3.511 \(\int (a^{k x}-a^{l x})^n \, dx\)

Optimal. Leaf size=74 \[ \frac{\left (1-a^{x (k-l)}\right ) \left (a^{k x}-a^{l x}\right )^n \, _2F_1\left (1,\frac{k n}{k-l}+1;\frac{l n}{k-l}+1;a^{(k-l) x}\right )}{l n \log (a)} \]

[Out]

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

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Rubi [A]  time = 0.0623792, antiderivative size = 82, normalized size of antiderivative = 1.11, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2285, 2251} \[ \frac{\left (1-a^{x (-(k-l))}\right )^{-n} \left (a^{k x}-a^{l x}\right )^n \text{Hypergeometric2F1}\left (-n,-\frac{k n}{k-l},1-\frac{k n}{k-l},a^{x (-(k-l))}\right )}{k n \log (a)} \]

Antiderivative was successfully verified.

[In]

Int[(a^(k*x) - a^(l*x))^n,x]

[Out]

((a^(k*x) - a^(l*x))^n*Hypergeometric2F1[-n, -((k*n)/(k - l)), 1 - (k*n)/(k - l), a^(-((k - l)*x))])/((1 - a^(
-((k - l)*x)))^n*k*n*Log[a])

Rule 2285

Int[(u_.)*((a_.)*(F_)^(v_) + (b_.)*(F_)^(w_))^(n_), x_Symbol] :> Dist[(a*F^v + b*F^w)^n/(F^(n*v)*(a + b*F^Expa
ndToSum[w - v, x])^n), Int[u*F^(n*v)*(a + b*F^ExpandToSum[w - v, x])^n, x], x] /; FreeQ[{F, a, b, n}, x] &&  !
IntegerQ[n] && LinearQ[{v, w}, x]

Rule 2251

Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_)*(G_)^((h_.)*((f_.) + (g_.)*(x_))), x_Symbol] :> Simp
[(a^p*G^(h*(f + g*x))*Hypergeometric2F1[-p, (g*h*Log[G])/(d*e*Log[F]), (g*h*Log[G])/(d*e*Log[F]) + 1, Simplify
[-((b*F^(e*(c + d*x)))/a)]])/(g*h*Log[G]), x] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x] && (ILtQ[p, 0] ||
GtQ[a, 0])

Rubi steps

\begin{align*} \int \left (a^{k x}-a^{l x}\right )^n \, dx &=\left (a^{-k n x} \left (1-a^{-(k-l) x}\right )^{-n} \left (a^{k x}-a^{l x}\right )^n\right ) \int a^{k n x} \left (1-a^{-(k-l) x}\right )^n \, dx\\ &=\frac{\left (1-a^{-(k-l) x}\right )^{-n} \left (a^{k x}-a^{l x}\right )^n \, _2F_1\left (-n,-\frac{k n}{k-l};1-\frac{k n}{k-l};a^{-(k-l) x}\right )}{k n \log (a)}\\ \end{align*}

Mathematica [A]  time = 0.0260134, size = 75, normalized size = 1.01 \[ \frac{\left (1-a^{x (l-k)}\right ) \left (a^{k x}-a^{l x}\right )^n \, _2F_1\left (1,\frac{k n}{l-k}+n+1;\frac{k n}{l-k}+1;a^{(l-k) x}\right )}{k n \log (a)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a^(k*x) - a^(l*x))^n,x]

[Out]

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

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Maple [F]  time = 0.048, size = 0, normalized size = 0. \begin{align*} \int \left ({a}^{kx}-{a}^{lx} \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^(k*x)-a^(l*x))^n,x)

[Out]

int((a^(k*x)-a^(l*x))^n,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a^{k x} - a^{l x}\right )}^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^(k*x)-a^(l*x))^n,x, algorithm="maxima")

[Out]

integrate((a^(k*x) - a^(l*x))^n, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{k x} - a^{l x}\right )}^{n}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^(k*x)-a^(l*x))^n,x, algorithm="fricas")

[Out]

integral((a^(k*x) - a^(l*x))^n, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a^{k x} - a^{l x}\right )^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a**(k*x)-a**(l*x))**n,x)

[Out]

Integral((a**(k*x) - a**(l*x))**n, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a^{k x} - a^{l x}\right )}^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^(k*x)-a^(l*x))^n,x, algorithm="giac")

[Out]

integrate((a^(k*x) - a^(l*x))^n, x)