\[ \int \frac {\left (c+d x^2\right )^3}{\sqrt {x} \left (a+b x^2\right )^2} \, dx \]
Optimal antiderivative \[ \frac {2 d^{3} x^{\frac {5}{2}}}{5 b^{2}}-\frac {3 \left (-a d +b c \right )^{2} \left (3 a d +b c \right ) \arctan \left (1-\frac {b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{a^{\frac {1}{4}}}\right ) \sqrt {2}}{8 a^{\frac {7}{4}} b^{\frac {13}{4}}}+\frac {3 \left (-a d +b c \right )^{2} \left (3 a d +b c \right ) \arctan \left (1+\frac {b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{a^{\frac {1}{4}}}\right ) \sqrt {2}}{8 a^{\frac {7}{4}} b^{\frac {13}{4}}}-\frac {3 \left (-a d +b c \right )^{2} \left (3 a d +b c \right ) \ln \left (\sqrt {a}+x \sqrt {b}-a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{16 a^{\frac {7}{4}} b^{\frac {13}{4}}}+\frac {3 \left (-a d +b c \right )^{2} \left (3 a d +b c \right ) \ln \left (\sqrt {a}+x \sqrt {b}+a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{16 a^{\frac {7}{4}} b^{\frac {13}{4}}}+\frac {2 d^{2} \left (-2 a d +3 b c \right ) \sqrt {x}}{b^{3}}+\frac {\left (-a d +b c \right )^{3} \sqrt {x}}{2 a \,b^{3} \left (b \,x^{2}+a \right )} \]
command
integrate((d*x**2+c)**3/(b*x**2+a)**2/x**(1/2),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 \[ \text {Timed out} \]_____________________________________________________