\[ \int \frac {\left (c+d x^2\right )^3}{x^{7/2} \left (a+b x^2\right )} \, dx \]
Optimal antiderivative \[ -\frac {2 c^{3}}{5 a \,x^{\frac {5}{2}}}+\frac {2 d^{3} x^{\frac {3}{2}}}{3 b}-\frac {\left (-a d +b c \right )^{3} \arctan \left (1-\frac {b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{a^{\frac {1}{4}}}\right ) \sqrt {2}}{2 a^{\frac {9}{4}} b^{\frac {7}{4}}}+\frac {\left (-a d +b c \right )^{3} \arctan \left (1+\frac {b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{a^{\frac {1}{4}}}\right ) \sqrt {2}}{2 a^{\frac {9}{4}} b^{\frac {7}{4}}}+\frac {\left (-a d +b c \right )^{3} \ln \left (\sqrt {a}+x \sqrt {b}-a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{4 a^{\frac {9}{4}} b^{\frac {7}{4}}}-\frac {\left (-a d +b c \right )^{3} \ln \left (\sqrt {a}+x \sqrt {b}+a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{4 a^{\frac {9}{4}} b^{\frac {7}{4}}}+\frac {2 c^{2} \left (-3 a d +b c \right )}{a^{2} \sqrt {x}} \]
command
integrate((d*x**2+c)**3/x**(7/2)/(b*x**2+a),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ c^{3} \left (\begin {cases} \frac {\tilde {\infty }}{x^{\frac {9}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{9 b x^{\frac {9}{2}}} & \text {for}\: a = 0 \\- \frac {2}{5 a x^{\frac {5}{2}}} & \text {for}\: b = 0 \\- \frac {2}{5 a x^{\frac {5}{2}}} + \frac {b \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 a^{2} \sqrt [4]{- \frac {a}{b}}} - \frac {b \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 a^{2} \sqrt [4]{- \frac {a}{b}}} + \frac {b \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{a^{2} \sqrt [4]{- \frac {a}{b}}} + \frac {2 b}{a^{2} \sqrt {x}} & \text {otherwise} \end {cases}\right ) + 3 c^{2} d \left (\begin {cases} \frac {\tilde {\infty }}{x^{\frac {5}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{5 b x^{\frac {5}{2}}} & \text {for}\: a = 0 \\- \frac {2}{a \sqrt {x}} & \text {for}\: b = 0 \\- \frac {\log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 a \sqrt [4]{- \frac {a}{b}}} + \frac {\log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 a \sqrt [4]{- \frac {a}{b}}} - \frac {\operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{a \sqrt [4]{- \frac {a}{b}}} - \frac {2}{a \sqrt {x}} & \text {otherwise} \end {cases}\right ) + 6 c d^{2} \operatorname {RootSum} {\left (256 t^{4} a b^{3} + 1, \left ( t \mapsto t \log {\left (64 t^{3} a b^{2} + \sqrt {x} \right )} \right )\right )} + d^{3} \left (\begin {cases} \tilde {\infty } x^{\frac {3}{2}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 x^{\frac {7}{2}}}{7 a} & \text {for}\: b = 0 \\\frac {2 x^{\frac {3}{2}}}{3 b} & \text {for}\: a = 0 \\- \frac {a \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{2} \sqrt [4]{- \frac {a}{b}}} + \frac {a \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{2} \sqrt [4]{- \frac {a}{b}}} - \frac {a \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{b^{2} \sqrt [4]{- \frac {a}{b}}} + \frac {2 x^{\frac {3}{2}}}{3 b} & \text {otherwise} \end {cases}\right ) \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________