\[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx \]
Optimal antiderivative \[ -\frac {\left (A b -B a \right ) \left (e x +d \right )^{\frac {3}{2}}}{3 b \left (-a e +b d \right ) \left (b x +a \right )^{3}}+\frac {e^{2} \left (-A b e -B a e +2 B b d \right ) \arctanh \left (\frac {\sqrt {b}\, \sqrt {e x +d}}{\sqrt {-a e +b d}}\right )}{8 b^{\frac {5}{2}} \left (-a e +b d \right )^{\frac {5}{2}}}-\frac {\left (-A b e -B a e +2 B b d \right ) \sqrt {e x +d}}{4 b^{2} \left (-a e +b d \right ) \left (b x +a \right )^{2}}-\frac {e \left (-A b e -B a e +2 B b d \right ) \sqrt {e x +d}}{8 b^{2} \left (-a e +b d \right )^{2} \left (b x +a \right )} \]
command
integrate((B*x+A)*(e*x+d)**(1/2)/(b**2*x**2+2*a*b*x+a**2)**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 \[ \text {Timed out} \]_____________________________________________________