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