\[ \int \frac {(a+b x) \sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx \]
Optimal antiderivative \[ \frac {e^{2} \arctanh \left (\frac {\sqrt {b}\, \sqrt {e x +d}}{\sqrt {-a e +b d}}\right )}{4 b^{\frac {3}{2}} \left (-a e +b d \right )^{\frac {3}{2}}}-\frac {\sqrt {e x +d}}{2 b \left (b x +a \right )^{2}}-\frac {e \sqrt {e x +d}}{4 b \left (-a e +b d \right ) \left (b x +a \right )} \]
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 \[ \text {Timed out} \]_____________________________________________________