\[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx \]
Optimal antiderivative \[ \frac {\left (A b -B a \right ) x^{\frac {5}{2}}}{4 a b \left (b \,x^{2}+a \right )^{2}}-\frac {\left (3 A b +5 B a \right ) \arctan \left (1-\frac {b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{a^{\frac {1}{4}}}\right ) \sqrt {2}}{64 a^{\frac {7}{4}} b^{\frac {9}{4}}}+\frac {\left (3 A b +5 B a \right ) \arctan \left (1+\frac {b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{a^{\frac {1}{4}}}\right ) \sqrt {2}}{64 a^{\frac {7}{4}} b^{\frac {9}{4}}}-\frac {\left (3 A b +5 B a \right ) \ln \left (\sqrt {a}+x \sqrt {b}-a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{128 a^{\frac {7}{4}} b^{\frac {9}{4}}}+\frac {\left (3 A b +5 B a \right ) \ln \left (\sqrt {a}+x \sqrt {b}+a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{128 a^{\frac {7}{4}} b^{\frac {9}{4}}}-\frac {\left (3 A b +5 B a \right ) \sqrt {x}}{16 a \,b^{2} \left (b \,x^{2}+a \right )} \]
command
integrate(x**(3/2)*(B*x**2+A)/(b*x**2+a)**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 \[ \text {Timed out} \]_____________________________________________________