\[ \int \frac {\left (c+d x^2\right )^3}{x^{9/2} \left (a+b x^2\right )} \, dx \]
Optimal antiderivative \[ -\frac {2 c^{3}}{7 a \,x^{\frac {7}{2}}}+\frac {2 c^{2} \left (-3 a d +b c \right )}{3 a^{2} x^{\frac {3}{2}}}-\frac {\left (-a d +b c \right )^{3} \arctan \left (1-\frac {b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{a^{\frac {1}{4}}}\right ) \sqrt {2}}{2 a^{\frac {11}{4}} b^{\frac {5}{4}}}+\frac {\left (-a d +b c \right )^{3} \arctan \left (1+\frac {b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{a^{\frac {1}{4}}}\right ) \sqrt {2}}{2 a^{\frac {11}{4}} b^{\frac {5}{4}}}-\frac {\left (-a d +b c \right )^{3} \ln \left (\sqrt {a}+x \sqrt {b}-a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{4 a^{\frac {11}{4}} b^{\frac {5}{4}}}+\frac {\left (-a d +b c \right )^{3} \ln \left (\sqrt {a}+x \sqrt {b}+a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{4 a^{\frac {11}{4}} b^{\frac {5}{4}}}+\frac {2 d^{3} \sqrt {x}}{b} \]
command
integrate((d*x**2+c)**3/x**(9/2)/(b*x**2+a),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \tilde {\infty } \left (- \frac {2 c^{3}}{11 x^{\frac {11}{2}}} - \frac {6 c^{2} d}{7 x^{\frac {7}{2}}} - \frac {2 c d^{2}}{x^{\frac {3}{2}}} + 2 d^{3} \sqrt {x}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {- \frac {2 c^{3}}{11 x^{\frac {11}{2}}} - \frac {6 c^{2} d}{7 x^{\frac {7}{2}}} - \frac {2 c d^{2}}{x^{\frac {3}{2}}} + 2 d^{3} \sqrt {x}}{b} & \text {for}\: a = 0 \\\frac {- \frac {2 c^{3}}{7 x^{\frac {7}{2}}} - \frac {2 c^{2} d}{x^{\frac {3}{2}}} + 6 c d^{2} \sqrt {x} + \frac {2 d^{3} x^{\frac {5}{2}}}{5}}{a} & \text {for}\: b = 0 \\\frac {2 d^{3} \sqrt {x}}{b} + \frac {d^{3} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 b} - \frac {d^{3} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 b} - \frac {d^{3} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{b} - \frac {2 c^{3}}{7 a x^{\frac {7}{2}}} - \frac {2 c^{2} d}{a x^{\frac {3}{2}}} - \frac {3 c d^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 a} + \frac {3 c d^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 a} + \frac {3 c d^{2} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{a} + \frac {2 b c^{3}}{3 a^{2} x^{\frac {3}{2}}} + \frac {3 b c^{2} d \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 a^{2}} - \frac {3 b c^{2} d \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 a^{2}} - \frac {3 b c^{2} d \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{a^{2}} - \frac {b^{2} c^{3} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 a^{3}} + \frac {b^{2} c^{3} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 a^{3}} + \frac {b^{2} c^{3} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{a^{3}} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________