\[ \int \frac {-b^5+a^5 x^5}{\sqrt {b^2 x+a^2 x^3} \left (b^5+a^5 x^5\right )} \, dx \]
Optimal antiderivative \[ -\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {a}\, \sqrt {b}\, \sqrt {a^{2} x^{3}+b^{2} x}}{a^{2} x^{2}+b^{2}}\right )}{5 \sqrt {a}\, \sqrt {b}}-\frac {2 \sqrt {2+2 \sqrt {5}}\, \arctan \left (\frac {\sqrt {-2+2 \sqrt {5}}\, \sqrt {a}\, \sqrt {b}\, \sqrt {a^{2} x^{3}+b^{2} x}}{2 a^{2} x^{2}+2 b^{2}}\right )}{5 \sqrt {a}\, \sqrt {b}}-\frac {2 \sqrt {-2+2 \sqrt {5}}\, \operatorname {arctanh}\left (\frac {\sqrt {2+2 \sqrt {5}}\, \sqrt {a}\, \sqrt {b}\, \sqrt {a^{2} x^{3}+b^{2} x}}{2 a^{2} x^{2}+2 b^{2}}\right )}{5 \sqrt {a}\, \sqrt {b}} \]
command
Int[(-b^5 + a^5*x^5)/(Sqrt[b^2*x + a^2*x^3]*(b^5 + a^5*x^5)),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \text {output too large to display} \]
Rubi 4.16.1 under Mathematica 13.3.1 output \[ \int \frac {-b^5+a^5 x^5}{\sqrt {b^2 x+a^2 x^3} \left (b^5+a^5 x^5\right )} \, dx \]__________________________________________