\[ \int \frac {x^{3/2} (A+B x)}{\left (b x+c x^2\right )^3} \, dx \]
Optimal antiderivative \[ \frac {3 \left (-5 A c +b B \right ) \arctan \left (\frac {\sqrt {c}\, \sqrt {x}}{\sqrt {b}}\right )}{4 b^{\frac {7}{2}} \sqrt {c}}+\frac {-\frac {15 A c}{4}+\frac {3 b B}{4}}{b^{3} c \sqrt {x}}+\frac {A c -b B}{2 b c \left (c x +b \right )^{2} \sqrt {x}}+\frac {5 A c -b B}{4 b^{2} c \left (c x +b \right ) \sqrt {x}} \]
command
integrate(x**(3/2)*(B*x+A)/(c*x**2+b*x)**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 \[ \text {Timed out} \]_____________________________________________________