\[ \int (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx \]
Optimal antiderivative \[ \frac {256 \left (-a \,e^{2}+c \,d^{2}\right )^{4} \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {7}{2}}}{45045 c^{5} d^{5} \left (e x +d \right )^{\frac {7}{2}}}+\frac {128 \left (-a \,e^{2}+c \,d^{2}\right )^{3} \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {7}{2}}}{6435 c^{4} d^{4} \left (e x +d \right )^{\frac {5}{2}}}+\frac {32 \left (-a \,e^{2}+c \,d^{2}\right )^{2} \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {7}{2}}}{715 c^{3} d^{3} \left (e x +d \right )^{\frac {3}{2}}}+\frac {16 \left (-a \,e^{2}+c \,d^{2}\right ) \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {7}{2}}}{195 c^{2} d^{2} \sqrt {e x +d}}+\frac {2 \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {7}{2}} \sqrt {e x +d}}{15 c d} \]
command
integrate((e*x+d)^(3/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(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 \[ \text {Timed out} \]_______________________________________________________