\[ x^2 \left (y'(x)^2+1\right )^3-a^2=0 \] ✓ Mathematica : cpu = 0.231281 (sec), leaf count = 216
DSolve[-a^2 + x^2*(1 + Derivative[1][y][x]^2)^3 == 0,y[x],x]
\[\left \{\left \{y(x)\to -x \left (\frac {a^{2/3}}{x^{2/3}}-1\right )^{3/2}+c_1\right \},\left \{y(x)\to x \left (\frac {a^{2/3}}{x^{2/3}}-1\right )^{3/2}+c_1\right \},\left \{y(x)\to c_1-x \left (-1-\frac {i \left (\sqrt {3}-i\right ) a^{2/3}}{2 x^{2/3}}\right )^{3/2}\right \},\left \{y(x)\to x \left (-1-\frac {i \left (\sqrt {3}-i\right ) a^{2/3}}{2 x^{2/3}}\right )^{3/2}+c_1\right \},\left \{y(x)\to c_1-x \left (-1+\frac {i \left (\sqrt {3}+i\right ) a^{2/3}}{2 x^{2/3}}\right )^{3/2}\right \},\left \{y(x)\to x \left (-1+\frac {i \left (\sqrt {3}+i\right ) a^{2/3}}{2 x^{2/3}}\right )^{3/2}+c_1\right \}\right \}\] ✓ Maple : cpu = 0.818 (sec), leaf count = 553
dsolve(x^2*(diff(y(x),x)^2+1)^3-a^2=0,y(x))
\[y \left (x \right ) = \frac {\sqrt {-\frac {\left (a^{2} x \right )^{\frac {4}{3}} \left (\left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{a^{4}}}\, \left (\left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{\left (a^{2} x \right )^{\frac {2}{3}}}+c_{1}\]