\[ \int \frac {1}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^3} \, dx \]
Optimal antiderivative \[ -\frac {x \left (c \,x^{n}\right )^{-\frac {1}{n}}}{2 b \left (a +b \left (c \,x^{n}\right )^{\frac {1}{n}}\right )^{2}} \]
command
integrate(1/(a+b*(c*x**n)**(1/n))**3,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \frac {2 a x}{2 a^{4} + 4 a^{3} b \left (c x^{n}\right )^{\frac {1}{n}} + 2 a^{2} b^{2} \left (c x^{n}\right )^{\frac {2}{n}}} + \frac {b x \left (c x^{n}\right )^{\frac {1}{n}}}{2 a^{4} + 4 a^{3} b \left (c x^{n}\right )^{\frac {1}{n}} + 2 a^{2} b^{2} \left (c x^{n}\right )^{\frac {2}{n}}} & \text {for}\: a \neq 0 \\- \frac {x \left (c x^{n}\right )^{- \frac {3}{n}}}{2 b^{3}} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________