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