\[ \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (b c d -b^{2} e +2 a c e +c \left (-b e +2 c d \right ) x \right )}{3 \left (-4 a c +b^{2}\right ) \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (e x +d \right )^{2} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {5 e^{4} \left (24 c^{2} d^{2}+7 b^{2} e^{2}-4 c e \left (a e +6 b d \right )\right ) \arctanh \left (\frac {b d -2 a e +\left (-b e +2 c d \right ) x}{2 \sqrt {a \,e^{2}-b d e +c \,d^{2}}\, \sqrt {c \,x^{2}+b x +a}}\right )}{8 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{\frac {9}{2}}}-\frac {2 \left (8 a c e \left (-b e +2 c d \right )^{2}-\left (2 a c e -b^{2} e +b c d \right ) \left (20 a c \,e^{2}-7 b^{2} e^{2}+8 c^{2} d^{2}\right )-c \left (-b e +2 c d \right ) \left (8 c^{2} d^{2}-7 b^{2} e^{2}-4 c e \left (-9 a e +2 b d \right )\right ) x \right )}{3 \left (-4 a c +b^{2}\right )^{2} \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (e x +d \right )^{2} \sqrt {c \,x^{2}+b x +a}}+\frac {e \left (64 c^{4} d^{4}-35 b^{4} e^{4}-128 c^{3} d^{2} e \left (-3 a e +b d \right )-48 a \,c^{2} e^{3} \left (5 a e +8 b d \right )+8 b^{2} c \,e^{3} \left (27 a e +8 b d \right )\right ) \sqrt {c \,x^{2}+b x +a}}{6 \left (-4 a c +b^{2}\right )^{2} \left (a \,e^{2}-b d e +c \,d^{2}\right )^{3} \left (e x +d \right )^{2}}+\frac {e \left (-b e +2 c d \right ) \left (64 c^{4} d^{4}-105 b^{4} e^{4}-64 c^{3} d^{2} e \left (-7 a e +2 b d \right )+40 b^{2} c \,e^{3} \left (19 a e +2 b d \right )-16 c^{2} e^{2} \left (81 a^{2} e^{2}+28 a b d e +b^{2} d^{2}\right )\right ) \sqrt {c \,x^{2}+b x +a}}{12 \left (-4 a c +b^{2}\right )^{2} \left (a \,e^{2}-b d e +c \,d^{2}\right )^{4} \left (e x +d \right )} \]
command
integrate(1/(e*x+d)^3/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \text {Timed out} \]_______________________________________________________