\[ \int \frac {x^{-1+4 n}}{\left (a+b x^n\right )^2} \, dx \]
Optimal antiderivative \[ -\frac {2 a \,x^{n}}{b^{3} n}+\frac {x^{2 n}}{2 b^{2} n}+\frac {a^{3}}{b^{4} n \left (a +b \,x^{n}\right )}+\frac {3 a^{2} \ln \left (a +b \,x^{n}\right )}{b^{4} n} \]
command
integrate(x**(-1+4*n)/(a+b*x**n)**2,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \frac {\log {\left (x \right )}}{a^{2}} & \text {for}\: b = 0 \wedge n = 0 \\\frac {x^{4 n}}{4 a^{2} n} & \text {for}\: b = 0 \\\frac {\log {\left (x \right )}}{\left (a + b\right )^{2}} & \text {for}\: n = 0 \\\frac {6 a^{3} \log {\left (\frac {a}{b} + x^{n} \right )}}{2 a b^{4} n + 2 b^{5} n x^{n}} + \frac {6 a^{3}}{2 a b^{4} n + 2 b^{5} n x^{n}} + \frac {6 a^{2} b x^{n} \log {\left (\frac {a}{b} + x^{n} \right )}}{2 a b^{4} n + 2 b^{5} n x^{n}} - \frac {3 a b^{2} x^{2 n}}{2 a b^{4} n + 2 b^{5} n x^{n}} + \frac {b^{3} x^{3 n}}{2 a b^{4} n + 2 b^{5} n x^{n}} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Exception raised: HeuristicGCDFailed} \]________________________________________________________________________________________