\[ \int x^{-1-4 n} \left (a+b x^n\right )^5 \, dx \]
Optimal antiderivative \[ -\frac {a^{5} x^{-4 n}}{4 n}-\frac {5 a^{4} b \,x^{-3 n}}{3 n}-\frac {5 a^{3} b^{2} x^{-2 n}}{n}-\frac {10 a^{2} b^{3} x^{-n}}{n}+\frac {b^{5} x^{n}}{n}+5 a \,b^{4} \ln \left (x \right ) \]
command
integrate(x**(-1-4*n)*(a+b*x**n)**5,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} a^{5} x + \frac {20 a^{4} b x^{\frac {3}{4}}}{3} + 20 a^{3} b^{2} \sqrt {x} + 40 a^{2} b^{3} \sqrt [4]{x} - 20 a b^{4} \log {\left (\frac {1}{\sqrt [4]{x}} \right )} - \frac {4 b^{5}}{\sqrt [4]{x}} & \text {for}\: n = - \frac {1}{4} \\\left (a + b\right )^{5} \log {\left (x \right )} & \text {for}\: n = 0 \\- \frac {12 a^{5} n}{48 n^{2} x^{4 n} + 12 n x^{4 n}} - \frac {3 a^{5}}{48 n^{2} x^{4 n} + 12 n x^{4 n}} - \frac {80 a^{4} b n x^{n}}{48 n^{2} x^{4 n} + 12 n x^{4 n}} - \frac {20 a^{4} b x^{n}}{48 n^{2} x^{4 n} + 12 n x^{4 n}} - \frac {240 a^{3} b^{2} n x^{2 n}}{48 n^{2} x^{4 n} + 12 n x^{4 n}} - \frac {60 a^{3} b^{2} x^{2 n}}{48 n^{2} x^{4 n} + 12 n x^{4 n}} - \frac {480 a^{2} b^{3} n x^{3 n}}{48 n^{2} x^{4 n} + 12 n x^{4 n}} - \frac {120 a^{2} b^{3} x^{3 n}}{48 n^{2} x^{4 n} + 12 n x^{4 n}} + \frac {240 a b^{4} n x^{4 n} \log {\left (x^{n} \right )}}{48 n^{2} x^{4 n} + 12 n x^{4 n}} + \frac {60 a b^{4} n x^{4 n}}{48 n^{2} x^{4 n} + 12 n x^{4 n}} + \frac {60 a b^{4} x^{4 n} \log {\left (x^{n} \right )}}{48 n^{2} x^{4 n} + 12 n x^{4 n}} + \frac {48 b^{5} n x^{5 n}}{48 n^{2} x^{4 n} + 12 n x^{4 n}} + \frac {12 b^{5} x^{5 n}}{48 n^{2} x^{4 n} + 12 n x^{4 n}} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________