\[ y'(x)=\frac {4 (a-1) (a+1) x \left (a^2 x^2-x^2-y(x)^2-2\right )}{a^8 x^6-4 a^6 x^6-3 a^6 x^4 y(x)^2+6 a^4 x^6+9 a^4 x^4 y(x)^2+3 a^4 x^2 y(x)^4-4 a^2 x^6-9 a^2 x^4 y(x)^2-6 a^2 x^2 y(x)^4+4 a^2 x^2 y(x)-a^2 y(x)^6+x^6+3 x^4 y(x)^2+3 x^2 y(x)^4-4 x^2 y(x)+y(x)^6-4 y(x)^3-8 y(x)} \] ✓ Mathematica : cpu = 3.94572 (sec), leaf count = 1191
DSolve[Derivative[1][y][x] == (4*(-1 + a)*(1 + a)*x*(-2 - x^2 + a^2*x^2 - y[x]^2))/(x^6 - 4*a^2*x^6 + 6*a^4*x^6 - 4*a^6*x^6 + a^8*x^6 - 8*y[x] - 4*x^2*y[x] + 4*a^2*x^2*y[x] + 3*x^4*y[x]^2 - 9*a^2*x^4*y[x]^2 + 9*a^4*x^4*y[x]^2 - 3*a^6*x^4*y[x]^2 - 4*y[x]^3 + 3*x^2*y[x]^4 - 6*a^2*x^2*y[x]^4 + 3*a^4*x^2*y[x]^4 + y[x]^6 - a^2*y[x]^6),y[x],x]
\[\left \{\left \{y(x)\to \text {Root}\left [2 x^4 a^8-8 x^4 a^6+e^{c_1} x^4 a^4+11 x^4 a^4-2 e^{c_1} x^4 a^2-6 x^4 a^2+4 x^2 a^2+\left (2 a^2-2\right ) \text {$\#$1}^5+e^{c_1} x^4+x^4+\left (2 a^4-4 a^2+e^{c_1}+1\right ) \text {$\#$1}^4+\left (-4 x^2 a^4+8 x^2 a^2-4 x^2\right ) \text {$\#$1}^3-4 x^2+\left (-4 x^2 a^6+12 x^2 a^4-2 e^{c_1} x^2 a^2-10 x^2 a^2+2 e^{c_1} x^2+2 x^2-4\right ) \text {$\#$1}^2+\left (2 x^4 a^6-6 x^4 a^4+6 x^4 a^2-2 x^4\right ) \text {$\#$1}-4\& ,1\right ]\right \},\left \{y(x)\to \text {Root}\left [2 x^4 a^8-8 x^4 a^6+e^{c_1} x^4 a^4+11 x^4 a^4-2 e^{c_1} x^4 a^2-6 x^4 a^2+4 x^2 a^2+\left (2 a^2-2\right ) \text {$\#$1}^5+e^{c_1} x^4+x^4+\left (2 a^4-4 a^2+e^{c_1}+1\right ) \text {$\#$1}^4+\left (-4 x^2 a^4+8 x^2 a^2-4 x^2\right ) \text {$\#$1}^3-4 x^2+\left (-4 x^2 a^6+12 x^2 a^4-2 e^{c_1} x^2 a^2-10 x^2 a^2+2 e^{c_1} x^2+2 x^2-4\right ) \text {$\#$1}^2+\left (2 x^4 a^6-6 x^4 a^4+6 x^4 a^2-2 x^4\right ) \text {$\#$1}-4\& ,2\right ]\right \},\left \{y(x)\to \text {Root}\left [2 x^4 a^8-8 x^4 a^6+e^{c_1} x^4 a^4+11 x^4 a^4-2 e^{c_1} x^4 a^2-6 x^4 a^2+4 x^2 a^2+\left (2 a^2-2\right ) \text {$\#$1}^5+e^{c_1} x^4+x^4+\left (2 a^4-4 a^2+e^{c_1}+1\right ) \text {$\#$1}^4+\left (-4 x^2 a^4+8 x^2 a^2-4 x^2\right ) \text {$\#$1}^3-4 x^2+\left (-4 x^2 a^6+12 x^2 a^4-2 e^{c_1} x^2 a^2-10 x^2 a^2+2 e^{c_1} x^2+2 x^2-4\right ) \text {$\#$1}^2+\left (2 x^4 a^6-6 x^4 a^4+6 x^4 a^2-2 x^4\right ) \text {$\#$1}-4\& ,3\right ]\right \},\left \{y(x)\to \text {Root}\left [2 x^4 a^8-8 x^4 a^6+e^{c_1} x^4 a^4+11 x^4 a^4-2 e^{c_1} x^4 a^2-6 x^4 a^2+4 x^2 a^2+\left (2 a^2-2\right ) \text {$\#$1}^5+e^{c_1} x^4+x^4+\left (2 a^4-4 a^2+e^{c_1}+1\right ) \text {$\#$1}^4+\left (-4 x^2 a^4+8 x^2 a^2-4 x^2\right ) \text {$\#$1}^3-4 x^2+\left (-4 x^2 a^6+12 x^2 a^4-2 e^{c_1} x^2 a^2-10 x^2 a^2+2 e^{c_1} x^2+2 x^2-4\right ) \text {$\#$1}^2+\left (2 x^4 a^6-6 x^4 a^4+6 x^4 a^2-2 x^4\right ) \text {$\#$1}-4\& ,4\right ]\right \},\left \{y(x)\to \text {Root}\left [2 x^4 a^8-8 x^4 a^6+e^{c_1} x^4 a^4+11 x^4 a^4-2 e^{c_1} x^4 a^2-6 x^4 a^2+4 x^2 a^2+\left (2 a^2-2\right ) \text {$\#$1}^5+e^{c_1} x^4+x^4+\left (2 a^4-4 a^2+e^{c_1}+1\right ) \text {$\#$1}^4+\left (-4 x^2 a^4+8 x^2 a^2-4 x^2\right ) \text {$\#$1}^3-4 x^2+\left (-4 x^2 a^6+12 x^2 a^4-2 e^{c_1} x^2 a^2-10 x^2 a^2+2 e^{c_1} x^2+2 x^2-4\right ) \text {$\#$1}^2+\left (2 x^4 a^6-6 x^4 a^4+6 x^4 a^2-2 x^4\right ) \text {$\#$1}-4\& ,5\right ]\right \}\right \}\] ✓ Maple : cpu = 1.407 (sec), leaf count = 79
dsolve(diff(y(x),x) = 4*x*(a-1)*(a+1)*(-y(x)^2+a^2*x^2-x^2-2)/(-4*y(x)^3+4*a^2*x^2*y(x)-4*x^2*y(x)-8*y(x)-a^2*y(x)^6+3*a^4*y(x)^4*x^2-6*y(x)^4*a^2*x^2-3*a^6*y(x)^2*x^4+9*y(x)^2*a^4*x^4-9*y(x)^2*a^2*x^4+a^8*x^6-4*a^6*x^6+6*a^4*x^6-4*a^2*x^6+y(x)^6+3*x^2*y(x)^4+3*x^4*y(x)^2+x^6),y(x))
\[-\frac {y \left (x \right )}{\left (a -1\right ) \left (a +1\right )}+\frac {2}{\left (a^{2}-1\right )^{2} \left (a^{2} x^{2}-x^{2}-y \left (x \right )^{2}\right )^{2}}-\frac {2}{\left (a^{2}-1\right )^{2} \left (a^{2} x^{2}-x^{2}-y \left (x \right )^{2}\right )}+c_{1} = 0\]