\[ \int \frac {1}{x^{7/2} \left (b x^2+c x^4\right )} \, dx \]
Optimal antiderivative \[ -\frac {2}{9 b \,x^{\frac {9}{2}}}+\frac {2 c}{5 b^{2} x^{\frac {5}{2}}}+\frac {c^{\frac {9}{4}} \arctan \left (1-\frac {c^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{b^{\frac {1}{4}}}\right ) \sqrt {2}}{2 b^{\frac {13}{4}}}-\frac {c^{\frac {9}{4}} \arctan \left (1+\frac {c^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{b^{\frac {1}{4}}}\right ) \sqrt {2}}{2 b^{\frac {13}{4}}}-\frac {c^{\frac {9}{4}} \ln \left (\sqrt {b}+x \sqrt {c}-b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{4 b^{\frac {13}{4}}}+\frac {c^{\frac {9}{4}} \ln \left (\sqrt {b}+x \sqrt {c}+b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{4 b^{\frac {13}{4}}}-\frac {2 c^{2}}{b^{3} \sqrt {x}} \]
command
integrate(1/x**(7/2)/(c*x**4+b*x**2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \frac {\tilde {\infty }}{x^{\frac {13}{2}}} & \text {for}\: b = 0 \wedge c = 0 \\- \frac {2}{13 c x^{\frac {13}{2}}} & \text {for}\: b = 0 \\- \frac {2}{9 b x^{\frac {9}{2}}} & \text {for}\: c = 0 \\- \frac {2}{9 b x^{\frac {9}{2}}} + \frac {2 c}{5 b^{2} x^{\frac {5}{2}}} - \frac {c^{2} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {b}{c}} \right )}}{2 b^{3} \sqrt [4]{- \frac {b}{c}}} + \frac {c^{2} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {b}{c}} \right )}}{2 b^{3} \sqrt [4]{- \frac {b}{c}}} - \frac {c^{2} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {b}{c}}} \right )}}{b^{3} \sqrt [4]{- \frac {b}{c}}} - \frac {2 c^{2}}{b^{3} \sqrt {x}} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________