\[ \int \frac {\sqrt {a+b x} (A+B x)}{\sqrt {d+e x}} \, dx \]
Optimal antiderivative \[ \frac {\left (-a e +b d \right ) \left (-4 A b e +B a e +3 B b d \right ) \arctanh \left (\frac {\sqrt {e}\, \sqrt {b x +a}}{\sqrt {b}\, \sqrt {e x +d}}\right )}{4 b^{\frac {3}{2}} e^{\frac {5}{2}}}+\frac {B \left (b x +a \right )^{\frac {3}{2}} \sqrt {e x +d}}{2 b e}-\frac {\left (-4 A b e +B a e +3 B b d \right ) \sqrt {b x +a}\, \sqrt {e x +d}}{4 b \,e^{2}} \]
command
integrate((B*x+A)*(b*x+a)**(1/2)/(e*x+d)**(1/2),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 \[ \text {Timed out} \]_____________________________________________________