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