\[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (b x+c x^2\right )^3} \, dx \]
Optimal antiderivative \[ -\frac {\left (48 A \,c^{2} d^{2}+b^{2} e \left (-A e +4 B d \right )-12 b c d \left (A e +2 B d \right )\right ) \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{4 b^{5} d^{\frac {3}{2}}}+\frac {\left (48 A \,c^{3} d^{2}-15 b^{3} B \,e^{2}-12 b \,c^{2} d \left (7 A e +2 B d \right )+5 b^{2} c e \left (7 A e +8 B d \right )\right ) \arctanh \left (\frac {\sqrt {c}\, \sqrt {e x +d}}{\sqrt {-b e +c d}}\right ) \sqrt {c}}{4 b^{5} \left (-b e +c d \right )^{\frac {3}{2}}}-\frac {\left (A b -\left (-2 A c +b B \right ) x \right ) \sqrt {e x +d}}{2 b^{2} \left (c \,x^{2}+b x \right )^{2}}-\frac {\left (b \left (-b e +c d \right ) \left (A b e -12 A c d +6 B b d \right )-c \left (24 A \,c^{2} d^{2}+b^{2} e \left (A e +11 B d \right )-12 b c d \left (2 A e +B d \right )\right ) x \right ) \sqrt {e x +d}}{4 b^{4} d \left (-b e +c d \right ) \left (c \,x^{2}+b x \right )} \]
command
integrate((B*x+A)*(e*x+d)**(1/2)/(c*x**2+b*x)**3,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 \[ \text {Timed out} \]_____________________________________________________