\[ \int \frac {\sqrt {x} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx \]
Optimal antiderivative \[ \frac {\left (A b -B a \right ) x^{\frac {3}{2}}}{5 a b \left (b x +a \right )^{5}}+\frac {\left (7 A b +3 B a \right ) \arctan \left (\frac {\sqrt {b}\, \sqrt {x}}{\sqrt {a}}\right )}{128 a^{\frac {9}{2}} b^{\frac {5}{2}}}-\frac {\left (7 A b +3 B a \right ) \sqrt {x}}{40 a \,b^{2} \left (b x +a \right )^{4}}+\frac {\left (7 A b +3 B a \right ) \sqrt {x}}{240 a^{2} b^{2} \left (b x +a \right )^{3}}+\frac {\left (7 A b +3 B a \right ) \sqrt {x}}{192 a^{3} b^{2} \left (b x +a \right )^{2}}+\frac {\left (7 A b +3 B a \right ) \sqrt {x}}{128 a^{4} b^{2} \left (b x +a \right )} \]
command
integrate((B*x+A)*x**(1/2)/(b**2*x**2+2*a*b*x+a**2)**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 \[ \text {Timed out} \]_____________________________________________________