Optimal. Leaf size=53 \[ \frac{2 a^{x (k+l)}}{\log (a) (k+l)}+\frac{a^{2 k x}}{2 k \log (a)}+\frac{a^{2 l x}}{2 l \log (a)} \]
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Rubi [A] time = 0.0784878, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {6742, 2194} \[ \frac{2 a^{x (k+l)}}{\log (a) (k+l)}+\frac{a^{2 k x}}{2 k \log (a)}+\frac{a^{2 l x}}{2 l \log (a)} \]
Antiderivative was successfully verified.
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Rule 6742
Rule 2194
Rubi steps
\begin{align*} \int \left (a^{k x}+a^{l x}\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \left (e^{k x}+e^{l x}\right )^2 \, dx,x,x \log (a)\right )}{\log (a)}\\ &=\frac{\operatorname{Subst}\left (\int \left (e^{2 k x}+e^{2 l x}+2 e^{(k+l) x}\right ) \, dx,x,x \log (a)\right )}{\log (a)}\\ &=\frac{\operatorname{Subst}\left (\int e^{2 k x} \, dx,x,x \log (a)\right )}{\log (a)}+\frac{\operatorname{Subst}\left (\int e^{2 l x} \, dx,x,x \log (a)\right )}{\log (a)}+\frac{2 \operatorname{Subst}\left (\int e^{(k+l) x} \, dx,x,x \log (a)\right )}{\log (a)}\\ &=\frac{a^{2 k x}}{2 k \log (a)}+\frac{a^{2 l x}}{2 l \log (a)}+\frac{2 a^{(k+l) x}}{(k+l) \log (a)}\\ \end{align*}
Mathematica [A] time = 0.0422056, size = 53, normalized size = 1. \[ \frac{2 a^{x (k+l)}}{\log (a) (k+l)}+\frac{a^{2 k x}}{2 k \log (a)}+\frac{a^{2 l x}}{2 l \log (a)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 59, normalized size = 1.1 \begin{align*}{\frac{ \left ({{\rm e}^{kx\ln \left ( a \right ) }} \right ) ^{2}}{2\,k\ln \left ( a \right ) }}+{\frac{ \left ({{\rm e}^{lx\ln \left ( a \right ) }} \right ) ^{2}}{2\,l\ln \left ( a \right ) }}+2\,{\frac{{{\rm e}^{kx\ln \left ( a \right ) }}{{\rm e}^{lx\ln \left ( a \right ) }}}{\ln \left ( a \right ) \left ( k+l \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.922317, size = 69, normalized size = 1.3 \begin{align*} \frac{2 \, a^{k x + l x}}{{\left (k + l\right )} \log \left (a\right )} + \frac{a^{2 \, k x}}{2 \, k \log \left (a\right )} + \frac{a^{2 \, l x}}{2 \, l \log \left (a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.92044, size = 138, normalized size = 2.6 \begin{align*} \frac{4 \, a^{k x} a^{l x} k l +{\left (k l + l^{2}\right )} a^{2 \, k x} +{\left (k^{2} + k l\right )} a^{2 \, l x}}{2 \,{\left (k^{2} l + k l^{2}\right )} \log \left (a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.0471, size = 250, normalized size = 4.72 \begin{align*} \begin{cases} 4 x & \text{for}\: a = 1 \wedge \left (a = 1 \vee k = 0\right ) \wedge \left (a = 1 \vee l = 0\right ) \\\frac{a^{2 l x}}{2 l \log{\left (a \right )}} + \frac{2 a^{l x}}{l \log{\left (a \right )}} + x & \text{for}\: k = 0 \\\frac{a^{2 l x}}{2 l \log{\left (a \right )}} + 2 x - \frac{a^{- 2 l x}}{2 l \log{\left (a \right )}} & \text{for}\: k = - l \\\frac{a^{2 k x}}{2 k \log{\left (a \right )}} + \frac{2 a^{k x}}{k \log{\left (a \right )}} + x & \text{for}\: l = 0 \\\frac{a^{2 k x} k l}{2 k^{2} l \log{\left (a \right )} + 2 k l^{2} \log{\left (a \right )}} + \frac{a^{2 k x} l^{2}}{2 k^{2} l \log{\left (a \right )} + 2 k l^{2} \log{\left (a \right )}} + \frac{4 a^{k x} a^{l x} k l}{2 k^{2} l \log{\left (a \right )} + 2 k l^{2} \log{\left (a \right )}} + \frac{a^{2 l x} k^{2}}{2 k^{2} l \log{\left (a \right )} + 2 k l^{2} \log{\left (a \right )}} + \frac{a^{2 l x} k l}{2 k^{2} l \log{\left (a \right )} + 2 k l^{2} \log{\left (a \right )}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.15727, size = 933, normalized size = 17.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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