\[ \int \left (d+e x^n\right ) \left (a+c x^{2 n}\right )^p \, dx \]
Optimal antiderivative \[ d x \left (a +c \,x^{2 n}\right )^{p} \hypergeom \left (\left [-p , \frac {1}{2 n}\right ], \left [1+\frac {1}{2 n}\right ], -\frac {c \,x^{2 n}}{a}\right ) \left (1+\frac {c \,x^{2 n}}{a}\right )^{-p}+\frac {e \,x^{1+n} \left (a +c \,x^{2 n}\right )^{p} \hypergeom \left (\left [-p , \frac {1+n}{2 n}\right ], \left [\frac {3}{2}+\frac {1}{2 n}\right ], -\frac {c \,x^{2 n}}{a}\right ) \left (1+\frac {c \,x^{2 n}}{a}\right )^{-p}}{1+n} \]
command
integrate((d+e*x**n)*(a+c*x**(2*n))**p,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \frac {a^{p} d x \Gamma \left (\frac {1}{2 n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2 n}, - p \\ 1 + \frac {1}{2 n} \end {matrix}\middle | {\frac {c x^{2 n} e^{i \pi }}{a}} \right )}}{2 n \Gamma \left (1 + \frac {1}{2 n}\right )} + \frac {a^{p} e x x^{n} \Gamma \left (\frac {1}{2} + \frac {1}{2 n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {1}{2} + \frac {1}{2 n} \\ \frac {3}{2} + \frac {1}{2 n} \end {matrix}\middle | {\frac {c x^{2 n} e^{i \pi }}{a}} \right )}}{2 n \Gamma \left (\frac {3}{2} + \frac {1}{2 n}\right )} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________