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