6.9 Problem number 2991

\[ \int \frac {1}{\sqrt {a+b \sqrt {\frac {c}{x}}} x^2} \, dx \]

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

command

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

Giac 1.9.0-11 via sagemath 9.6 output

\[ \text {Exception raised: TypeError} \]

Giac 1.7.0 via sagemath 9.3 output

\[ -\frac {4 \, {\left ({\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {3}{2}} b - 3 \, \sqrt {b \sqrt {\frac {c}{x}} + a} a b\right )} \mathrm {sgn}\left ({\left (b \sqrt {\frac {c}{x}} + a\right )} b - a b\right )}{3 \, b^{3} c} \]