\[ \int \frac {x^{9/2}}{a+c x^4} \, dx \]
Optimal antiderivative \[ \frac {2 x^{\frac {3}{2}}}{3 c}+\frac {\left (-a \right )^{\frac {3}{8}} \arctan \left (\frac {c^{\frac {1}{8}} \sqrt {x}}{\left (-a \right )^{\frac {1}{8}}}\right )}{2 c^{\frac {11}{8}}}-\frac {\left (-a \right )^{\frac {3}{8}} \arctanh \left (\frac {c^{\frac {1}{8}} \sqrt {x}}{\left (-a \right )^{\frac {1}{8}}}\right )}{2 c^{\frac {11}{8}}}-\frac {\left (-a \right )^{\frac {3}{8}} \arctan \left (-1+\frac {c^{\frac {1}{8}} \sqrt {2}\, \sqrt {x}}{\left (-a \right )^{\frac {1}{8}}}\right ) \sqrt {2}}{4 c^{\frac {11}{8}}}-\frac {\left (-a \right )^{\frac {3}{8}} \arctan \left (1+\frac {c^{\frac {1}{8}} \sqrt {2}\, \sqrt {x}}{\left (-a \right )^{\frac {1}{8}}}\right ) \sqrt {2}}{4 c^{\frac {11}{8}}}-\frac {\left (-a \right )^{\frac {3}{8}} \ln \left (\left (-a \right )^{\frac {1}{4}}+c^{\frac {1}{4}} x -\left (-a \right )^{\frac {1}{8}} c^{\frac {1}{8}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{8 c^{\frac {11}{8}}}+\frac {\left (-a \right )^{\frac {3}{8}} \ln \left (\left (-a \right )^{\frac {1}{4}}+c^{\frac {1}{4}} x +\left (-a \right )^{\frac {1}{8}} c^{\frac {1}{8}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{8 c^{\frac {11}{8}}} \]
command
integrate(x**(9/2)/(c*x**4+a),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \tilde {\infty } x^{\frac {3}{2}} & \text {for}\: a = 0 \wedge c = 0 \\\frac {2 x^{\frac {11}{2}}}{11 a} & \text {for}\: c = 0 \\\frac {2 x^{\frac {3}{2}}}{3 c} & \text {for}\: a = 0 \\\frac {2 x^{\frac {3}{2}}}{3 c} + \frac {\left (- \frac {a}{c}\right )^{\frac {3}{8}} \log {\left (\sqrt {x} - \sqrt [8]{- \frac {a}{c}} \right )}}{4 c} - \frac {\left (- \frac {a}{c}\right )^{\frac {3}{8}} \log {\left (\sqrt {x} + \sqrt [8]{- \frac {a}{c}} \right )}}{4 c} - \frac {\sqrt {2} \left (- \frac {a}{c}\right )^{\frac {3}{8}} \log {\left (- 4 \sqrt {2} \sqrt {x} \sqrt [8]{- \frac {a}{c}} + 4 x + 4 \sqrt [4]{- \frac {a}{c}} \right )}}{8 c} + \frac {\sqrt {2} \left (- \frac {a}{c}\right )^{\frac {3}{8}} \log {\left (4 \sqrt {2} \sqrt {x} \sqrt [8]{- \frac {a}{c}} + 4 x + 4 \sqrt [4]{- \frac {a}{c}} \right )}}{8 c} + \frac {\left (- \frac {a}{c}\right )^{\frac {3}{8}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [8]{- \frac {a}{c}}} \right )}}{2 c} - \frac {\sqrt {2} \left (- \frac {a}{c}\right )^{\frac {3}{8}} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt [8]{- \frac {a}{c}}} - 1 \right )}}{4 c} - \frac {\sqrt {2} \left (- \frac {a}{c}\right )^{\frac {3}{8}} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt [8]{- \frac {a}{c}}} + 1 \right )}}{4 c} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________