\[ \int x^{14 (-1+n)} \left (b+2 c x^n\right ) \left (b x+c x^{1+n}\right )^{13} \, dx \]
Optimal antiderivative \[ \frac {x^{14 n} \left (b +c \,x^{n}\right )^{14}}{14 n} \]
command
integrate(x^(-14+14*n)*(b+2*c*x^n)*(b*x+c*x^(1+n))^13,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {could not integrate} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \frac {c^{14} x^{28 \, n} + 14 \, b c^{13} x^{27 \, n} + 91 \, b^{2} c^{12} x^{26 \, n} + 364 \, b^{3} c^{11} x^{25 \, n} + 1001 \, b^{4} c^{10} x^{24 \, n} + 2002 \, b^{5} c^{9} x^{23 \, n} + 3003 \, b^{6} c^{8} x^{22 \, n} + 3432 \, b^{7} c^{7} x^{21 \, n} + 3003 \, b^{8} c^{6} x^{20 \, n} + 2002 \, b^{9} c^{5} x^{19 \, n} + 1001 \, b^{10} c^{4} x^{18 \, n} + 364 \, b^{11} c^{3} x^{17 \, n} + 91 \, b^{12} c^{2} x^{16 \, n} + 14 \, b^{13} c x^{15 \, n} + b^{14} x^{14 \, n}}{14 \, n} \]