\[ \int \frac {d+e x^n}{\left (a+c x^{2 n}\right )^2} \, dx \]
Optimal antiderivative \[ \frac {x \left (d +e \,x^{n}\right )}{2 a n \left (a +c \,x^{2 n}\right )}-\frac {d \left (1-2 n \right ) x \hypergeom \left (\left [1, \frac {1}{2 n}\right ], \left [1+\frac {1}{2 n}\right ], -\frac {c \,x^{2 n}}{a}\right )}{2 a^{2} n}-\frac {e \left (1-n \right ) x^{1+n} \hypergeom \left (\left [1, \frac {1+n}{2 n}\right ], \left [\frac {3}{2}+\frac {1}{2 n}\right ], -\frac {c \,x^{2 n}}{a}\right )}{2 a^{2} n \left (1+n \right )} \]
command
integrate((d+e*x**n)/(a+c*x**(2*n))**2,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ d \left (\frac {2 n x \Phi \left (\frac {c x^{2 n} e^{i \pi }}{a}, 1, \frac {1}{2 n}\right ) \Gamma \left (\frac {1}{2 n}\right )}{a \left (8 a n^{3} \Gamma \left (1 + \frac {1}{2 n}\right ) + 8 c n^{3} x^{2 n} \Gamma \left (1 + \frac {1}{2 n}\right )\right )} + \frac {2 n x \Gamma \left (\frac {1}{2 n}\right )}{a \left (8 a n^{3} \Gamma \left (1 + \frac {1}{2 n}\right ) + 8 c n^{3} x^{2 n} \Gamma \left (1 + \frac {1}{2 n}\right )\right )} - \frac {x \Phi \left (\frac {c x^{2 n} e^{i \pi }}{a}, 1, \frac {1}{2 n}\right ) \Gamma \left (\frac {1}{2 n}\right )}{a \left (8 a n^{3} \Gamma \left (1 + \frac {1}{2 n}\right ) + 8 c n^{3} x^{2 n} \Gamma \left (1 + \frac {1}{2 n}\right )\right )} + \frac {2 c n x x^{2 n} \Phi \left (\frac {c x^{2 n} e^{i \pi }}{a}, 1, \frac {1}{2 n}\right ) \Gamma \left (\frac {1}{2 n}\right )}{a^{2} \cdot \left (8 a n^{3} \Gamma \left (1 + \frac {1}{2 n}\right ) + 8 c n^{3} x^{2 n} \Gamma \left (1 + \frac {1}{2 n}\right )\right )} - \frac {c x x^{2 n} \Phi \left (\frac {c x^{2 n} e^{i \pi }}{a}, 1, \frac {1}{2 n}\right ) \Gamma \left (\frac {1}{2 n}\right )}{a^{2} \cdot \left (8 a n^{3} \Gamma \left (1 + \frac {1}{2 n}\right ) + 8 c n^{3} x^{2 n} \Gamma \left (1 + \frac {1}{2 n}\right )\right )}\right ) + e \left (\frac {n^{2} x x^{n} \Phi \left (\frac {c x^{2 n} e^{i \pi }}{a}, 1, \frac {1}{2} + \frac {1}{2 n}\right ) \Gamma \left (\frac {1}{2} + \frac {1}{2 n}\right )}{a \left (8 a n^{3} \Gamma \left (\frac {3}{2} + \frac {1}{2 n}\right ) + 8 c n^{3} x^{2 n} \Gamma \left (\frac {3}{2} + \frac {1}{2 n}\right )\right )} + \frac {2 n^{2} x x^{n} \Gamma \left (\frac {1}{2} + \frac {1}{2 n}\right )}{a \left (8 a n^{3} \Gamma \left (\frac {3}{2} + \frac {1}{2 n}\right ) + 8 c n^{3} x^{2 n} \Gamma \left (\frac {3}{2} + \frac {1}{2 n}\right )\right )} + \frac {2 n x x^{n} \Gamma \left (\frac {1}{2} + \frac {1}{2 n}\right )}{a \left (8 a n^{3} \Gamma \left (\frac {3}{2} + \frac {1}{2 n}\right ) + 8 c n^{3} x^{2 n} \Gamma \left (\frac {3}{2} + \frac {1}{2 n}\right )\right )} - \frac {x x^{n} \Phi \left (\frac {c x^{2 n} e^{i \pi }}{a}, 1, \frac {1}{2} + \frac {1}{2 n}\right ) \Gamma \left (\frac {1}{2} + \frac {1}{2 n}\right )}{a \left (8 a n^{3} \Gamma \left (\frac {3}{2} + \frac {1}{2 n}\right ) + 8 c n^{3} x^{2 n} \Gamma \left (\frac {3}{2} + \frac {1}{2 n}\right )\right )} + \frac {c n^{2} x x^{3 n} \Phi \left (\frac {c x^{2 n} e^{i \pi }}{a}, 1, \frac {1}{2} + \frac {1}{2 n}\right ) \Gamma \left (\frac {1}{2} + \frac {1}{2 n}\right )}{a^{2} \cdot \left (8 a n^{3} \Gamma \left (\frac {3}{2} + \frac {1}{2 n}\right ) + 8 c n^{3} x^{2 n} \Gamma \left (\frac {3}{2} + \frac {1}{2 n}\right )\right )} - \frac {c x x^{3 n} \Phi \left (\frac {c x^{2 n} e^{i \pi }}{a}, 1, \frac {1}{2} + \frac {1}{2 n}\right ) \Gamma \left (\frac {1}{2} + \frac {1}{2 n}\right )}{a^{2} \cdot \left (8 a n^{3} \Gamma \left (\frac {3}{2} + \frac {1}{2 n}\right ) + 8 c n^{3} x^{2 n} \Gamma \left (\frac {3}{2} + \frac {1}{2 n}\right )\right )}\right ) \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________