\[ \int \frac {x^2}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \, dx \]
Optimal antiderivative \[ -\frac {\left (-231 b^{6} d^{3}+1260 a \,b^{4} c \,d^{2}-1680 a^{2} b^{2} c^{2} d +320 a^{3} c^{3}\right ) \arctanh \left (\frac {2 a +b \sqrt {\frac {d}{x}}}{2 \sqrt {a}\, \sqrt {a +\frac {c}{x}+b \sqrt {\frac {d}{x}}}}\right )}{512 a^{\frac {13}{2}}}-\frac {11 b \,d^{3} \sqrt {a +\frac {c}{x}+b \sqrt {\frac {d}{x}}}}{30 a^{2} \left (\frac {d}{x}\right )^{\frac {5}{2}}}+\frac {b \,d^{2} \left (-77 b^{2} d +156 a c \right ) \sqrt {a +\frac {c}{x}+b \sqrt {\frac {d}{x}}}}{160 a^{4} \left (\frac {d}{x}\right )^{\frac {3}{2}}}+\frac {\left (385 b^{4} d^{2}-1176 a \,b^{2} c d +400 a^{2} c^{2}\right ) x \sqrt {a +\frac {c}{x}+b \sqrt {\frac {d}{x}}}}{640 a^{5}}-\frac {\left (-99 b^{2} d +100 a c \right ) x^{2} \sqrt {a +\frac {c}{x}+b \sqrt {\frac {d}{x}}}}{240 a^{3}}+\frac {x^{3} \sqrt {a +\frac {c}{x}+b \sqrt {\frac {d}{x}}}}{3 a}-\frac {7 b d \left (165 b^{4} d^{2}-680 a \,b^{2} c d +528 a^{2} c^{2}\right ) \sqrt {a +\frac {c}{x}+b \sqrt {\frac {d}{x}}}}{1280 a^{6} \sqrt {\frac {d}{x}}} \]
command
integrate(x^2/(a+c/x+b*(d/x)^(1/2))^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {2 \, \sqrt {a d^{2} x + \sqrt {d x} b d^{2} + c d^{2}} {\left (2 \, \sqrt {d x} {\left (4 \, \sqrt {d x} {\left (2 \, \sqrt {d x} {\left (8 \, \sqrt {d x} {\left (\frac {11 \, b}{a^{2} d^{2}} - \frac {10 \, \sqrt {d x}}{a d^{3}}\right )} - \frac {99 \, a^{3} b^{2} d^{2} - 100 \, a^{4} c d}{a^{6} d^{3}}\right )} + \frac {3 \, {\left (77 \, a^{2} b^{3} d^{3} - 156 \, a^{3} b c d^{2}\right )}}{a^{6} d^{3}}\right )} - \frac {3 \, {\left (385 \, a b^{4} d^{4} - 1176 \, a^{2} b^{2} c d^{3} + 400 \, a^{3} c^{2} d^{2}\right )}}{a^{6} d^{3}}\right )} + \frac {21 \, {\left (165 \, b^{5} d^{5} - 680 \, a b^{3} c d^{4} + 528 \, a^{2} b c^{2} d^{3}\right )}}{a^{6} d^{3}}\right )} + \frac {15 \, {\left (231 \, b^{6} d^{4} - 1260 \, a b^{4} c d^{3} + 1680 \, a^{2} b^{2} c^{2} d^{2} - 320 \, a^{3} c^{3} d\right )} \log \left ({\left | -b d^{2} - 2 \, \sqrt {a d} {\left (\sqrt {a d} \sqrt {d x} - \sqrt {a d^{2} x + \sqrt {d x} b d^{2} + c d^{2}}\right )} \right |}\right )}{\sqrt {a d} a^{6}} - \frac {3 \, {\left (1155 \, b^{6} d^{4} \log \left ({\left | -b d^{2} + 2 \, \sqrt {c d^{2}} \sqrt {a d} \right |}\right ) - 6300 \, a b^{4} c d^{3} \log \left ({\left | -b d^{2} + 2 \, \sqrt {c d^{2}} \sqrt {a d} \right |}\right ) + 8400 \, a^{2} b^{2} c^{2} d^{2} \log \left ({\left | -b d^{2} + 2 \, \sqrt {c d^{2}} \sqrt {a d} \right |}\right ) + 2310 \, \sqrt {c d^{2}} \sqrt {a d} b^{5} d^{2} - 1600 \, a^{3} c^{3} d \log \left ({\left | -b d^{2} + 2 \, \sqrt {c d^{2}} \sqrt {a d} \right |}\right ) - 9520 \, \sqrt {c d^{2}} \sqrt {a d} a b^{3} c d + 7392 \, \sqrt {c d^{2}} \sqrt {a d} a^{2} b c^{2}\right )}}{\sqrt {a d} a^{6}}}{7680 \, \sqrt {d} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________