\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{\sqrt {d+e x}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (-a \,e^{2}+c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {7}{2}}}{7 e^{4}}+\frac {2 c d \left (-a \,e^{2}+c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {9}{2}}}{3 e^{4}}-\frac {6 c^{2} d^{2} \left (-a \,e^{2}+c \,d^{2}\right ) \left (e x +d \right )^{\frac {11}{2}}}{11 e^{4}}+\frac {2 c^{3} d^{3} \left (e x +d \right )^{\frac {13}{2}}}{13 e^{4}} \]
command
integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3/(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 \[ \text {Timed out} \]_____________________________________________________