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