\[ \int \frac {x^4 \left (3+x^5\right )}{\sqrt {1+x^5} \left (-1+a-(1+2 a) x^5+a x^{10}\right )} \, dx \]
Optimal antiderivative \[ \frac {\sqrt {2}\, \left (1+8 a +\sqrt {1+8 a}\right ) \arctan \left (\frac {\sqrt {2}\, \sqrt {a}\, \sqrt {x^{5}+1}}{\sqrt {-1-4 a -\sqrt {1+8 a}}}\right )}{5 \sqrt {a}\, \sqrt {1+8 a}\, \sqrt {-1-4 a -\sqrt {1+8 a}}}+\frac {\sqrt {2}\, \left (-1-8 a +\sqrt {1+8 a}\right ) \arctan \left (\frac {\sqrt {2}\, \sqrt {a}\, \sqrt {x^{5}+1}}{\sqrt {-1-4 a +\sqrt {1+8 a}}}\right )}{5 \sqrt {a}\, \sqrt {1+8 a}\, \sqrt {-1-4 a +\sqrt {1+8 a}}} \]
command
integrate(x^4*(x^5+3)/(x^5+1)^(1/2)/(-1+a-(1+2*a)*x^5+a*x^10),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {2 \, \sqrt {-a} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x^{5} + 1}}{\sqrt {-\frac {4 \, a + \sqrt {{\left (4 \, a + 1\right )}^{2} - 16 \, a^{2}} + 1}{a}}}\right )}{5 \, {\left | a \right |}} - \frac {{\left (8 \, \sqrt {-8 \, a^{2} - a} \sqrt {-a} a^{2} \mathrm {sgn}\left (a\right ) + 8 \, \sqrt {-8 \, a^{2} - a} \sqrt {8 \, a + 1} a^{2} \mathrm {sgn}\left (a\right ) - 8 \, \sqrt {-a} a^{3} - 8 \, \sqrt {8 \, a + 1} a^{3} + 3 \, \sqrt {-8 \, a^{2} - a} \sqrt {-a} \sqrt {8 \, a + 1} a \mathrm {sgn}\left (a\right ) - 10 \, \sqrt {-8 \, a^{2} - a} a^{2} \mathrm {sgn}\left (a\right ) + 8 \, \sqrt {-a} \sqrt {8 \, a + 1} a^{2} - 24 \, a^{3} + 11 \, \sqrt {-8 \, a^{2} - a} \sqrt {-a} a \mathrm {sgn}\left (a\right ) + 11 \, \sqrt {-8 \, a^{2} - a} \sqrt {8 \, a + 1} a \mathrm {sgn}\left (a\right ) - 2 \, \sqrt {-a} a^{2} - 11 \, \sqrt {8 \, a + 1} a^{2} - 3 \, \sqrt {-8 \, a^{2} - a} \sqrt {-a} \sqrt {8 \, a + 1} \mathrm {sgn}\left (a\right ) - 9 \, \sqrt {-8 \, a^{2} - a} a \mathrm {sgn}\left (a\right ) + 11 \, \sqrt {-a} \sqrt {8 \, a + 1} a + 21 \, a^{2} + \sqrt {-8 \, a^{2} - a} \sqrt {-a} \mathrm {sgn}\left (a\right ) + \sqrt {-8 \, a^{2} - a} \sqrt {8 \, a + 1} \mathrm {sgn}\left (a\right ) - 9 \, \sqrt {-a} a - \sqrt {8 \, a + 1} a - \sqrt {-8 \, a^{2} - a} \mathrm {sgn}\left (a\right ) + \sqrt {-a} \sqrt {8 \, a + 1} + 3 \, a - \sqrt {-a}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x^{5} + 1}}{\sqrt {-\frac {4 \, a - \sqrt {{\left (4 \, a + 1\right )}^{2} - 16 \, a^{2}} + 1}{a}}}\right )}{20 \, {\left (9 \, a^{3} + 10 \, a^{2} + a\right )} {\left | a \right |}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________