\[ y^{(3)}(x) y''(x)-a \sqrt {b^2 y''(x)^2+1}=0 \] ✓ Mathematica : cpu = 0.645312 (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 = 0.388 (sec), leaf count = 197
\[\left \{y \left (x \right ) = c_{2} x +c_{3}+\int \frac {-\frac {\ln \left (\frac {\left (c_{1}+x \right ) a^{2} b^{4}}{\sqrt {a^{2} b^{4}}}+\sqrt {\left (\left (c_{1}+x \right ) a \,b^{2}+1\right ) \left (\left (c_{1}+x \right ) a \,b^{2}-1\right )}\right )}{\sqrt {a^{2} b^{4}}}+\sqrt {\left (\left (c_{1}+x \right ) a \,b^{2}+1\right ) \left (\left (c_{1}+x \right ) a \,b^{2}-1\right )}\, \left (c_{1}+x \right )}{2 b}d x, y \left (x \right ) = c_{2} x +c_{3}+\int \frac {\frac {\ln \left (\frac {\left (c_{1}+x \right ) a^{2} b^{4}}{\sqrt {a^{2} b^{4}}}+\sqrt {\left (\left (c_{1}+x \right ) a \,b^{2}+1\right ) \left (\left (c_{1}+x \right ) a \,b^{2}-1\right )}\right )}{\sqrt {a^{2} b^{4}}}-\sqrt {\left (\left (c_{1}+x \right ) a \,b^{2}+1\right ) \left (\left (c_{1}+x \right ) a \,b^{2}-1\right )}\, \left (c_{1}+x \right )}{2 b}d x\right \}\]