\[ \int \frac {x^{5/2} \left (c+d x^2\right )^3}{a+b x^2} \, dx \]
Optimal antiderivative \[ \frac {2 \left (-a d +b c \right )^{3} x^{\frac {3}{2}}}{3 b^{4}}+\frac {2 d \left (a^{2} d^{2}-3 a b c d +3 b^{2} c^{2}\right ) x^{\frac {7}{2}}}{7 b^{3}}+\frac {2 d^{2} \left (-a d +3 b c \right ) x^{\frac {11}{2}}}{11 b^{2}}+\frac {2 d^{3} x^{\frac {15}{2}}}{15 b}+\frac {a^{\frac {3}{4}} \left (-a d +b c \right )^{3} \arctan \left (1-\frac {b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{a^{\frac {1}{4}}}\right ) \sqrt {2}}{2 b^{\frac {19}{4}}}-\frac {a^{\frac {3}{4}} \left (-a d +b c \right )^{3} \arctan \left (1+\frac {b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{a^{\frac {1}{4}}}\right ) \sqrt {2}}{2 b^{\frac {19}{4}}}-\frac {a^{\frac {3}{4}} \left (-a d +b c \right )^{3} \ln \left (\sqrt {a}+x \sqrt {b}-a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{4 b^{\frac {19}{4}}}+\frac {a^{\frac {3}{4}} \left (-a d +b c \right )^{3} \ln \left (\sqrt {a}+x \sqrt {b}+a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{4 b^{\frac {19}{4}}} \]
command
integrate(x**(5/2)*(d*x**2+c)**3/(b*x**2+a),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \tilde {\infty } \left (\frac {2 c^{3} x^{\frac {3}{2}}}{3} + \frac {6 c^{2} d x^{\frac {7}{2}}}{7} + \frac {6 c d^{2} x^{\frac {11}{2}}}{11} + \frac {2 d^{3} x^{\frac {15}{2}}}{15}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {\frac {2 c^{3} x^{\frac {7}{2}}}{7} + \frac {6 c^{2} d x^{\frac {11}{2}}}{11} + \frac {2 c d^{2} x^{\frac {15}{2}}}{5} + \frac {2 d^{3} x^{\frac {19}{2}}}{19}}{a} & \text {for}\: b = 0 \\\frac {\frac {2 c^{3} x^{\frac {3}{2}}}{3} + \frac {6 c^{2} d x^{\frac {7}{2}}}{7} + \frac {6 c d^{2} x^{\frac {11}{2}}}{11} + \frac {2 d^{3} x^{\frac {15}{2}}}{15}}{b} & \text {for}\: a = 0 \\- \frac {2 a^{3} d^{3} x^{\frac {3}{2}}}{3 b^{4}} - \frac {a^{3} d^{3} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{4}} + \frac {a^{3} d^{3} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{4}} - \frac {a^{3} d^{3} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{b^{4}} + \frac {2 a^{2} c d^{2} x^{\frac {3}{2}}}{b^{3}} + \frac {3 a^{2} c d^{2} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{3}} - \frac {3 a^{2} c d^{2} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{3}} + \frac {3 a^{2} c d^{2} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{b^{3}} + \frac {2 a^{2} d^{3} x^{\frac {7}{2}}}{7 b^{3}} - \frac {2 a c^{2} d x^{\frac {3}{2}}}{b^{2}} - \frac {3 a c^{2} d \left (- \frac {a}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{2}} + \frac {3 a c^{2} d \left (- \frac {a}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{2}} - \frac {3 a c^{2} d \left (- \frac {a}{b}\right )^{\frac {3}{4}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{b^{2}} - \frac {6 a c d^{2} x^{\frac {7}{2}}}{7 b^{2}} - \frac {2 a d^{3} x^{\frac {11}{2}}}{11 b^{2}} + \frac {2 c^{3} x^{\frac {3}{2}}}{3 b} + \frac {c^{3} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 b} - \frac {c^{3} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 b} + \frac {c^{3} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{b} + \frac {6 c^{2} d x^{\frac {7}{2}}}{7 b} + \frac {6 c d^{2} x^{\frac {11}{2}}}{11 b} + \frac {2 d^{3} x^{\frac {15}{2}}}{15 b} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________