Optimal. Leaf size=74 \[ \frac {\left (1-a^{x (k-l)}\right ) \left (a^{k x}-a^{l x}\right )^n \, _2F_1\left (1,\frac {k n}{k-l}+1;\frac {l n}{k-l}+1;a^{(k-l) x}\right )}{l n \log (a)} \]
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Rubi [A] time = 0.06, 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.
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Rule 2251
Rule 2285
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 \[ \frac {\left (1-a^{x (l-k)}\right ) \left (a^{k x}-a^{l x}\right )^n \, _2F_1\left (1,\frac {k n}{l-k}+n+1;\frac {k n}{l-k}+1;a^{(l-k) x}\right )}{k n \log (a)} \]
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a^{k x}-a^{l x}\right )^n \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.98, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{k x} - a^{l x}\right )}^{n}, x\right ) \]
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 \[ \int {\left (a^{k x} - a^{l x}\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, 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 \[ \int {\left (a^{k x} - a^{l x}\right )}^{n}\,{d x} \]
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 \[ \int {\left (a^{k\,x}-a^{l\,x}\right )}^n \,d x \]
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 \[ \int \left (a^{k x} - a^{l x}\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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