\[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^7} \, dx \]
Optimal antiderivative \[ -\frac {5 b \left (A b +6 B a \right ) \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{48 a \,x^{2}}-\frac {\left (A b +6 B a \right ) \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{24 a \,x^{4}}-\frac {A \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{6 a \,x^{6}}-\frac {5 b^{2} \left (A b +6 B a \right ) \arctanh \left (\frac {\sqrt {b \,x^{2}+a}}{\sqrt {a}}\right )}{16 \sqrt {a}}+\frac {5 b^{2} \left (A b +6 B a \right ) \sqrt {b \,x^{2}+a}}{16 a} \]
command
integrate((b*x**2+a)**(5/2)*(B*x**2+A)/x**7,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ - \frac {A a^{3}}{6 \sqrt {b} x^{7} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {17 A a^{2} \sqrt {b}}{24 x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {35 A a b^{\frac {3}{2}}}{48 x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {A b^{\frac {5}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{2 x} - \frac {3 A b^{\frac {5}{2}}}{16 x \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {5 A b^{3} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{16 \sqrt {a}} - \frac {15 B \sqrt {a} b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{8} - \frac {B a^{3}}{4 \sqrt {b} x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {3 B a^{2} \sqrt {b}}{8 x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {B a b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{x} + \frac {7 B a b^{\frac {3}{2}}}{8 x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {B b^{\frac {5}{2}} x}{\sqrt {\frac {a}{b x^{2}} + 1}} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________