\[ \int \frac {\left (a+b x^2\right )^2}{x^{5/2} \left (c+d x^2\right )^2} \, dx \]
Optimal antiderivative \[ -\frac {2 a^{2}}{3 c \,x^{\frac {3}{2}} \left (d \,x^{2}+c \right )}-\frac {\left (-a d +b c \right ) \left (7 a d +b c \right ) \arctan \left (1-\frac {d^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{c^{\frac {1}{4}}}\right ) \sqrt {2}}{8 c^{\frac {11}{4}} d^{\frac {5}{4}}}+\frac {\left (-a d +b c \right ) \left (7 a d +b c \right ) \arctan \left (1+\frac {d^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{c^{\frac {1}{4}}}\right ) \sqrt {2}}{8 c^{\frac {11}{4}} d^{\frac {5}{4}}}-\frac {\left (-a d +b c \right ) \left (7 a d +b c \right ) \ln \left (\sqrt {c}+x \sqrt {d}-c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{16 c^{\frac {11}{4}} d^{\frac {5}{4}}}+\frac {\left (-a d +b c \right ) \left (7 a d +b c \right ) \ln \left (\sqrt {c}+x \sqrt {d}+c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{16 c^{\frac {11}{4}} d^{\frac {5}{4}}}-\frac {\left (7 a^{2} d^{2}-6 a b c d +3 b^{2} c^{2}\right ) \sqrt {x}}{6 c^{2} d \left (d \,x^{2}+c \right )} \]
command
integrate((b*x**2+a)**2/x**(5/2)/(d*x**2+c)**2,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {output too large to display} \]
Sympy 1.8 under Python 3.8.8 output \[ \text {Timed out} \]_____________________________________________________