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