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