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

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 = 0.125999, 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 \[ \frac{\left (1-a^{x (-(k-l))}\right )^{-n} \left (a^{k x}-a^{l x}\right )^n \, _2F_1\left (-n,-\frac{k n}{k-l};1-\frac{k n}{k-l};a^{-(k-l) x}\right )}{k n \log (a)} \]

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])

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Rubi in Sympy [A]  time = 7.66933, size = 63, normalized size = 0.85 \[ \frac{\left (a^{k x} - a^{l x}\right )^{n} \left (- a^{x \left (- k + l\right )} + 1\right )^{- n} \left (- a^{x \left (- k + l\right )} + 1\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{k n}{- k + l} + n + 1 \\ \frac{k n}{- k + l} + 1 \end{matrix}\middle |{a^{x \left (- k + l\right )}} \right )}}{k n \log{\left (a \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a**(k*x)-a**(l*x))**n,x)

[Out]

(a**(k*x) - a**(l*x))**n*(-a**(x*(-k + l)) + 1)**(-n)*(-a**(x*(-k + l)) + 1)**(n
+ 1)*hyper((1, k*n/(-k + l) + n + 1), (k*n/(-k + l) + 1,), a**(x*(-k + l)))/(k*
n*log(a))

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Mathematica [A]  time = 0.0367328, size = 0, normalized size = 0. \[ \int \left (a^{k x}-a^{l x}\right )^n \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[(a^(k*x) - a^(l*x))^n,x]

[Out]

Integrate[(a^(k*x) - a^(l*x))^n, x]

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Maple [F]  time = 0.082, size = 0, normalized size = 0. \[ \int \left ({a}^{kx}-{a}^{lx} \right ) ^{n}\, dx \]

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. \[ \int{\left (a^{k x} - a^{l x}\right )}^{n}\,{d x} \]

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. \[{\rm integral}\left ({\left (a^{k x} - a^{l x}\right )}^{n}, x\right ) \]

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. \[ \int \left (a^{k x} - a^{l x}\right )^{n}\, dx \]

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/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (a^{k x} - a^{l x}\right )}^{n}\,{d x} \]

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)