\[ \int \frac {(d+e x)^{7/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx \]
Optimal antiderivative \[ \frac {2 \left (-a \,e^{2}+c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3 c^{2} d^{2}}+\frac {2 \left (e x +d \right )^{\frac {5}{2}}}{5 c d}-\frac {2 \left (-a \,e^{2}+c \,d^{2}\right )^{\frac {5}{2}} \arctanh \left (\frac {\sqrt {c}\, \sqrt {d}\, \sqrt {e x +d}}{\sqrt {-a \,e^{2}+c \,d^{2}}}\right )}{c^{\frac {7}{2}} d^{\frac {7}{2}}}+\frac {2 \left (-a \,e^{2}+c \,d^{2}\right )^{2} \sqrt {e x +d}}{c^{3} d^{3}} \]
command
integrate((e*x+d)**(7/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \frac {2 \left (d + e x\right )^{\frac {5}{2}}}{5 c d} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (- 2 a e^{2} + 2 c d^{2}\right )}{3 c^{2} d^{2}} + \frac {\sqrt {d + e x} \left (2 a^{2} e^{4} - 4 a c d^{2} e^{2} + 2 c^{2} d^{4}\right )}{c^{3} d^{3}} - \frac {2 \left (a e^{2} - c d^{2}\right )^{3} \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {a e^{2} - c d^{2}}{c d}}} \right )}}{c^{4} d^{4} \sqrt {\frac {a e^{2} - c d^{2}}{c d}}} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________