\[ \int \frac {x^{7/2} \left (a+b x^2\right )^2}{c+d x^2} \, dx \]
Optimal antiderivative \[ \frac {2 \left (-a d +b c \right )^{2} x^{\frac {5}{2}}}{5 d^{3}}-\frac {2 b \left (-2 a d +b c \right ) x^{\frac {9}{2}}}{9 d^{2}}+\frac {2 b^{2} x^{\frac {13}{2}}}{13 d}-\frac {c^{\frac {5}{4}} \left (-a d +b c \right )^{2} \arctan \left (1-\frac {d^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{c^{\frac {1}{4}}}\right ) \sqrt {2}}{2 d^{\frac {17}{4}}}+\frac {c^{\frac {5}{4}} \left (-a d +b c \right )^{2} \arctan \left (1+\frac {d^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{c^{\frac {1}{4}}}\right ) \sqrt {2}}{2 d^{\frac {17}{4}}}-\frac {c^{\frac {5}{4}} \left (-a d +b c \right )^{2} \ln \left (\sqrt {c}+x \sqrt {d}-c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{4 d^{\frac {17}{4}}}+\frac {c^{\frac {5}{4}} \left (-a d +b c \right )^{2} \ln \left (\sqrt {c}+x \sqrt {d}+c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{4 d^{\frac {17}{4}}}-\frac {2 c \left (-a d +b c \right )^{2} \sqrt {x}}{d^{4}} \]
command
integrate(x**(7/2)*(b*x**2+a)**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} x^{\frac {5}{2}}}{5} + \frac {4 a b x^{\frac {9}{2}}}{9} + \frac {2 b^{2} x^{\frac {13}{2}}}{13}\right ) & \text {for}\: c = 0 \wedge d = 0 \\\frac {\frac {2 a^{2} x^{\frac {5}{2}}}{5} + \frac {4 a b x^{\frac {9}{2}}}{9} + \frac {2 b^{2} x^{\frac {13}{2}}}{13}}{d} & \text {for}\: c = 0 \\\frac {\frac {2 a^{2} x^{\frac {9}{2}}}{9} + \frac {4 a b x^{\frac {13}{2}}}{13} + \frac {2 b^{2} x^{\frac {17}{2}}}{17}}{c} & \text {for}\: d = 0 \\- \frac {2 a^{2} c \sqrt {x}}{d^{2}} - \frac {a^{2} c \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{2 d^{2}} + \frac {a^{2} c \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{2 d^{2}} + \frac {a^{2} c \sqrt [4]{- \frac {c}{d}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{d^{2}} + \frac {2 a^{2} x^{\frac {5}{2}}}{5 d} + \frac {4 a b c^{2} \sqrt {x}}{d^{3}} + \frac {a b c^{2} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{d^{3}} - \frac {a b c^{2} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{d^{3}} - \frac {2 a b c^{2} \sqrt [4]{- \frac {c}{d}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{d^{3}} - \frac {4 a b c x^{\frac {5}{2}}}{5 d^{2}} + \frac {4 a b x^{\frac {9}{2}}}{9 d} - \frac {2 b^{2} c^{3} \sqrt {x}}{d^{4}} - \frac {b^{2} c^{3} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{2 d^{4}} + \frac {b^{2} c^{3} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{2 d^{4}} + \frac {b^{2} c^{3} \sqrt [4]{- \frac {c}{d}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{d^{4}} + \frac {2 b^{2} c^{2} x^{\frac {5}{2}}}{5 d^{3}} - \frac {2 b^{2} c x^{\frac {9}{2}}}{9 d^{2}} + \frac {2 b^{2} x^{\frac {13}{2}}}{13 d} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________