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