\[ \int \frac {-b+a x^8}{\sqrt [4]{b+a x^8} \left (b-c x^4+a x^8\right )} \, dx \]
Optimal antiderivative \[ -\frac {\arctan \left (\frac {c^{\frac {1}{4}} x}{\left (a \,x^{8}+b \right )^{\frac {1}{4}}}\right )}{2 c^{\frac {1}{4}}}-\frac {\arctanh \left (\frac {c^{\frac {1}{4}} x}{\left (a \,x^{8}+b \right )^{\frac {1}{4}}}\right )}{2 c^{\frac {1}{4}}} \]
command
Integrate[(-b + a*x^8)/((b + a*x^8)^(1/4)*(b - c*x^4 + a*x^8)),x]
Mathematica 13.1 output
\[ -\frac {\text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{b+a x^8}}\right )+\tanh ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{b+a x^8}}\right )}{2 \sqrt [4]{c}} \]
Mathematica 12.3 output
\[ \int \frac {-b+a x^8}{\sqrt [4]{b+a x^8} \left (b-c x^4+a x^8\right )} \, dx \]________________________________________________________________________________________