\[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^7} \, dx \]
Optimal antiderivative \[ -\frac {\left (24 b^{2} c^{2}+a d \left (-a d +12 b c \right )\right ) \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{48 c^{2} x^{2}}-\frac {a^{2} \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{6 c \,x^{6}}-\frac {a \left (-a d +12 b c \right ) \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{24 c^{2} x^{4}}-\frac {d \left (24 b^{2} c^{2}+a d \left (-a d +12 b c \right )\right ) \arctanh \left (\frac {\sqrt {d \,x^{2}+c}}{\sqrt {c}}\right )}{16 c^{\frac {3}{2}}}+\frac {d \left (24 b^{2} c^{2}+a d \left (-a d +12 b c \right )\right ) \sqrt {d \,x^{2}+c}}{16 c^{2}} \]
command
integrate((b*x**2+a)**2*(d*x**2+c)**(3/2)/x**7,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ - \frac {a^{2} c^{2}}{6 \sqrt {d} x^{7} \sqrt {\frac {c}{d x^{2}} + 1}} - \frac {11 a^{2} c \sqrt {d}}{24 x^{5} \sqrt {\frac {c}{d x^{2}} + 1}} - \frac {17 a^{2} d^{\frac {3}{2}}}{48 x^{3} \sqrt {\frac {c}{d x^{2}} + 1}} - \frac {a^{2} d^{\frac {5}{2}}}{16 c x \sqrt {\frac {c}{d x^{2}} + 1}} + \frac {a^{2} d^{3} \operatorname {asinh}{\left (\frac {\sqrt {c}}{\sqrt {d} x} \right )}}{16 c^{\frac {3}{2}}} - \frac {a b c^{2}}{2 \sqrt {d} x^{5} \sqrt {\frac {c}{d x^{2}} + 1}} - \frac {3 a b c \sqrt {d}}{4 x^{3} \sqrt {\frac {c}{d x^{2}} + 1}} - \frac {a b d^{\frac {3}{2}} \sqrt {\frac {c}{d x^{2}} + 1}}{x} - \frac {a b d^{\frac {3}{2}}}{4 x \sqrt {\frac {c}{d x^{2}} + 1}} - \frac {3 a b d^{2} \operatorname {asinh}{\left (\frac {\sqrt {c}}{\sqrt {d} x} \right )}}{4 \sqrt {c}} - \frac {3 b^{2} \sqrt {c} d \operatorname {asinh}{\left (\frac {\sqrt {c}}{\sqrt {d} x} \right )}}{2} - \frac {b^{2} c \sqrt {d} \sqrt {\frac {c}{d x^{2}} + 1}}{2 x} + \frac {b^{2} c \sqrt {d}}{x \sqrt {\frac {c}{d x^{2}} + 1}} + \frac {b^{2} d^{\frac {3}{2}} x}{\sqrt {\frac {c}{d x^{2}} + 1}} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________