\[ x^2 \left (y'(x)^2+1\right )^3-a^2=0 \] ✓ Mathematica : cpu = 19.1034 (sec), leaf count = 376
DSolve[-a^2 + x^2*(1 + Derivative[1][y][x]^2)^3 == 0,y[x],x]
\[\left \{\left \{y(x)\to -\sqrt [3]{x} \sqrt {\frac {a^{2/3}}{x^{2/3}}-1} \left (a^{2/3}-x^{2/3}\right )+c_1\right \},\left \{y(x)\to \sqrt [3]{x} \sqrt {\frac {a^{2/3}}{x^{2/3}}-1} \left (a^{2/3}-x^{2/3}\right )+c_1\right \},\left \{y(x)\to c_1-\frac {1}{2} \sqrt [3]{x} \sqrt {-1+\frac {i \left (\sqrt {3}+i\right ) a^{2/3}}{2 x^{2/3}}} \left (2 x^{2/3}+\left (1-i \sqrt {3}\right ) a^{2/3}\right )\right \},\left \{y(x)\to \frac {1}{2} \sqrt [3]{x} \sqrt {-1+\frac {i \left (\sqrt {3}+i\right ) a^{2/3}}{2 x^{2/3}}} \left (2 x^{2/3}+\left (1-i \sqrt {3}\right ) a^{2/3}\right )+c_1\right \},\left \{y(x)\to c_1-\frac {1}{2} \sqrt [3]{x} \sqrt {-1-\frac {i \left (\sqrt {3}-i\right ) a^{2/3}}{2 x^{2/3}}} \left (2 x^{2/3}+\left (1+i \sqrt {3}\right ) a^{2/3}\right )\right \},\left \{y(x)\to \frac {1}{2} \sqrt [3]{x} \sqrt {-1-\frac {i \left (\sqrt {3}-i\right ) a^{2/3}}{2 x^{2/3}}} \left (2 x^{2/3}+\left (1+i \sqrt {3}\right ) a^{2/3}\right )+c_1\right \}\right \}\] ✓ Maple : cpu = 1.006 (sec), leaf count = 545
dsolve(x^2*(diff(y(x),x)^2+1)^3-a^2=0,y(x))
\[y \left (x \right ) = \frac {\sqrt {-\frac {\left (a^{2} x \right )^{\frac {4}{3}} \left (\left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{a^{4}}}\, \left (\left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{\left (a^{2} x \right )^{\frac {2}{3}}}+c_{1}\]