(x)^2+1}=0 \] ✓ Mathematica : cpu = 0.573837 (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 = 0.952 (sec), leaf count = 337
\[ \left \{ y \left ( x \right ) =\int \!-{\frac {x}{2\,b}\sqrt {{{\it \_C1}}^{2}{a}^{2}{b}^{4}+2\,{\it \_C1}\,{a}^{2}{b}^{4}x+{a}^{2}{b}^{4}{x}^{2}-1}}-{\frac {{\it \_C1}}{2\,b}\sqrt {{{\it \_C1}}^{2}{a}^{2}{b}^{4}+2\,{\it \_C1}\,{a}^{2}{b}^{4}x+{a}^{2}{b}^{4}{x}^{2}-1}}+{\frac {1}{2\,b}\ln \left ( {({\it \_C1}\,{a}^{2}{b}^{4}+{a}^{2}{b}^{4}x){\frac {1}{\sqrt {{a}^{2}{b}^{4}}}}}+\sqrt {{{\it \_C1}}^{2}{a}^{2}{b}^{4}+2\,{\it \_C1}\,{a}^{2}{b}^{4}x+{a}^{2}{b}^{4}{x}^{2}-1} \right ) {\frac {1}{\sqrt {{a}^{2}{b}^{4}}}}}\,{\rm d}x+{\it \_C2}\,x+{\it \_C3},y \left ( x \right ) =\int \!{\frac {x}{2\,b}\sqrt {{{\it \_C1}}^{2}{a}^{2}{b}^{4}+2\,{\it \_C1}\,{a}^{2}{b}^{4}x+{a}^{2}{b}^{4}{x}^{2}-1}}+{\frac {{\it \_C1}}{2\,b}\sqrt {{{\it \_C1}}^{2}{a}^{2}{b}^{4}+2\,{\it \_C1}\,{a}^{2}{b}^{4}x+{a}^{2}{b}^{4}{x}^{2}-1}}-{\frac {1}{2\,b}\ln \left ( {({\it \_C1}\,{a}^{2}{b}^{4}+{a}^{2}{b}^{4}x){\frac {1}{\sqrt {{a}^{2}{b}^{4}}}}}+\sqrt {{{\it \_C1}}^{2}{a}^{2}{b}^{4}+2\,{\it \_C1}\,{a}^{2}{b}^{4}x+{a}^{2}{b}^{4}{x}^{2}-1} \right ) {\frac {1}{\sqrt {{a}^{2}{b}^{4}}}}}\,{\rm d}x+{\it \_C2}\,x+{\it \_C3} \right \} \]