\[ \int \frac {x^{3/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx \]
Optimal antiderivative \[ \frac {\left (-a d +b c \right )^{2} x^{\frac {5}{2}}}{4 c \,d^{2} \left (d \,x^{2}+c \right )^{2}}-\frac {\left (-a d +b c \right ) \left (3 a d +13 b c \right ) x^{\frac {5}{2}}}{16 c^{2} d^{2} \left (d \,x^{2}+c \right )}+\frac {\left (-3 a^{2} d^{2}-10 a b c d +45 b^{2} c^{2}\right ) \arctan \left (1-\frac {d^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{c^{\frac {1}{4}}}\right ) \sqrt {2}}{64 c^{\frac {7}{4}} d^{\frac {13}{4}}}-\frac {\left (-3 a^{2} d^{2}-10 a b c d +45 b^{2} c^{2}\right ) \arctan \left (1+\frac {d^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{c^{\frac {1}{4}}}\right ) \sqrt {2}}{64 c^{\frac {7}{4}} d^{\frac {13}{4}}}+\frac {\left (-3 a^{2} d^{2}-10 a b c d +45 b^{2} c^{2}\right ) \ln \left (\sqrt {c}+x \sqrt {d}-c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{128 c^{\frac {7}{4}} d^{\frac {13}{4}}}-\frac {\left (-3 a^{2} d^{2}-10 a b c d +45 b^{2} c^{2}\right ) \ln \left (\sqrt {c}+x \sqrt {d}+c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{128 c^{\frac {7}{4}} d^{\frac {13}{4}}}-\frac {\left (10 a b -\frac {45 b^{2} c}{d}+\frac {3 a^{2} d}{c}\right ) \sqrt {x}}{16 c \,d^{2}} \]
command
integrate(x**(3/2)*(b*x**2+a)**2/(d*x**2+c)**3,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 \[ \text {Timed out} \]_____________________________________________________