\[ \int \frac {\left (4 b+a x^5\right ) \left (-b+c x^4+a x^5\right )}{x^2 \left (-b+a x^5\right )^{3/4} \left (-b-c x^4+a x^5\right )} \, dx \]
Optimal antiderivative \[ \frac {4 \left (a \,x^{5}-b \right )^{\frac {1}{4}}}{x}-4 c^{\frac {1}{4}} \arctan \left (\frac {c^{\frac {1}{4}} x \left (a \,x^{5}-b \right )^{\frac {3}{4}}}{-a \,x^{5}+b}\right )+4 c^{\frac {1}{4}} \arctanh \left (\frac {c^{\frac {1}{4}} x \left (a \,x^{5}-b \right )^{\frac {3}{4}}}{-a \,x^{5}+b}\right ) \]
command
Integrate[((4*b + a*x^5)*(-b + c*x^4 + a*x^5))/(x^2*(-b + a*x^5)^(3/4)*(-b - c*x^4 + a*x^5)),x]
Mathematica 13.1 output
\[ \frac {4 \sqrt [4]{-b+a x^5}}{x}+4 \sqrt [4]{c} \text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+a x^5}}\right )-4 \sqrt [4]{c} \tanh ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+a x^5}}\right ) \]
Mathematica 12.3 output
\[ \int \frac {\left (4 b+a x^5\right ) \left (-b+c x^4+a x^5\right )}{x^2 \left (-b+a x^5\right )^{3/4} \left (-b-c x^4+a x^5\right )} \, dx \]________________________________________________________________________________________