\(\int (1-a^{m x})^n \, dx\) [521]

Optimal result

Integrand size = 11, antiderivative size = 44 \[ \int \left (1-a^{m x}\right )^n \, dx=-\frac {\left (1-a^{m x}\right )^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1-a^{m x}\right )}{m (1+n) \log (a)} \]

[Out]

-(1-a^(m*x))^(1+n)*hypergeom([1, 1+n],[2+n],1-a^(m*x))/m/(1+n)/ln(a)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2320, 67} \[ \int \left (1-a^{m x}\right )^n \, dx=-\frac {\left (1-a^{m x}\right )^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,1-a^{m x}\right )}{m (n+1) \log (a)} \]

[In]

Int[(1 - a^(m*x))^n,x]

[Out]

-(((1 - a^(m*x))^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 - a^(m*x)])/(m*(1 + n)*Log[a]))

Rule 67

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x)^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))
*Hypergeometric2F1[-m, n + 1, n + 2, 1 + d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (Intege
rQ[m] || GtQ[-d/(b*c), 0])

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {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} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1-a^{m x}\right )}{m (1+n) \log (a)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00 \[ \int \left (1-a^{m x}\right )^n \, dx=-\frac {\left (1-a^{m x}\right )^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1-a^{m x}\right )}{m (1+n) \log (a)} \]

[In]

Integrate[(1 - a^(m*x))^n,x]

[Out]

-(((1 - a^(m*x))^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 - a^(m*x)])/(m*(1 + n)*Log[a]))

Maple [F]

\[\int \left (1-a^{m x}\right )^{n}d x\]

[In]

int((1-a^(m*x))^n,x)

[Out]

int((1-a^(m*x))^n,x)

Fricas [F]

\[ \int \left (1-a^{m x}\right )^n \, dx=\int { {\left (-a^{m x} + 1\right )}^{n} \,d x } \]

[In]

integrate((1-a^(m*x))^n,x, algorithm="fricas")

[Out]

integral((-a^(m*x) + 1)^n, x)

Sympy [F]

\[ \int \left (1-a^{m x}\right )^n \, dx=\int \left (1 - a^{m x}\right )^{n}\, dx \]

[In]

integrate((1-a**(m*x))**n,x)

[Out]

Integral((1 - a**(m*x))**n, x)

Maxima [F]

\[ \int \left (1-a^{m x}\right )^n \, dx=\int { {\left (-a^{m x} + 1\right )}^{n} \,d x } \]

[In]

integrate((1-a^(m*x))^n,x, algorithm="maxima")

[Out]

integrate((-a^(m*x) + 1)^n, x)

Giac [F]

\[ \int \left (1-a^{m x}\right )^n \, dx=\int { {\left (-a^{m x} + 1\right )}^{n} \,d x } \]

[In]

integrate((1-a^(m*x))^n,x, algorithm="giac")

[Out]

integrate((-a^(m*x) + 1)^n, x)

Mupad [B] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.30 \[ \int \left (1-a^{m x}\right )^n \, dx=\frac {{\left (1-a^{m\,x}\right )}^n\,{{}}_2{\mathrm {F}}_1\left (-n,-n;\ 1-n;\ \frac {1}{a^{m\,x}}\right )}{m\,n\,\ln \left (a\right )\,{\left (1-\frac {1}{a^{m\,x}}\right )}^n} \]

[In]

int((1 - a^(m*x))^n,x)

[Out]

((1 - a^(m*x))^n*hypergeom([-n, -n], 1 - n, 1/a^(m*x)))/(m*n*log(a)*(1 - 1/a^(m*x))^n)