\[ \int \frac {x^{3/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx \]
Optimal antiderivative \[ \frac {2 b^{2} x^{\frac {5}{2}}}{5 d^{2}}+\frac {\left (-a d +b c \right )^{2} x^{\frac {5}{2}}}{2 c \,d^{2} \left (d \,x^{2}+c \right )}-\frac {\left (-a d +b c \right ) \left (-a d +9 b c \right ) \arctan \left (1-\frac {d^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{c^{\frac {1}{4}}}\right ) \sqrt {2}}{8 c^{\frac {3}{4}} d^{\frac {13}{4}}}+\frac {\left (-a d +b c \right ) \left (-a d +9 b c \right ) \arctan \left (1+\frac {d^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{c^{\frac {1}{4}}}\right ) \sqrt {2}}{8 c^{\frac {3}{4}} d^{\frac {13}{4}}}-\frac {\left (-a d +b c \right ) \left (-a d +9 b c \right ) \ln \left (\sqrt {c}+x \sqrt {d}-c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{16 c^{\frac {3}{4}} d^{\frac {13}{4}}}+\frac {\left (-a d +b c \right ) \left (-a d +9 b c \right ) \ln \left (\sqrt {c}+x \sqrt {d}+c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{16 c^{\frac {3}{4}} d^{\frac {13}{4}}}-\frac {\left (-a d +b c \right ) \left (-a d +9 b c \right ) \sqrt {x}}{2 c \,d^{3}} \]
command
integrate(x**(3/2)*(b*x**2+a)**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 \[ \text {Timed out} \]_____________________________________________________