\[ \int \frac {x^2}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \, dx \]
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/Sqrt[a + b*Sqrt[d/x] + c/x],x]
Mathematica 13.1 output
\[ \frac {\sqrt {a} d \left (-3465 b^5 d^2 \left (b d+c \sqrt {\frac {d}{x}}\right )+1280 a^6 x^3-64 a^5 x^2 \left (5 c+2 b \sqrt {\frac {d}{x}} x\right )+16 a^4 x \left (50 c^2+11 b^2 d x+46 b c \sqrt {\frac {d}{x}} x\right )-105 a b^3 d \left (-158 b c d-136 c^2 \sqrt {\frac {d}{x}}+11 b^2 d \sqrt {\frac {d}{x}} x\right )+42 a^2 b \left (-432 b c^2 d-264 c^3 \sqrt {\frac {d}{x}}+11 b^3 d^2 x+128 b^2 c d \sqrt {\frac {d}{x}} x\right )+24 a^3 \left (100 c^3-72 b^2 c d x-206 b c^2 \sqrt {\frac {d}{x}} x-11 b^3 \left (\frac {d}{x}\right )^{3/2} x^3\right )\right )-15 \sqrt {d} \left (-320 a^3 c^3+1680 a^2 b^2 c^2 d-1260 a b^4 c d^2+231 b^6 d^3\right ) \sqrt {\frac {d \left (c+\left (a+b \sqrt {\frac {d}{x}}\right ) x\right )}{x}} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {d}{x}}-\sqrt {\frac {d \left (c+a x+b \sqrt {\frac {d}{x}} x\right )}{x}}}{\sqrt {a} \sqrt {d}}\right )}{3840 a^{13/2} d \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \]
Mathematica 12.3 output
\[ \int \frac {x^2}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \, dx \]________________________________________________________________________________________