\[ \int \frac {\left (2 b+a x^6\right ) \left (-b-c x^4+a x^6\right )}{x^2 \left (-b+a x^6\right )^{3/4} \left (-b+c x^4+a x^6\right )} \, dx \]
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
\[ \int \frac {\left (2 b+a x^6\right ) \left (-b-c x^4+a x^6\right )}{x^2 \left (-b+a x^6\right )^{3/4} \left (-b+c x^4+a x^6\right )} \, dx \]________________________________________________________________________________________