\[ \int \frac {A+B x}{\sqrt {x} \left (b x+c x^2\right )^3} \, dx \]
Optimal antiderivative \[ \frac {-\frac {63 A c}{20}+\frac {7 b B}{4}}{b^{3} c \,x^{\frac {5}{2}}}-\frac {7 \left (-9 A c +5 b B \right )}{12 b^{4} x^{\frac {3}{2}}}+\frac {A c -b B}{2 b c \,x^{\frac {5}{2}} \left (c x +b \right )^{2}}+\frac {9 A c -5 b B}{4 b^{2} c \,x^{\frac {5}{2}} \left (c x +b \right )}+\frac {7 c^{\frac {3}{2}} \left (-9 A c +5 b B \right ) \arctan \left (\frac {\sqrt {c}\, \sqrt {x}}{\sqrt {b}}\right )}{4 b^{\frac {11}{2}}}+\frac {7 c \left (-9 A c +5 b B \right )}{4 b^{5} \sqrt {x}} \]
command
integrate((B*x+A)/(c*x**2+b*x)**3/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 \[ \text {Timed out} \]_____________________________________________________