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