\[ \int \frac {x^4 \sqrt [4]{b x^2+a x^4}}{b+a x^2+x^4} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
integrate(x^4*(a*x^4+b*x^2)^(1/4)/(x^4+a*x^2+b),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{2} \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}} x^{2} + \frac {\sqrt {2} {\left (4 \, a^{2} - b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{8 \, \left (-a\right )^{\frac {3}{4}}} + \frac {\sqrt {2} {\left (4 \, a^{2} - b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{8 \, \left (-a\right )^{\frac {3}{4}}} + \frac {\sqrt {2} {\left (4 \, a^{2} - b\right )} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x^{2}}}\right )}{16 \, \left (-a\right )^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (4 \, a^{2} - b\right )} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x^{2}}}\right )}{16 \, \left (-a\right )^{\frac {3}{4}}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________