\[ \int \frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{x^{10}} \, dx \]
Optimal antiderivative \[ \frac {b^{2} c^{3} \arctanh \left (\frac {\sqrt {a +b \left (c \,x^{3}\right )^{\frac {3}{2}}}}{\sqrt {a}}\right )}{18 a^{\frac {3}{2}}}-\frac {\sqrt {a +b \left (c \,x^{3}\right )^{\frac {3}{2}}}}{9 x^{9}}-\frac {b \,c^{3} \sqrt {a +b \left (c \,x^{3}\right )^{\frac {3}{2}}}}{18 a \left (c \,x^{3}\right )^{\frac {3}{2}}} \]
command
Integrate[Sqrt[a + b*(c*x^3)^(3/2)]/x^10,x]
Mathematica 13.1 output
\[ -\frac {\sqrt {a+b \left (c x^3\right )^{3/2}} \left (2 a+b \left (c x^3\right )^{3/2}\right )}{18 a x^9}+\frac {b^2 c^3 \tanh ^{-1}\left (\frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{\sqrt {a}}\right )}{18 a^{3/2}} \]
Mathematica 12.3 output
\[ \int \frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{x^{10}} \, dx \]________________________________________________________________________________________