\[ \int \frac {A+B x}{\sqrt {x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx \]
Optimal antiderivative \[ \frac {7 \left (9 A b +B a \right ) \arctan \left (\frac {\sqrt {b}\, \sqrt {x}}{\sqrt {a}}\right )}{128 a^{\frac {11}{2}} b^{\frac {3}{2}}}+\frac {\left (A b -B a \right ) \sqrt {x}}{5 a b \left (b x +a \right )^{5}}+\frac {\left (9 A b +B a \right ) \sqrt {x}}{40 a^{2} b \left (b x +a \right )^{4}}+\frac {7 \left (9 A b +B a \right ) \sqrt {x}}{240 a^{3} b \left (b x +a \right )^{3}}+\frac {7 \left (9 A b +B a \right ) \sqrt {x}}{192 a^{4} b \left (b x +a \right )^{2}}+\frac {7 \left (9 A b +B a \right ) \sqrt {x}}{128 a^{5} b \left (b x +a \right )} \]
command
integrate((B*x+A)/(b**2*x**2+2*a*b*x+a**2)**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 \[ \text {Timed out} \]_____________________________________________________