\[ \int \frac {1}{\sqrt {a+b \sqrt {\frac {c}{x}}} x^3} \, dx \]
Optimal antiderivative \[ -\frac {4 a^{2} \left (a +b \sqrt {\frac {c}{x}}\right )^{\frac {3}{2}}}{b^{4} c^{2}}+\frac {12 a \left (a +b \sqrt {\frac {c}{x}}\right )^{\frac {5}{2}}}{5 b^{4} c^{2}}-\frac {4 \left (a +b \sqrt {\frac {c}{x}}\right )^{\frac {7}{2}}}{7 b^{4} c^{2}}+\frac {4 a^{3} \sqrt {a +b \sqrt {\frac {c}{x}}}}{b^{4} c^{2}} \]
command
integrate(1/x^3/(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
\[ -\frac {4 \, {\left (5 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {7}{2}} \mathrm {sgn}\left ({\left (b \sqrt {\frac {c}{x}} + a\right )} b - a b\right ) - 21 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {5}{2}} a \mathrm {sgn}\left ({\left (b \sqrt {\frac {c}{x}} + a\right )} b - a b\right ) + 35 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {3}{2}} a^{2} \mathrm {sgn}\left ({\left (b \sqrt {\frac {c}{x}} + a\right )} b - a b\right ) - 35 \, \sqrt {b \sqrt {\frac {c}{x}} + a} a^{3} \mathrm {sgn}\left ({\left (b \sqrt {\frac {c}{x}} + a\right )} b - a b\right )\right )}}{35 \, b^{4} c^{2}} \]