6.8 Problem number 1751

\[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^7} \, dx \]

Optimal antiderivative \[ -\frac {2 a^{5}}{3 b^{6} \left (a +\frac {b}{x}\right )^{\frac {3}{2}}}-\frac {20 a^{2} \left (a +\frac {b}{x}\right )^{\frac {3}{2}}}{3 b^{6}}+\frac {2 a \left (a +\frac {b}{x}\right )^{\frac {5}{2}}}{b^{6}}-\frac {2 \left (a +\frac {b}{x}\right )^{\frac {7}{2}}}{7 b^{6}}+\frac {10 a^{4}}{b^{6} \sqrt {a +\frac {b}{x}}}+\frac {20 a^{3} \sqrt {a +\frac {b}{x}}}{b^{6}} \]

command

integrate(1/(a+b/x)^(5/2)/x^7,x, algorithm="giac")

Giac 1.9.0-11 via sagemath 9.6 output

\[ \text {could not integrate} \]

Giac 1.7.0 via sagemath 9.3 output

\[ \frac {2 \, {\left (210 \, a^{3} \sqrt {\frac {a x + b}{x}} - \frac {70 \, {\left (a x + b\right )} a^{2} \sqrt {\frac {a x + b}{x}}}{x} + \frac {21 \, {\left (a x + b\right )}^{2} a \sqrt {\frac {a x + b}{x}}}{x^{2}} - \frac {7 \, {\left (a^{5} - \frac {15 \, {\left (a x + b\right )} a^{4}}{x}\right )} x}{{\left (a x + b\right )} \sqrt {\frac {a x + b}{x}}} - \frac {3 \, {\left (a x + b\right )}^{3} \sqrt {\frac {a x + b}{x}}}{x^{3}}\right )}}{21 \, b^{6}} \]