6.2 Problem number 1738

\[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^4} \, dx \]

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

command

integrate(1/(a+b/x)^(3/2)/x^4,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

\[ \frac {2 \, {\left (\frac {3 \, a^{2}}{\sqrt {\frac {a x + b}{x}}} + 6 \, a \sqrt {\frac {a x + b}{x}} - \frac {{\left (a x + b\right )} \sqrt {\frac {a x + b}{x}}}{x}\right )}}{3 \, b^{3}} \]