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