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