\[ \int \frac {x^{5/2} \left (a+b x^2\right )^2}{c+d x^2} \, dx \]
Optimal antiderivative \[ \frac {2 \left (-a d +b c \right )^{2} x^{\frac {3}{2}}}{3 d^{3}}-\frac {2 b \left (-2 a d +b c \right ) x^{\frac {7}{2}}}{7 d^{2}}+\frac {2 b^{2} x^{\frac {11}{2}}}{11 d}+\frac {c^{\frac {3}{4}} \left (-a d +b c \right )^{2} \arctan \left (1-\frac {d^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{c^{\frac {1}{4}}}\right ) \sqrt {2}}{2 d^{\frac {15}{4}}}-\frac {c^{\frac {3}{4}} \left (-a d +b c \right )^{2} \arctan \left (1+\frac {d^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{c^{\frac {1}{4}}}\right ) \sqrt {2}}{2 d^{\frac {15}{4}}}-\frac {c^{\frac {3}{4}} \left (-a d +b c \right )^{2} \ln \left (\sqrt {c}+x \sqrt {d}-c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{4 d^{\frac {15}{4}}}+\frac {c^{\frac {3}{4}} \left (-a d +b c \right )^{2} \ln \left (\sqrt {c}+x \sqrt {d}+c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{4 d^{\frac {15}{4}}} \]
command
integrate(x**(5/2)*(b*x**2+a)**2/(d*x**2+c),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ a^{2} \left (\begin {cases} \tilde {\infty } x^{\frac {3}{2}} & \text {for}\: c = 0 \wedge d = 0 \\\frac {2 x^{\frac {3}{2}}}{3 d} & \text {for}\: c = 0 \\\frac {2 x^{\frac {7}{2}}}{7 c} & \text {for}\: d = 0 \\- \frac {c \log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{2 d^{2} \sqrt [4]{- \frac {c}{d}}} + \frac {c \log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{2 d^{2} \sqrt [4]{- \frac {c}{d}}} - \frac {c \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{d^{2} \sqrt [4]{- \frac {c}{d}}} + \frac {2 x^{\frac {3}{2}}}{3 d} & \text {otherwise} \end {cases}\right ) + 2 a b \left (\begin {cases} \tilde {\infty } x^{\frac {7}{2}} & \text {for}\: c = 0 \wedge d = 0 \\\frac {2 x^{\frac {7}{2}}}{7 d} & \text {for}\: c = 0 \\\frac {2 x^{\frac {11}{2}}}{11 c} & \text {for}\: d = 0 \\\frac {c^{2} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{2 d^{3} \sqrt [4]{- \frac {c}{d}}} - \frac {c^{2} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{2 d^{3} \sqrt [4]{- \frac {c}{d}}} + \frac {c^{2} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{d^{3} \sqrt [4]{- \frac {c}{d}}} - \frac {2 c x^{\frac {3}{2}}}{3 d^{2}} + \frac {2 x^{\frac {7}{2}}}{7 d} & \text {otherwise} \end {cases}\right ) + b^{2} \left (\begin {cases} \tilde {\infty } x^{\frac {11}{2}} & \text {for}\: c = 0 \wedge d = 0 \\\frac {2 x^{\frac {11}{2}}}{11 d} & \text {for}\: c = 0 \\\frac {2 x^{\frac {15}{2}}}{15 c} & \text {for}\: d = 0 \\- \frac {c^{3} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{2 d^{4} \sqrt [4]{- \frac {c}{d}}} + \frac {c^{3} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{2 d^{4} \sqrt [4]{- \frac {c}{d}}} - \frac {c^{3} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{d^{4} \sqrt [4]{- \frac {c}{d}}} + \frac {2 c^{2} x^{\frac {3}{2}}}{3 d^{3}} - \frac {2 c x^{\frac {7}{2}}}{7 d^{2}} + \frac {2 x^{\frac {11}{2}}}{11 d} & \text {otherwise} \end {cases}\right ) \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________