Optimal. Leaf size=44 \[ -\frac{\left (1-a^{m x}\right )^{n+1} \, _2F_1\left (1,n+1;n+2;1-a^{m x}\right )}{m (n+1) \log (a)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0187026, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2282, 65} \[ -\frac{\left (1-a^{m x}\right )^{n+1} \text{Hypergeometric2F1}\left (1,n+1,n+2,1-a^{m x}\right )}{m (n+1) \log (a)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2282
Rule 65
Rubi steps
\begin{align*} \int \left (1-a^{m x}\right )^n \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(1-x)^n}{x} \, dx,x,a^{m x}\right )}{m \log (a)}\\ &=-\frac{\left (1-a^{m x}\right )^{1+n} \, _2F_1\left (1,1+n;2+n;1-a^{m x}\right )}{m (1+n) \log (a)}\\ \end{align*}
Mathematica [A] time = 0.0138956, size = 44, normalized size = 1. \[ -\frac{\left (1-a^{m x}\right )^{n+1} \, _2F_1\left (1,n+1;n+2;1-a^{m x}\right )}{m (n+1) \log (a)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.032, size = 0, normalized size = 0. \begin{align*} \int \left ( 1-{a}^{mx} \right ) ^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-a^{m x} + 1\right )}^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (-a^{m x} + 1\right )}^{n}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (1 - a^{m x}\right )^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-a^{m x} + 1\right )}^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]