\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) =-8\,{\frac {x \left ( a-1 \right ) \left ( a+1 \right ) }{8-8\,y \left ( x \right ) +2\,{x}^{4}+3\,{x}^{4} \left ( y \left ( x \right ) \right ) ^{2}+4\,{x}^{2} \left ( y \left ( x \right ) \right ) ^{2}+2\, \left ( y \left ( x \right ) \right ) ^{4}+3\,{a}^{4} \left ( y \left ( x \right ) \right ) ^{4}{x}^{2}-3\,{a}^{6} \left ( y \left ( x \right ) \right ) ^{2}{x}^{4}+9\, \left ( y \left ( x \right ) \right ) ^{2}{a}^{4}{x}^{4}-9\, \left ( y \left ( x \right ) \right ) ^{2}{a}^{2}{x}^{4}-{a}^{2} \left ( y \left ( x \right ) \right ) ^{6}+{a}^{8}{x}^{6}-4\,{a}^{6}{x}^{6}+6\,{a}^{4}{x}^{6}-2\,{a}^{2} \left ( y \left ( x \right ) \right ) ^{4}-2\,{a}^{6}{x}^{4}+6\,{a}^{4}{x}^{4}-6\,{a}^{2}{x}^{4}-4\,{a}^{2}{x}^{6}-6\, \left ( y \left ( x \right ) \right ) ^{4}{a}^{2}{x}^{2}+4\,{a}^{4} \left ( y \left ( x \right ) \right ) ^{2}{x}^{2}-8\, \left ( y \left ( x \right ) \right ) ^{2}{a}^{2}{x}^{2}+3\,{x}^{2} \left ( y \left ( x \right ) \right ) ^{4}+{x}^{6}+ \left ( y \left ( x \right ) \right ) ^{6}-8\,{a}^{2}}}=0} \]
Mathematica: cpu = 4.810111 (sec), leaf count = 264 \[ \text {Solve}\left [\frac {y(x)}{(a-1) (a+1)}-\frac {8 \text {RootSum}\left [-\text {$\#$1}^3 a^6+3 \text {$\#$1}^3 a^4-3 \text {$\#$1}^3 a^2+\text {$\#$1}^3+3 \text {$\#$1}^2 a^4 y(x)^2+2 \text {$\#$1}^2 a^4-6 \text {$\#$1}^2 a^2 y(x)^2-4 \text {$\#$1}^2 a^2+3 \text {$\#$1}^2 y(x)^2+2 \text {$\#$1}^2-3 \text {$\#$1} a^2 y(x)^4-4 \text {$\#$1} a^2 y(x)^2+3 \text {$\#$1} y(x)^4+4 \text {$\#$1} y(x)^2+y(x)^6+2 y(x)^4+8\& ,\frac {\log \left (x^2-\text {$\#$1}\right )}{3 \text {$\#$1}^2 a^4-6 \text {$\#$1}^2 a^2+3 \text {$\#$1}^2-6 \text {$\#$1} a^2 y(x)^2-4 \text {$\#$1} a^2+6 \text {$\#$1} y(x)^2+4 \text {$\#$1}+3 y(x)^4+4 y(x)^2}\& \right ]}{(a-1) (a+1) \left (2-2 a^2\right )}=c_1,y(x)\right ] \]
Maple: cpu = 2.012 (sec), leaf count = 80 \[ \left \{ {\frac {y \left ( x \right ) }{ \left ( a-1 \right ) \left ( a+1 \right ) }}+4\,{\frac {1}{{a}^{4}-2\,{a}^{2}+1}\sum _{{\it \_R}={\it RootOf} \left ( {{\it \_Z}}^{3}+2\,{{\it \_Z}}^{2}+8 \right ) }{\frac { \ln \left ( -{a}^{2}{x}^{2}+{x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}-{\it \_R} \right ) }{3\,{{\it \_R}}^{2}+4\,{\it \_R}}}}-{ \it \_C1}=0 \right \} \]