\[ \int F^{c+d x} \left (a+b F^{c+d x}\right )^n \, dx \]
Optimal antiderivative \[ \frac {\left (a +b \,F^{d x +c}\right )^{1+n}}{b d \left (1+n \right ) \ln \left (F \right )} \]
command
integrate(F**(d*x+c)*(a+b*F**(d*x+c))**n,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \frac {\begin {cases} \left (a + b\right )^{n} \left (c + d x\right ) & \text {for}\: \log {\left (F \right )} = 0 \\\frac {\begin {cases} F^{c + d x} a^{n} & \text {for}\: b = 0 \\\frac {\begin {cases} \frac {\left (F^{c + d x} b + a\right )^{n + 1}}{n + 1} & \text {for}\: n \neq -1 \\\log {\left (F^{c + d x} b + a \right )} & \text {otherwise} \end {cases}}{b} & \text {otherwise} \end {cases}}{\log {\left (F \right )}} & \text {otherwise} \end {cases}}{d} & \text {for}\: d \neq 0 \\F^{c} x \left (F^{c} b + a\right )^{n} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________