\[ \int \frac {x^{11/2}}{b x^2+c x^4} \, dx \]
Optimal antiderivative \[ \frac {2 x^{\frac {5}{2}}}{5 c}-\frac {b^{\frac {5}{4}} \arctan \left (1-\frac {c^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{b^{\frac {1}{4}}}\right ) \sqrt {2}}{2 c^{\frac {9}{4}}}+\frac {b^{\frac {5}{4}} \arctan \left (1+\frac {c^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{b^{\frac {1}{4}}}\right ) \sqrt {2}}{2 c^{\frac {9}{4}}}-\frac {b^{\frac {5}{4}} \ln \left (\sqrt {b}+x \sqrt {c}-b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{4 c^{\frac {9}{4}}}+\frac {b^{\frac {5}{4}} \ln \left (\sqrt {b}+x \sqrt {c}+b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{4 c^{\frac {9}{4}}}-\frac {2 b \sqrt {x}}{c^{2}} \]
command
integrate(x**(11/2)/(c*x**4+b*x**2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \tilde {\infty } x^{\frac {5}{2}} & \text {for}\: b = 0 \wedge c = 0 \\\frac {2 x^{\frac {9}{2}}}{9 b} & \text {for}\: c = 0 \\\frac {2 x^{\frac {5}{2}}}{5 c} & \text {for}\: b = 0 \\- \frac {2 b \sqrt {x}}{c^{2}} - \frac {b \sqrt [4]{- \frac {b}{c}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {b}{c}} \right )}}{2 c^{2}} + \frac {b \sqrt [4]{- \frac {b}{c}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {b}{c}} \right )}}{2 c^{2}} + \frac {b \sqrt [4]{- \frac {b}{c}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {b}{c}}} \right )}}{c^{2}} + \frac {2 x^{\frac {5}{2}}}{5 c} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________