\[ \int \frac {\left (-2+x^5\right ) \sqrt {-1+x^5}}{x^6 \left (1-x^5+a x^{10}\right )} \, dx \]
Optimal antiderivative \[ \frac {2 \sqrt {x^{5}-1}}{5 x^{5}}+\frac {\left (\sqrt {2}\, \sqrt {a}+\sqrt {2}\, \sqrt {1-4 a}\, \sqrt {a}-4 \sqrt {2}\, a^{\frac {3}{2}}\right ) \arctan \left (\frac {\sqrt {2}\, \sqrt {a}\, \sqrt {x^{5}-1}}{\sqrt {-1-\sqrt {1-4 a}+2 a}}\right )}{5 \sqrt {1-4 a}\, \sqrt {-1-\sqrt {1-4 a}+2 a}}+\frac {\left (-\sqrt {2}\, \sqrt {a}+\sqrt {2}\, \sqrt {1-4 a}\, \sqrt {a}+4 \sqrt {2}\, a^{\frac {3}{2}}\right ) \arctan \left (\frac {\sqrt {2}\, \sqrt {a}\, \sqrt {x^{5}-1}}{\sqrt {-1+\sqrt {1-4 a}+2 a}}\right )}{5 \sqrt {1-4 a}\, \sqrt {-1+\sqrt {1-4 a}+2 a}} \]
command
integrate((x^5-2)*(x^5-1)^(1/2)/x^6/(a*x^10-x^5+1),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {{\left (\pi + 2 \, \arctan \left (\frac {{\left (x^{5} - 1\right )} a + a}{\sqrt {x^{5} - 1} \sqrt {-a}}\right )\right )} a}{5 \, \sqrt {-a}} + \frac {2}{5 \, {\left (\sqrt {x^{5} - 1} + \frac {1}{\sqrt {x^{5} - 1}}\right )}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________