\[ y'(x)=\frac {4 (a-1) (a+1) x}{a^6 x^4-3 a^4 x^4-2 a^4 x^2 y(x)^2+3 a^2 x^4+4 a^2 x^2 y(x)^2+a^2 y(x)^4-x^4-2 x^2 y(x)^2-y(x)^4+4 y(x)} \] ✓ Mathematica : cpu = 0.903235 (sec), leaf count = 1269
DSolve[Derivative[1][y][x] == (4*(-1 + a)*(1 + a)*x)/(-x^4 + 3*a^2*x^4 - 3*a^4*x^4 + a^6*x^4 + 4*y[x] - 2*x^2*y[x]^2 + 4*a^2*x^2*y[x]^2 - 2*a^4*x^2*y[x]^2 - y[x]^4 + a^2*y[x]^4),y[x],x]
\[\left \{\left \{y(x)\to \frac {c_1}{3 \left (a^2-1\right )}+\frac {\sqrt [3]{-18 x^2 c_1 a^6+54 x^2 c_1 a^4+54 a^4-54 x^2 c_1 a^2-108 a^2+2 c_1{}^3+18 x^2 c_1+\sqrt {4 \left (-3 x^2 a^6+9 x^2 a^4-9 x^2 a^2+3 x^2-c_1{}^2\right ){}^3+\left (-18 x^2 c_1 a^6+54 x^2 c_1 a^4+54 a^4-54 x^2 c_1 a^2-108 a^2+2 c_1{}^3+18 x^2 c_1+54\right ){}^2}+54}}{3 \sqrt [3]{2} \left (a^2-1\right )}-\frac {\sqrt [3]{2} \left (-3 x^2 a^6+9 x^2 a^4-9 x^2 a^2+3 x^2-c_1{}^2\right )}{3 \left (a^2-1\right ) \sqrt [3]{-18 x^2 c_1 a^6+54 x^2 c_1 a^4+54 a^4-54 x^2 c_1 a^2-108 a^2+2 c_1{}^3+18 x^2 c_1+\sqrt {4 \left (-3 x^2 a^6+9 x^2 a^4-9 x^2 a^2+3 x^2-c_1{}^2\right ){}^3+\left (-18 x^2 c_1 a^6+54 x^2 c_1 a^4+54 a^4-54 x^2 c_1 a^2-108 a^2+2 c_1{}^3+18 x^2 c_1+54\right ){}^2}+54}}\right \},\left \{y(x)\to \frac {c_1}{3 \left (a^2-1\right )}-\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-18 x^2 c_1 a^6+54 x^2 c_1 a^4+54 a^4-54 x^2 c_1 a^2-108 a^2+2 c_1{}^3+18 x^2 c_1+\sqrt {4 \left (-3 x^2 a^6+9 x^2 a^4-9 x^2 a^2+3 x^2-c_1{}^2\right ){}^3+\left (-18 x^2 c_1 a^6+54 x^2 c_1 a^4+54 a^4-54 x^2 c_1 a^2-108 a^2+2 c_1{}^3+18 x^2 c_1+54\right ){}^2}+54}}{6 \sqrt [3]{2} \left (a^2-1\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (-3 x^2 a^6+9 x^2 a^4-9 x^2 a^2+3 x^2-c_1{}^2\right )}{3\ 2^{2/3} \left (a^2-1\right ) \sqrt [3]{-18 x^2 c_1 a^6+54 x^2 c_1 a^4+54 a^4-54 x^2 c_1 a^2-108 a^2+2 c_1{}^3+18 x^2 c_1+\sqrt {4 \left (-3 x^2 a^6+9 x^2 a^4-9 x^2 a^2+3 x^2-c_1{}^2\right ){}^3+\left (-18 x^2 c_1 a^6+54 x^2 c_1 a^4+54 a^4-54 x^2 c_1 a^2-108 a^2+2 c_1{}^3+18 x^2 c_1+54\right ){}^2}+54}}\right \},\left \{y(x)\to \frac {c_1}{3 \left (a^2-1\right )}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-18 x^2 c_1 a^6+54 x^2 c_1 a^4+54 a^4-54 x^2 c_1 a^2-108 a^2+2 c_1{}^3+18 x^2 c_1+\sqrt {4 \left (-3 x^2 a^6+9 x^2 a^4-9 x^2 a^2+3 x^2-c_1{}^2\right ){}^3+\left (-18 x^2 c_1 a^6+54 x^2 c_1 a^4+54 a^4-54 x^2 c_1 a^2-108 a^2+2 c_1{}^3+18 x^2 c_1+54\right ){}^2}+54}}{6 \sqrt [3]{2} \left (a^2-1\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (-3 x^2 a^6+9 x^2 a^4-9 x^2 a^2+3 x^2-c_1{}^2\right )}{3\ 2^{2/3} \left (a^2-1\right ) \sqrt [3]{-18 x^2 c_1 a^6+54 x^2 c_1 a^4+54 a^4-54 x^2 c_1 a^2-108 a^2+2 c_1{}^3+18 x^2 c_1+\sqrt {4 \left (-3 x^2 a^6+9 x^2 a^4-9 x^2 a^2+3 x^2-c_1{}^2\right ){}^3+\left (-18 x^2 c_1 a^6+54 x^2 c_1 a^4+54 a^4-54 x^2 c_1 a^2-108 a^2+2 c_1{}^3+18 x^2 c_1+54\right ){}^2}+54}}\right \}\right \}\] ✓ Maple : cpu = 0.313 (sec), leaf count = 1742
dsolve(diff(y(x),x) = 4*x*(a-1)*(a+1)/(4*y(x)+a^2*y(x)^4-2*a^4*y(x)^2*x^2+4*y(x)^2*a^2*x^2+a^6*x^4-3*a^4*x^4+3*a^2*x^4-y(x)^4-2*x^2*y(x)^2-x^4),y(x))
\[y \left (x \right ) = \frac {\left (\left (-a^{2} c_{1}+c_{1}\right ) 9^{\frac {1}{3}} \left (\left (\frac {\sqrt {-3 \left (a -1\right )^{5} \left (a +1\right )^{5} x^{6}+6 c_{1}^{2} \left (a -1\right )^{4} \left (a +1\right )^{4} x^{4}-3 c_{1} \left (a -1\right )^{2} \left (a +1\right )^{2} \left (c_{1}^{3} a^{2}-c_{1}^{3}-18\right ) x^{2}-6 c_{1}^{3} a^{2}+6 c_{1}^{3}+81}}{3}+3+\left (-\frac {a^{2}}{9}+\frac {1}{9}\right ) c_{1}^{3}+x^{2} \left (a -1\right )^{2} \left (a +1\right )^{2} c_{1}\right ) \left (a +1\right )^{2} \left (a -1\right )^{2}\right )^{\frac {1}{3}}+3 a^{6} x^{2}+\left (-9 x^{2}+c_{1}^{2}\right ) a^{4}+\left (9 x^{2}-2 c_{1}^{2}\right ) a^{2}+\left (\left (9 x^{2} c_{1} a^{4}-c_{1}^{3} a^{2}-18 c_{1} a^{2} x^{2}+c_{1}^{3}+9 c_{1} x^{2}+3 \sqrt {-3 \left (a -1\right )^{5} \left (a +1\right )^{5} x^{6}+6 c_{1}^{2} \left (a -1\right )^{4} \left (a +1\right )^{4} x^{4}-3 c_{1} \left (a -1\right )^{2} \left (a +1\right )^{2} \left (c_{1}^{3} a^{2}-c_{1}^{3}-18\right ) x^{2}-6 c_{1}^{3} a^{2}+6 c_{1}^{3}+81}+27\right ) \left (a^{2}-1\right )^{2}\right )^{\frac {2}{3}}-3 x^{2}+c_{1}^{2}\right ) 9^{\frac {2}{3}}}{9 \left (\left (\frac {\sqrt {-3 \left (a -1\right )^{5} \left (a +1\right )^{5} x^{6}+6 c_{1}^{2} \left (a -1\right )^{4} \left (a +1\right )^{4} x^{4}-3 c_{1} \left (a -1\right )^{2} \left (a +1\right )^{2} \left (c_{1}^{3} a^{2}-c_{1}^{3}-18\right ) x^{2}-6 c_{1}^{3} a^{2}+6 c_{1}^{3}+81}}{3}+3+\left (-\frac {a^{2}}{9}+\frac {1}{9}\right ) c_{1}^{3}+x^{2} \left (a -1\right )^{2} \left (a +1\right )^{2} c_{1}\right ) \left (a +1\right )^{2} \left (a -1\right )^{2}\right )^{\frac {1}{3}} \left (3 a^{2}-3\right )}\]