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