\[ \int \frac {x^{3/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx \]
Optimal antiderivative \[ -\frac {\left (-13 a d +b c \right ) \left (-a d +b c \right )^{2} \arctan \left (1-\frac {b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{a^{\frac {1}{4}}}\right ) \sqrt {2}}{8 a^{\frac {3}{4}} b^{\frac {17}{4}}}+\frac {\left (-13 a d +b c \right ) \left (-a d +b c \right )^{2} \arctan \left (1+\frac {b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{a^{\frac {1}{4}}}\right ) \sqrt {2}}{8 a^{\frac {3}{4}} b^{\frac {17}{4}}}-\frac {\left (-13 a d +b c \right ) \left (-a d +b c \right )^{2} \ln \left (\sqrt {a}+x \sqrt {b}-a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{16 a^{\frac {3}{4}} b^{\frac {17}{4}}}+\frac {\left (-13 a d +b c \right ) \left (-a d +b c \right )^{2} \ln \left (\sqrt {a}+x \sqrt {b}+a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{16 a^{\frac {3}{4}} b^{\frac {17}{4}}}+\frac {d \left (585 a^{2} d^{2}-1098 a b c d +497 b^{2} c^{2}\right ) \sqrt {x}}{90 b^{4}}+\frac {d \left (-117 a d +113 b c \right ) \left (d \,x^{2}+c \right ) \sqrt {x}}{90 b^{3}}+\frac {13 d \left (d \,x^{2}+c \right )^{2} \sqrt {x}}{18 b^{2}}-\frac {\left (d \,x^{2}+c \right )^{3} \sqrt {x}}{2 b \left (b \,x^{2}+a \right )} \]
command
integrate(x**(3/2)*(d*x**2+c)**3/(b*x**2+a)**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 \[ \text {Timed out} \]_____________________________________________________