\[ \boxed { {x}^{2} \left ( \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{2}+1 \right ) ^{3}-{a}^{2}=0} \]
Mathematica: cpu = 0.246031 (sec), leaf count = 406 \[ \left \{\left \{y(x)\to c_1-\frac {\sqrt [3]{x} \left (2 x^{2/3}+\left (1+i \sqrt {3}\right ) a^{2/3}\right ) \sqrt {\frac {-2 x^{2/3}+\left (-1-i \sqrt {3}\right ) a^{2/3}}{x^{2/3}}}}{2 \sqrt {2}}\right \},\left \{y(x)\to c_1+\frac {\sqrt [3]{x} \sqrt {\frac {-2 x^{2/3}+\left (-1-i \sqrt {3}\right ) a^{2/3}}{x^{2/3}}} \left (2 x^{2/3}+\left (1+i \sqrt {3}\right ) a^{2/3}\right )}{2 \sqrt {2}}\right \},\left \{y(x)\to c_1-\frac {\sqrt [3]{x} \left (2 x^{2/3}+\left (1-i \sqrt {3}\right ) a^{2/3}\right ) \sqrt {\frac {-2 x^{2/3}+i \left (\sqrt {3}+i\right ) a^{2/3}}{x^{2/3}}}}{2 \sqrt {2}}\right \},\left \{y(x)\to c_1+\frac {\sqrt [3]{x} \sqrt {\frac {-2 x^{2/3}+i \left (\sqrt {3}+i\right ) a^{2/3}}{x^{2/3}}} \left (2 x^{2/3}+\left (1-i \sqrt {3}\right ) a^{2/3}\right )}{2 \sqrt {2}}\right \},\left \{y(x)\to c_1-\sqrt {\frac {a^{2/3}}{x^{2/3}}-1} \left (x-a^{2/3} \sqrt [3]{x}\right )\right \},\left \{y(x)\to \sqrt {\frac {a^{2/3}}{x^{2/3}}-1} \left (x-a^{2/3} \sqrt [3]{x}\right )+c_1\right \}\right \} \]
Maple: cpu = 0.140 (sec), leaf count = 552 \[ \left \{ y \left ( x \right ) =-{1\sqrt {-{\frac {1}{{a}^{4}} \left ( {a} ^{2}x \right ) ^{{\frac {4}{3}}} \left ( \left ( {a}^{2}x \right ) ^{{ \frac {2}{3}}}-{a}^{2} \right ) }} \left ( \left ( {a}^{2}x \right ) ^{{ \frac {2}{3}}}-{a}^{2} \right ) \left ( {a}^{2}x \right ) ^{-{\frac {2}{ 3}}}}+{\it \_C1},y \left ( x \right ) ={1\sqrt {-{\frac {1}{{a}^{4}} \left ( {a}^{2}x \right ) ^{{\frac {4}{3}}} \left ( \left ( {a}^{2}x \right ) ^{{\frac {2}{3}}}-{a}^{2} \right ) }} \left ( \left ( {a}^{2}x \right ) ^{{\frac {2}{3}}}-{a}^{2} \right ) \left ( {a}^{2}x \right ) ^{ -{\frac {2}{3}}}}+{\it \_C1},y \left ( x \right ) ={-{\frac {i}{4}} \sqrt {2}\sqrt {{\frac {i}{{a}^{4}} \left ( {a}^{2}x \right ) ^{{\frac { 4}{3}}} \left ( \sqrt {3}{a}^{2}+2\,i \left ( {a}^{2}x \right ) ^{{\frac {2}{3}}}+i{a}^{2} \right ) }} \left ( \sqrt {3}{a}^{2}+2\,i \left ( {a}^{ 2}x \right ) ^{{\frac {2}{3}}}+i{a}^{2} \right ) \left ( {a}^{2}x \right ) ^{-{\frac {2}{3}}}}+{\it \_C1},y \left ( x \right ) ={{\frac {i }{4}}\sqrt {2}\sqrt {{\frac {i}{{a}^{4}} \left ( {a}^{2}x \right ) ^{{ \frac {4}{3}}} \left ( \sqrt {3}{a}^{2}+2\,i \left ( {a}^{2}x \right ) ^{ {\frac {2}{3}}}+i{a}^{2} \right ) }} \left ( \sqrt {3}{a}^{2}+2\,i \left ( {a}^{2}x \right ) ^{{\frac {2}{3}}}+i{a}^{2} \right ) \left ( {a }^{2}x \right ) ^{-{\frac {2}{3}}}}+{\it \_C1},y \left ( x \right ) ={-{ \frac {i}{4}}\sqrt {2}\sqrt {-i \left ( \sqrt {3}\sqrt [3]{{a}^{2}x}-i \sqrt [3]{{a}^{2}x}-2\,ix \right ) x}\sqrt {{\frac {1}{{a}^{4}} \left ( {a}^{2}x \right ) ^{{\frac {4}{3}}} \left ( \sqrt {3}{a}^{2}-2\,i \left ( {a}^{2}x \right ) ^{{\frac {2}{3}}}-i{a}^{2} \right ) }} \left ( \sqrt {3}{a}^{2}-2\,i \left ( {a}^{2}x \right ) ^{{\frac {2}{3}}}-i{a}^{ 2} \right ) {\frac {1}{\sqrt { \left ( \sqrt {3}\sqrt [3]{{a}^{2}x}-i \sqrt [3]{{a}^{2}x}-2\,ix \right ) x}}} \left ( {a}^{2}x \right ) ^{-{ \frac {2}{3}}}}+{\it \_C1},y \left ( x \right ) ={{\frac {i}{4}}\sqrt {2 }\sqrt {-i \left ( \sqrt {3}\sqrt [3]{{a}^{2}x}-i\sqrt [3]{{a}^{2}x}-2 \,ix \right ) x}\sqrt {{\frac {1}{{a}^{4}} \left ( {a}^{2}x \right ) ^{{ \frac {4}{3}}} \left ( \sqrt {3}{a}^{2}-2\,i \left ( {a}^{2}x \right ) ^{ {\frac {2}{3}}}-i{a}^{2} \right ) }} \left ( \sqrt {3}{a}^{2}-2\,i \left ( {a}^{2}x \right ) ^{{\frac {2}{3}}}-i{a}^{2} \right ) {\frac {1} {\sqrt { \left ( \sqrt {3}\sqrt [3]{{a}^{2}x}-i\sqrt [3]{{a}^{2}x}-2\,i x \right ) x}}} \left ( {a}^{2}x \right ) ^{-{\frac {2}{3}}}}+{\it \_C1} \right \} \]