\[ y^{(3)}(x) y''(x)-a \sqrt {b^2 y''(x)^2+1}=0 \] ✓ Mathematica : cpu = 0.551446 (sec), leaf count = 426
DSolve[-(a*Sqrt[1 + b^2*Derivative[2][y][x]^2]) + Derivative[2][y][x]*Derivative[3][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to \frac {\frac {\left (a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1{}^2-1\right ){}^{3/2}}{3 a b^2}+\frac {\sqrt {a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1{}^2-1}}{a b^2}-\frac {c_1 \log \left (\sqrt {a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1{}^2-1}+a b^2 x+b^2 c_1\right )}{a}-x \log \left (b^2 \left (\sqrt {a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1{}^2-1}+a b^2 x+b^2 c_1\right )\right )}{2 a b^3}+c_3 x+c_2\right \},\left \{y(x)\to \frac {-\frac {\left (a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1{}^2-1\right ){}^{3/2}}{3 a b^2}-\frac {\sqrt {a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1{}^2-1}}{a b^2}+\frac {c_1 \log \left (\sqrt {a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1{}^2-1}+a b^2 x+b^2 c_1\right )}{a}+x \log \left (b^2 \left (\sqrt {a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1{}^2-1}+a b^2 x+b^2 c_1\right )\right )}{2 a b^3}+c_3 x+c_2\right \}\right \}\] ✓ Maple : cpu = 0.308 (sec), leaf count = 197
dsolve(diff(diff(y(x),x),x)*diff(diff(diff(y(x),x),x),x)-a*(b^2*diff(diff(y(x),x),x)^2+1)^(1/2)=0,y(x))
\[y \left (x \right ) = x c_{2}+\int \frac {-\frac {\ln \left (\sqrt {\left (1+b^{2} \left (x +c_{1}\right ) a \right ) \left (-1+b^{2} \left (x +c_{1}\right ) a \right )}+\frac {\left (x +c_{1}\right ) b^{4} a^{2}}{\sqrt {a^{2} b^{4}}}\right )}{\sqrt {a^{2} b^{4}}}+\sqrt {\left (1+b^{2} \left (x +c_{1}\right ) a \right ) \left (-1+b^{2} \left (x +c_{1}\right ) a \right )}\, \left (x +c_{1}\right )}{2 b}d x +c_{3}\]