\[ \int \frac {x}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \, dx \]
Optimal antiderivative \[ \frac {\left (35 b^{4} d^{2}-120 a \,b^{2} c d +48 a^{2} c^{2}\right ) \arctanh \left (\frac {2 a +b \sqrt {\frac {d}{x}}}{2 \sqrt {a}\, \sqrt {a +\frac {c}{x}+b \sqrt {\frac {d}{x}}}}\right )}{64 a^{\frac {9}{2}}}-\frac {7 b \,d^{2} \sqrt {a +\frac {c}{x}+b \sqrt {\frac {d}{x}}}}{12 a^{2} \left (\frac {d}{x}\right )^{\frac {3}{2}}}-\frac {\left (-35 b^{2} d +36 a c \right ) x \sqrt {a +\frac {c}{x}+b \sqrt {\frac {d}{x}}}}{48 a^{3}}+\frac {x^{2} \sqrt {a +\frac {c}{x}+b \sqrt {\frac {d}{x}}}}{2 a}+\frac {5 b d \left (-21 b^{2} d +44 a c \right ) \sqrt {a +\frac {c}{x}+b \sqrt {\frac {d}{x}}}}{96 a^{4} \sqrt {\frac {d}{x}}} \]
command
integrate(x/(a+c/x+b*(d/x)^(1/2))^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {2 \, \sqrt {a d^{2} x + \sqrt {d x} b d^{2} + c d^{2}} {\left (2 \, \sqrt {d x} {\left (4 \, \sqrt {d x} {\left (\frac {7 \, b}{a^{2}} - \frac {6 \, \sqrt {d x}}{a d}\right )} - \frac {35 \, a b^{2} d^{2} - 36 \, a^{2} c d}{a^{4} d}\right )} + \frac {5 \, {\left (21 \, b^{3} d^{3} - 44 \, a b c d^{2}\right )}}{a^{4} d}\right )} + \frac {3 \, {\left (35 \, b^{4} d^{4} - 120 \, a b^{2} c d^{3} + 48 \, a^{2} c^{2} d^{2}\right )} \log \left ({\left | -b d^{2} - 2 \, \sqrt {a d} {\left (\sqrt {a d} \sqrt {d x} - \sqrt {a d^{2} x + \sqrt {d x} b d^{2} + c d^{2}}\right )} \right |}\right )}{\sqrt {a d} a^{4}} - \frac {105 \, b^{4} d^{4} \log \left ({\left | -b d^{2} + 2 \, \sqrt {c d^{2}} \sqrt {a d} \right |}\right ) - 360 \, a b^{2} c d^{3} \log \left ({\left | -b d^{2} + 2 \, \sqrt {c d^{2}} \sqrt {a d} \right |}\right ) + 144 \, a^{2} c^{2} d^{2} \log \left ({\left | -b d^{2} + 2 \, \sqrt {c d^{2}} \sqrt {a d} \right |}\right ) + 210 \, \sqrt {c d^{2}} \sqrt {a d} b^{3} d^{2} - 440 \, \sqrt {c d^{2}} \sqrt {a d} a b c d}{\sqrt {a d} a^{4}}}{192 \, d^{\frac {3}{2}} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________