\[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^5} \, dx \]
Optimal antiderivative \[ \frac {\left (d \left (3 A b \,e^{2}-B d \left (-5 b e +8 c d \right )\right )-e \left (B d \left (-11 b e +14 c d \right )-3 A e \left (-b e +2 c d \right )\right ) x \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{24 d \,e^{2} \left (-b e +c d \right ) \left (e x +d \right )^{4}}+\frac {2 B \,c^{\frac {3}{2}} \arctanh \! \left (\frac {x \sqrt {c}}{\sqrt {c \,x^{2}+b x}}\right )}{e^{5}}+\frac {\left (3 A \,b^{4} e^{5}-B d \left (-5 b^{4} e^{4}-40 b^{3} c d \,e^{3}+240 b^{2} c^{2} d^{2} e^{2}-320 b \,c^{3} d^{3} e +128 c^{4} d^{4}\right )\right ) \arctanh \! \left (\frac {b d +\left (-b e +2 c d \right ) x}{2 \sqrt {d}\, \sqrt {-b e +c d}\, \sqrt {c \,x^{2}+b x}}\right )}{128 d^{\frac {5}{2}} e^{5} \left (-b e +c d \right )^{\frac {5}{2}}}-\frac {\left (d \left (3 A \,b^{3} e^{4}+B d \left (5 b^{3} e^{3}+40 b^{2} c d \,e^{2}-112 b \,c^{2} d^{2} e +64 c^{3} d^{3}\right )\right )+e \left (3 A \,b^{2} e^{3} \left (-b e +2 c d \right )+B d \left (-5 b^{3} e^{3}+98 b^{2} c d \,e^{2}-192 b \,c^{2} d^{2} e +96 c^{3} d^{3}\right )\right ) x \right ) \sqrt {c \,x^{2}+b x}}{64 d^{2} e^{4} \left (-b e +c d \right )^{2} \left (e x +d \right )^{2}} \]
command
integrate((B*x+A)*(c*x^2+b*x)^(3/2)/(e*x+d)^5,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {Exception raised: TypeError} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {output too large to display} \]