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