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