\[ \int \frac {\left (c+d x^2\right )^3}{x^{5/2} \left (a+b x^2\right )^2} \, dx \]
Optimal antiderivative \[ -\frac {c^{2} \left (-3 a d +7 b c \right )}{6 a^{2} b \,x^{\frac {3}{2}}}+\frac {\left (-a d +b c \right ) \left (d \,x^{2}+c \right )^{2}}{2 a b \,x^{\frac {3}{2}} \left (b \,x^{2}+a \right )}+\frac {\left (-a d +b c \right )^{2} \left (5 a d +7 b c \right ) \arctan \left (1-\frac {b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{a^{\frac {1}{4}}}\right ) \sqrt {2}}{8 a^{\frac {11}{4}} b^{\frac {9}{4}}}-\frac {\left (-a d +b c \right )^{2} \left (5 a d +7 b c \right ) \arctan \left (1+\frac {b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{a^{\frac {1}{4}}}\right ) \sqrt {2}}{8 a^{\frac {11}{4}} b^{\frac {9}{4}}}+\frac {\left (-a d +b c \right )^{2} \left (5 a d +7 b c \right ) \ln \left (\sqrt {a}+x \sqrt {b}-a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{16 a^{\frac {11}{4}} b^{\frac {9}{4}}}-\frac {\left (-a d +b c \right )^{2} \left (5 a d +7 b c \right ) \ln \left (\sqrt {a}+x \sqrt {b}+a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{16 a^{\frac {11}{4}} b^{\frac {9}{4}}}-\frac {d^{2} \left (-5 a d +b c \right ) \sqrt {x}}{2 a \,b^{2}} \]
command
integrate((d*x**2+c)**3/x**(5/2)/(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 \[ \text {Timed out} \]_____________________________________________________