Optimal. Leaf size=72 \[ \frac{\left (a^{x (k-l)}+1\right ) \left (a^{k x}+a^{l x}\right )^n \, _2F_1\left (1,\frac{k n}{k-l}+1;\frac{l n}{k-l}+1;-a^{(k-l) x}\right )}{l n \log (a)} \]
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Rubi [A] time = 0.0771031, antiderivative size = 80, normalized size of antiderivative = 1.11, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2285, 2251} \[ \frac{\left (a^{x (-(k-l))}+1\right )^{-n} \left (a^{k x}+a^{l x}\right )^n \text{Hypergeometric2F1}\left (-n,-\frac{k n}{k-l},1-\frac{k n}{k-l},-a^{x (-(k-l))}\right )}{k n \log (a)} \]
Antiderivative was successfully verified.
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Rule 2285
Rule 2251
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 = 0.0286806, size = 73, normalized size = 1.01 \[ \frac{\left (a^{x (l-k)}+1\right ) \left (a^{k x}+a^{l x}\right )^n \, _2F_1\left (1,\frac{k n}{l-k}+n+1;\frac{k n}{l-k}+1;-a^{(l-k) x}\right )}{k n \log (a)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.05, 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.
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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.
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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.
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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.
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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.
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