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