\[ \int \frac {\left (-2 b+a x^6\right ) \left (b-c x^4+a x^6\right )}{x^2 \left (b+a x^6\right )^{3/4} \left (b+c x^4+a x^6\right )} \, dx \]
Optimal antiderivative \[ \frac {2 \left (a \,x^{6}+b \right )^{\frac {1}{4}}}{x}+\sqrt {2}\, c^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2}\, c^{\frac {1}{4}} x \left (a \,x^{6}+b \right )^{\frac {1}{4}}}{-x^{2} \sqrt {c}+\sqrt {a \,x^{6}+b}}\right )-\sqrt {2}\, c^{\frac {1}{4}} \arctanh \left (\frac {\frac {c^{\frac {1}{4}} x^{2} \sqrt {2}}{2}+\frac {\sqrt {a \,x^{6}+b}\, \sqrt {2}}{2 c^{\frac {1}{4}}}}{x \left (a \,x^{6}+b \right )^{\frac {1}{4}}}\right ) \]
command
Integrate[((-2*b + a*x^6)*(b - c*x^4 + a*x^6))/(x^2*(b + a*x^6)^(3/4)*(b + c*x^4 + a*x^6)),x]
Mathematica 13.1 output
\[ \frac {2 \sqrt [4]{b+a x^6}}{x}+\sqrt {2} \sqrt [4]{c} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{b+a x^6}}{-\sqrt {c} x^2+\sqrt {b+a x^6}}\right )-\sqrt {2} \sqrt [4]{c} \tanh ^{-1}\left (\frac {\sqrt {c} x^2+\sqrt {b+a x^6}}{\sqrt {2} \sqrt [4]{c} x \sqrt [4]{b+a x^6}}\right ) \]
Mathematica 12.3 output
\[ \int \frac {\left (-2 b+a x^6\right ) \left (b-c x^4+a x^6\right )}{x^2 \left (b+a x^6\right )^{3/4} \left (b+c x^4+a x^6\right )} \, dx \]________________________________________________________________________________________