\[ \boxed { \left ( {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) \right ) {\frac {{\rm d}^{3}}{{\rm d}{x}^{3}}}y \left ( x \right ) -a\sqrt {{b}^{2} \left ( {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) \right ) ^{2}+1}=0} \]
Mathematica: cpu = 0.591575 (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 = 0.125 (sec), leaf count = 337 \[ \left \{ y \left ( x \right ) =\int \!-{\frac {x}{2\,b}\sqrt {{{\it \_C1 }}^{2}{a}^{2}{b}^{4}+2\,{\it \_C1}\,{a}^{2}{b}^{4}x+{a}^{2}{b}^{4}{x}^ {2}-1}}-{\frac {{\it \_C1}}{2\,b}\sqrt {{{\it \_C1}}^{2}{a}^{2}{b}^{4} +2\,{\it \_C1}\,{a}^{2}{b}^{4}x+{a}^{2}{b}^{4}{x}^{2}-1}}+{\frac {1}{2 \,b}\ln \left ( {({\it \_C1}\,{a}^{2}{b}^{4}+{a}^{2}{b}^{4}x){\frac {1 }{\sqrt {{a}^{2}{b}^{4}}}}}+\sqrt {{{\it \_C1}}^{2}{a}^{2}{b}^{4}+2\,{ \it \_C1}\,{a}^{2}{b}^{4}x+{a}^{2}{b}^{4}{x}^{2}-1} \right ) {\frac {1} {\sqrt {{a}^{2}{b}^{4}}}}}\,{\rm d}x+{\it \_C2}\,x+{\it \_C3},y \left ( x \right ) =\int \!{\frac {x}{2\,b}\sqrt {{{\it \_C1}}^{2}{a}^{ 2}{b}^{4}+2\,{\it \_C1}\,{a}^{2}{b}^{4}x+{a}^{2}{b}^{4}{x}^{2}-1}}+{ \frac {{\it \_C1}}{2\,b}\sqrt {{{\it \_C1}}^{2}{a}^{2}{b}^{4}+2\,{\it \_C1}\,{a}^{2}{b}^{4}x+{a}^{2}{b}^{4}{x}^{2}-1}}-{\frac {1}{2\,b}\ln \left ( {({\it \_C1}\,{a}^{2}{b}^{4}+{a}^{2}{b}^{4}x){\frac {1}{\sqrt {{a}^{2}{b}^{4}}}}}+\sqrt {{{\it \_C1}}^{2}{a}^{2}{b}^{4}+2\,{\it \_C1 }\,{a}^{2}{b}^{4}x+{a}^{2}{b}^{4}{x}^{2}-1} \right ) {\frac {1}{\sqrt { {a}^{2}{b}^{4}}}}}\,{\rm d}x+{\it \_C2}\,x+{\it \_C3} \right \} \]