\[ \int \frac {(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx \]
Optimal antiderivative \[ \frac {e \arctanh \! \left (\frac {\sqrt {c}\, \sqrt {d}\, \sqrt {e x +d}}{\sqrt {-a \,e^{2}+c \,d^{2}}}\right )}{\left (-a \,e^{2}+c \,d^{2}\right )^{\frac {3}{2}} \sqrt {c}\, \sqrt {d}}-\frac {\sqrt {e x +d}}{\left (-a \,e^{2}+c \,d^{2}\right ) \left (c d x +a e \right )} \]
command
integrate((e*x+d)**(3/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ - \frac {e \sqrt {- \frac {1}{c d \left (a e^{2} - c d^{2}\right )^{3}}} \log {\left (- a^{2} e^{4} \sqrt {- \frac {1}{c d \left (a e^{2} - c d^{2}\right )^{3}}} + 2 a c d^{2} e^{2} \sqrt {- \frac {1}{c d \left (a e^{2} - c d^{2}\right )^{3}}} - c^{2} d^{4} \sqrt {- \frac {1}{c d \left (a e^{2} - c d^{2}\right )^{3}}} + \sqrt {d + e x} \right )}}{2} + \frac {e \sqrt {- \frac {1}{c d \left (a e^{2} - c d^{2}\right )^{3}}} \log {\left (a^{2} e^{4} \sqrt {- \frac {1}{c d \left (a e^{2} - c d^{2}\right )^{3}}} - 2 a c d^{2} e^{2} \sqrt {- \frac {1}{c d \left (a e^{2} - c d^{2}\right )^{3}}} + c^{2} d^{4} \sqrt {- \frac {1}{c d \left (a e^{2} - c d^{2}\right )^{3}}} + \sqrt {d + e x} \right )}}{2} + \frac {2 e \sqrt {d + e x}}{2 a^{2} e^{4} - 2 a c d^{2} e^{2} + 2 a c d e^{3} x - 2 c^{2} d^{3} e x} \]