\[ \int \frac {(c+d x)^{5/2}}{x^3 (a+b x)} \, dx \]
Optimal antiderivative \[ -\frac {c \left (d x +c \right )^{\frac {3}{2}}}{2 a \,x^{2}}+\frac {2 \left (-a d +b c \right )^{\frac {5}{2}} \arctanh \left (\frac {\sqrt {b}\, \sqrt {d x +c}}{\sqrt {-a d +b c}}\right )}{a^{3} \sqrt {b}}-\frac {\left (15 a^{2} d^{2}-20 a b c d +8 b^{2} c^{2}\right ) \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right ) \sqrt {c}}{4 a^{3}}+\frac {c \left (-7 a d +4 b c \right ) \sqrt {d x +c}}{4 a^{2} x} \]
command
integrate((d*x+c)**(5/2)/x**3/(b*x+a),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ - \frac {10 c^{4} d^{2} \sqrt {c + d x}}{- 8 a c^{4} - 16 a c^{3} d x + 8 a c^{2} \left (c + d x\right )^{2}} + \frac {6 c^{3} d^{2} \left (c + d x\right )^{\frac {3}{2}}}{- 8 a c^{4} - 16 a c^{3} d x + 8 a c^{2} \left (c + d x\right )^{2}} + \frac {2 d^{3} \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d}{b} - c}} \right )}}{b \sqrt {\frac {a d}{b} - c}} + \frac {3 c^{3} d^{2} \sqrt {\frac {1}{c^{5}}} \log {\left (- c^{3} \sqrt {\frac {1}{c^{5}}} + \sqrt {c + d x} \right )}}{8 a} - \frac {3 c^{3} d^{2} \sqrt {\frac {1}{c^{5}}} \log {\left (c^{3} \sqrt {\frac {1}{c^{5}}} + \sqrt {c + d x} \right )}}{8 a} - \frac {3 c^{2} d^{2} \sqrt {\frac {1}{c^{3}}} \log {\left (- c^{2} \sqrt {\frac {1}{c^{3}}} + \sqrt {c + d x} \right )}}{2 a} + \frac {3 c^{2} d^{2} \sqrt {\frac {1}{c^{3}}} \log {\left (c^{2} \sqrt {\frac {1}{c^{3}}} + \sqrt {c + d x} \right )}}{2 a} - \frac {6 c d^{2} \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d}{b} - c}} \right )}}{a \sqrt {\frac {a d}{b} - c}} + \frac {6 c d^{2} \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {- c}} \right )}}{a \sqrt {- c}} - \frac {3 c d \sqrt {c + d x}}{a x} + \frac {b c^{3} d \sqrt {\frac {1}{c^{3}}} \log {\left (- c^{2} \sqrt {\frac {1}{c^{3}}} + \sqrt {c + d x} \right )}}{2 a^{2}} - \frac {b c^{3} d \sqrt {\frac {1}{c^{3}}} \log {\left (c^{2} \sqrt {\frac {1}{c^{3}}} + \sqrt {c + d x} \right )}}{2 a^{2}} + \frac {6 b c^{2} 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 {6 b c^{2} d \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {- c}} \right )}}{a^{2} \sqrt {- c}} + \frac {b c^{2} \sqrt {c + d x}}{a^{2} x} - \frac {2 b^{2} c^{3} \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d}{b} - c}} \right )}}{a^{3} \sqrt {\frac {a d}{b} - c}} + \frac {2 b^{2} c^{3} \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {- c}} \right )}}{a^{3} \sqrt {- c}} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________