\[ \int \frac {A+B x}{x^5 (a+b x)^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {35 b^{3} \left (9 A b -8 B a \right ) \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{64 a^{\frac {11}{2}}}+\frac {35 b^{3} \left (9 A b -8 B a \right )}{64 a^{5} \sqrt {b x +a}}-\frac {A}{4 a \,x^{4} \sqrt {b x +a}}+\frac {9 A b -8 B a}{24 a^{2} x^{3} \sqrt {b x +a}}-\frac {7 b \left (9 A b -8 B a \right )}{96 a^{3} x^{2} \sqrt {b x +a}}+\frac {35 b^{2} \left (9 A b -8 B a \right )}{192 a^{4} x \sqrt {b x +a}} \]
command
integrate((B*x+A)/x**5/(b*x+a)**(3/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ A \left (- \frac {1}{4 a \sqrt {b} x^{\frac {9}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {3 \sqrt {b}}{8 a^{2} x^{\frac {7}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {21 b^{\frac {3}{2}}}{32 a^{3} x^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {105 b^{\frac {5}{2}}}{64 a^{4} x^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {315 b^{\frac {7}{2}}}{64 a^{5} \sqrt {x} \sqrt {\frac {a}{b x} + 1}} - \frac {315 b^{4} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{64 a^{\frac {11}{2}}}\right ) + B \left (- \frac {1}{3 a \sqrt {b} x^{\frac {7}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {7 \sqrt {b}}{12 a^{2} x^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {35 b^{\frac {3}{2}}}{24 a^{3} x^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {35 b^{\frac {5}{2}}}{8 a^{4} \sqrt {x} \sqrt {\frac {a}{b x} + 1}} + \frac {35 b^{3} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{8 a^{\frac {9}{2}}}\right ) \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________