\[ \int \frac {\left (-4 b+a x^5\right ) \left (b+a x^5\right )^{3/4}}{x^4 \left (2 b+c x^4+2 a x^5\right )} \, dx \]
Optimal antiderivative \[ \frac {2 \left (a \,x^{5}+b \right )^{\frac {3}{4}}}{3 x^{3}}+\frac {c^{\frac {3}{4}} \arctan \left (\frac {2^{\frac {3}{4}} c^{\frac {1}{4}} x \left (a \,x^{5}+b \right )^{\frac {1}{4}}}{-x^{2} \sqrt {c}+\sqrt {2}\, \sqrt {a \,x^{5}+b}}\right ) 2^{\frac {3}{4}}}{4}+\frac {c^{\frac {3}{4}} \arctanh \left (\frac {\frac {c^{\frac {1}{4}} x^{2} 2^{\frac {1}{4}}}{2}+\frac {\sqrt {a \,x^{5}+b}\, 2^{\frac {3}{4}}}{2 c^{\frac {1}{4}}}}{x \left (a \,x^{5}+b \right )^{\frac {1}{4}}}\right ) 2^{\frac {3}{4}}}{4} \]
command
Integrate[((-4*b + a*x^5)*(b + a*x^5)^(3/4))/(x^4*(2*b + c*x^4 + 2*a*x^5)),x]
Mathematica 13.1 output
\[ \frac {1}{12} \left (\frac {8 \left (b+a x^5\right )^{3/4}}{x^3}+3\ 2^{3/4} c^{3/4} \text {ArcTan}\left (\frac {\sqrt [4]{c} x}{2^{3/4} \sqrt [4]{b+a x^5}}-\frac {\sqrt [4]{b+a x^5}}{\sqrt [4]{2} \sqrt [4]{c} x}\right )+3\ 2^{3/4} c^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{c} x}{2^{3/4} \sqrt [4]{b+a x^5}}+\frac {\sqrt [4]{b+a x^5}}{\sqrt [4]{2} \sqrt [4]{c} x}\right )\right ) \]
Mathematica 12.3 output
\[ \int \frac {\left (-4 b+a x^5\right ) \left (b+a x^5\right )^{3/4}}{x^4 \left (2 b+c x^4+2 a x^5\right )} \, dx \]________________________________________________________________________________________