\[ \int \frac {x^4 \left (-2+x^5\right )}{\sqrt {-1+x^5} \left (1-x^5+a x^{10}\right )} \, dx \]
Optimal antiderivative \[ \frac {\sqrt {2}\, \left (1+\sqrt {1-4 a}-4 a \right ) \arctan \left (\frac {\sqrt {2}\, \sqrt {a}\, \sqrt {x^{5}-1}}{\sqrt {-1-\sqrt {1-4 a}+2 a}}\right )}{5 \sqrt {1-4 a}\, \sqrt {a}\, \sqrt {-1-\sqrt {1-4 a}+2 a}}+\frac {\sqrt {2}\, \left (-1+\sqrt {1-4 a}+4 a \right ) \arctan \left (\frac {\sqrt {2}\, \sqrt {a}\, \sqrt {x^{5}-1}}{\sqrt {-1+\sqrt {1-4 a}+2 a}}\right )}{5 \sqrt {1-4 a}\, \sqrt {a}\, \sqrt {-1+\sqrt {1-4 a}+2 a}} \]
command
integrate(x^4*(x^5-2)/(x^5-1)^(1/2)/(a*x^10-x^5+1),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\pi + 2 \, \arctan \left (\frac {{\left (x^{5} - 1\right )} a + a}{\sqrt {x^{5} - 1} \sqrt {-a}}\right )}{5 \, \sqrt {-a}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________