\[ \int \frac {\sqrt {1+x^5} \left (2+x^5\right )}{x^6 \left (-1-x^5+a x^{10}\right )} \, dx \]
Optimal antiderivative \[ \frac {2 \sqrt {x^{5}+1}}{5 x^{5}}+\frac {\left (\sqrt {2}\, \sqrt {a}+4 \sqrt {2}\, a^{\frac {3}{2}}+\sqrt {2}\, \sqrt {a}\, \sqrt {1+4 a}\right ) \arctan \left (\frac {\sqrt {2}\, \sqrt {a}\, \sqrt {x^{5}+1}}{\sqrt {-1-2 a -\sqrt {1+4 a}}}\right )}{5 \sqrt {1+4 a}\, \sqrt {-1-2 a -\sqrt {1+4 a}}}+\frac {\left (-\sqrt {2}\, \sqrt {a}-4 \sqrt {2}\, a^{\frac {3}{2}}+\sqrt {2}\, \sqrt {a}\, \sqrt {1+4 a}\right ) \arctan \left (\frac {\sqrt {2}\, \sqrt {a}\, \sqrt {x^{5}+1}}{\sqrt {-1-2 a +\sqrt {1+4 a}}}\right )}{5 \sqrt {1+4 a}\, \sqrt {-1-2 a +\sqrt {1+4 a}}} \]
command
integrate((x^5+1)^(1/2)*(x^5+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
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________