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