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