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