\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{7/2}} \, dx \]
Optimal antiderivative \[ \frac {2 c d \left (-a \,e^{2}+c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {3}{2}}}{e^{4}}-\frac {6 c^{2} d^{2} \left (-a \,e^{2}+c \,d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5 e^{4}}+\frac {2 c^{3} d^{3} \left (e x +d \right )^{\frac {7}{2}}}{7 e^{4}}-\frac {2 \left (-a \,e^{2}+c \,d^{2}\right )^{3} \sqrt {e x +d}}{e^{4}} \]
command
integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3/(e*x+d)**(7/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \frac {- \frac {2 a^{3} d e^{3}}{\sqrt {d + e x}} - 2 a^{3} e^{3} \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right ) - 6 a^{2} c d^{2} e \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right ) - 6 a^{2} c d e \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right ) - \frac {6 a c^{2} d^{3} \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{e} - \frac {6 a c^{2} d^{2} \left (- \frac {d^{3}}{\sqrt {d + e x}} - 3 d^{2} \sqrt {d + e x} + d \left (d + e x\right )^{\frac {3}{2}} - \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e} - \frac {2 c^{3} d^{4} \left (- \frac {d^{3}}{\sqrt {d + e x}} - 3 d^{2} \sqrt {d + e x} + d \left (d + e x\right )^{\frac {3}{2}} - \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{3}} - \frac {2 c^{3} d^{3} \left (\frac {d^{4}}{\sqrt {d + e x}} + 4 d^{3} \sqrt {d + e x} - 2 d^{2} \left (d + e x\right )^{\frac {3}{2}} + \frac {4 d \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{3}}}{e} & \text {for}\: e \neq 0 \\\frac {c^{3} d^{\frac {5}{2}} x^{4}}{4} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________