Optimal. Leaf size=74 \[ \frac {\left (1-a^{(k-l) x}\right ) \left (a^{k x}-a^{l x}\right )^n \, _2F_1\left (1,1+\frac {k n}{k-l};1+\frac {l n}{k-l};a^{(k-l) x}\right )}{l n \log (a)} \]
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Rubi [A]
time = 0.04, 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 = {2323, 2283}
\begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2283
Rule 2323
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*}
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Mathematica [A]
time = 0.03, size = 75, normalized size = 1.01 \begin {gather*} \frac {\left (a^{k x}-a^{l x}\right )^n \left (1-a^{(-k+l) x}\right ) \, _2F_1\left (1,1+n+\frac {k n}{-k+l};1+\frac {k n}{-k+l};a^{(-k+l) x}\right )}{k n \log (a)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \left (a^{k x}-a^{l x}\right )^{n}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a^{k x} - a^{l x}\right )^{n}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (a^{k\,x}-a^{l\,x}\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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