\[ \int \frac {\left (c x^n\right )^{\frac {1}{n}}}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^5} \, dx \]
Optimal antiderivative \[ \frac {a x \left (c \,x^{n}\right )^{-\frac {1}{n}}}{4 b^{2} \left (a +b \left (c \,x^{n}\right )^{\frac {1}{n}}\right )^{4}}-\frac {x \left (c \,x^{n}\right )^{-\frac {1}{n}}}{3 b^{2} \left (a +b \left (c \,x^{n}\right )^{\frac {1}{n}}\right )^{3}} \]
command
integrate((c*x**n)**(1/n)/(a+b*(c*x**n)**(1/n))**5,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \frac {6 a^{2} x \left (c x^{n}\right )^{\frac {1}{n}}}{12 a^{7} + 48 a^{6} b \left (c x^{n}\right )^{\frac {1}{n}} + 72 a^{5} b^{2} \left (c x^{n}\right )^{\frac {2}{n}} + 48 a^{4} b^{3} \left (c x^{n}\right )^{\frac {3}{n}} + 12 a^{3} b^{4} \left (c x^{n}\right )^{\frac {4}{n}}} + \frac {4 a b x \left (c x^{n}\right )^{\frac {2}{n}}}{12 a^{7} + 48 a^{6} b \left (c x^{n}\right )^{\frac {1}{n}} + 72 a^{5} b^{2} \left (c x^{n}\right )^{\frac {2}{n}} + 48 a^{4} b^{3} \left (c x^{n}\right )^{\frac {3}{n}} + 12 a^{3} b^{4} \left (c x^{n}\right )^{\frac {4}{n}}} + \frac {b^{2} x \left (c x^{n}\right )^{\frac {3}{n}}}{12 a^{7} + 48 a^{6} b \left (c x^{n}\right )^{\frac {1}{n}} + 72 a^{5} b^{2} \left (c x^{n}\right )^{\frac {2}{n}} + 48 a^{4} b^{3} \left (c x^{n}\right )^{\frac {3}{n}} + 12 a^{3} b^{4} \left (c x^{n}\right )^{\frac {4}{n}}} & \text {for}\: a \neq 0 \\- \frac {x \left (c x^{n}\right )^{- \frac {4}{n}}}{3 b^{5}} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________