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