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

Optimal. Leaf size=44 \[ -\frac {\left (1-a^{m x}\right )^{1+n} \, _2F_1\left (1,1+n;2+n;1-a^{m x}\right )}{m (1+n) \log (a)} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, 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 = {2320, 67} \begin {gather*} -\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)} \end {gather*}

Antiderivative was successfully verified.

[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*} \int \left (1-a^{m x}\right )^n \, dx &=\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} \, _2F_1\left (1,1+n;2+n;1-a^{m x}\right )}{m (1+n) \log (a)}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 44, normalized size = 1.00 \begin {gather*} -\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 {gather*}

Antiderivative was successfully verified.

[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]))

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \left (1-a^{m x}\right )^{n}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((-a^(m*x) + 1)^n, x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((-a^(m*x) + 1)^n, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (1 - a^{m x}\right )^{n}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((1 - a**(m*x))**n, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((-a^(m*x) + 1)^n, x)

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Mupad [B]
time = 0.32, size = 57, normalized size = 1.30 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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