\[ \int \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2 \, dx \]
Optimal antiderivative \[ \frac {\left (-b^{2} d +4 a c \right ) \left (21 b^{4} d^{2}-56 a \,b^{2} c d +16 a^{2} c^{2}\right ) \arctanh \left (\frac {2 a +b \sqrt {\frac {d}{x}}}{2 \sqrt {a}\, \sqrt {a +\frac {c}{x}+b \sqrt {\frac {d}{x}}}}\right )}{512 a^{\frac {11}{2}}}-\frac {3 b \,d^{3} \left (a +\frac {c}{x}+b \sqrt {\frac {d}{x}}\right )^{\frac {3}{2}}}{10 a^{2} \left (\frac {d}{x}\right )^{\frac {5}{2}}}+\frac {7 b \,d^{2} \left (-15 b^{2} d +28 a c \right ) \left (a +\frac {c}{x}+b \sqrt {\frac {d}{x}}\right )^{\frac {3}{2}}}{480 a^{4} \left (\frac {d}{x}\right )^{\frac {3}{2}}}-\frac {\left (-21 b^{2} d +20 a c \right ) x^{2} \left (a +\frac {c}{x}+b \sqrt {\frac {d}{x}}\right )^{\frac {3}{2}}}{80 a^{3}}+\frac {x^{3} \left (a +\frac {c}{x}+b \sqrt {\frac {d}{x}}\right )^{\frac {3}{2}}}{3 a}+\frac {\left (21 b^{4} d^{2}-56 a \,b^{2} c d +16 a^{2} c^{2}\right ) x \left (2 a +b \sqrt {\frac {d}{x}}\right ) \sqrt {a +\frac {c}{x}+b \sqrt {\frac {d}{x}}}}{256 a^{5}} \]
command
Integrate[Sqrt[a + b*Sqrt[d/x] + c/x]*x^2,x]
Mathematica 13.1 output
\[ \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} \left (\sqrt {a} x \left (-210 a b^3 d \left (b d+8 c \sqrt {\frac {d}{x}}\right )+315 b^5 d \left (\frac {d}{x}\right )^{3/2} x+1280 a^5 x^2+64 a^4 x \left (5 c+2 b \sqrt {\frac {d}{x}} x\right )-16 a^3 \left (30 c^2+9 b^2 d x+34 b c \sqrt {\frac {d}{x}} x\right )+8 a^2 b \left (112 b c d+226 c^2 \sqrt {\frac {d}{x}}+21 b^2 d \sqrt {\frac {d}{x}} x\right )\right )+\frac {15 \sqrt {d} \left (-64 a^3 c^3+240 a^2 b^2 c^2 d-140 a b^4 c d^2+21 b^6 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {d}{x}}-\sqrt {\frac {d \left (c+a x+b \sqrt {\frac {d}{x}} x\right )}{x}}}{\sqrt {a} \sqrt {d}}\right )}{\sqrt {\frac {d \left (c+\left (a+b \sqrt {\frac {d}{x}}\right ) x\right )}{x}}}\right )}{3840 a^{11/2}} \]
Mathematica 12.3 output
\[ \int \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2 \, dx \]________________________________________________________________________________________