Optimal. Leaf size=79 \[ \frac {a^{3 k x}}{3 k \log (a)}+\frac {a^{3 l x}}{3 l \log (a)}+\frac {3 a^{(2 k+l) x}}{(2 k+l) \log (a)}+\frac {3 a^{(k+2 l) x}}{(k+2 l) \log (a)} \]
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Rubi [A]
time = 0.07, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {6874, 2225}
\begin {gather*} \frac {3 a^{x (2 k+l)}}{\log (a) (2 k+l)}+\frac {3 a^{x (k+2 l)}}{\log (a) (k+2 l)}+\frac {a^{3 k x}}{3 k \log (a)}+\frac {a^{3 l x}}{3 l \log (a)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2225
Rule 6874
Rubi steps
\begin {align*} \int \left (a^{k x}+a^{l x}\right )^3 \, dx &=\frac {\text {Subst}\left (\int \left (e^{k x}+e^{l x}\right )^3 \, dx,x,x \log (a)\right )}{\log (a)}\\ &=\frac {\text {Subst}\left (\int \left (e^{3 k x}+e^{3 l x}+3 e^{(2 k+l) x}+3 e^{(k+2 l) x}\right ) \, dx,x,x \log (a)\right )}{\log (a)}\\ &=\frac {\text {Subst}\left (\int e^{3 k x} \, dx,x,x \log (a)\right )}{\log (a)}+\frac {\text {Subst}\left (\int e^{3 l x} \, dx,x,x \log (a)\right )}{\log (a)}+\frac {3 \text {Subst}\left (\int e^{(2 k+l) x} \, dx,x,x \log (a)\right )}{\log (a)}+\frac {3 \text {Subst}\left (\int e^{(k+2 l) x} \, dx,x,x \log (a)\right )}{\log (a)}\\ &=\frac {a^{3 k x}}{3 k \log (a)}+\frac {a^{3 l x}}{3 l \log (a)}+\frac {3 a^{(2 k+l) x}}{(2 k+l) \log (a)}+\frac {3 a^{(k+2 l) x}}{(k+2 l) \log (a)}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 65, normalized size = 0.82 \begin {gather*} \frac {\frac {a^{3 k x}}{k}+\frac {a^{3 l x}}{l}+\frac {9 a^{(2 k+l) x}}{2 k+l}+\frac {9 a^{(k+2 l) x}}{k+2 l}}{3 \log (a)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 84, normalized size = 1.06
method | result | size |
risch | \(\frac {a^{3 k x}}{3 k \ln \left (a \right )}+\frac {a^{3 l x}}{3 l \ln \left (a \right )}+\frac {3 a^{k x} a^{2 l x}}{\ln \left (a \right ) \left (k +2 l \right )}+\frac {3 a^{2 k x} a^{l x}}{\ln \left (a \right ) \left (2 k +l \right )}\) | \(84\) |
norman | \(\frac {{\mathrm e}^{3 k x \ln \left (a \right )}}{3 k \ln \left (a \right )}+\frac {{\mathrm e}^{3 l x \ln \left (a \right )}}{3 l \ln \left (a \right )}+\frac {3 \,{\mathrm e}^{k x \ln \left (a \right )} {\mathrm e}^{2 l x \ln \left (a \right )}}{\ln \left (a \right ) \left (k +2 l \right )}+\frac {3 \,{\mathrm e}^{2 k x \ln \left (a \right )} {\mathrm e}^{l x \ln \left (a \right )}}{\ln \left (a \right ) \left (2 k +l \right )}\) | \(90\) |
meijerg | \(-\frac {1-{\mathrm e}^{3 k x \ln \left (a \right )}}{3 k \ln \left (a \right )}-\frac {3 \left (1-{\mathrm e}^{x k \ln \left (a \right ) \left (2+\frac {l}{k}\right )}\right )}{k \ln \left (a \right ) \left (2+\frac {l}{k}\right )}-\frac {3 \left (1-{\mathrm e}^{x l \ln \left (a \right ) \left (1+\frac {k}{l}\right ) \left (1+\frac {1}{1+\frac {k}{l}}\right )}\right )}{l \ln \left (a \right ) \left (1+\frac {k}{l}\right ) \left (1+\frac {1}{1+\frac {k}{l}}\right )}-\frac {1-{\mathrm e}^{3 l x \ln \left (a \right )}}{3 l \ln \left (a \right )}\) | \(136\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.45, size = 77, normalized size = 0.97 \begin {gather*} \frac {3 \, a^{2 \, k x + l x}}{{\left (2 \, k + l\right )} \log \left (a\right )} + \frac {3 \, a^{k x + 2 \, l x}}{{\left (k + 2 \, l\right )} \log \left (a\right )} + \frac {a^{3 \, k x}}{3 \, k \log \left (a\right )} + \frac {a^{3 \, l x}}{3 \, l \log \left (a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.47, size = 130, normalized size = 1.65 \begin {gather*} \frac {9 \, {\left (2 \, k^{2} l + k l^{2}\right )} a^{k x} a^{2 \, l x} + 9 \, {\left (k^{2} l + 2 \, k l^{2}\right )} a^{2 \, k x} a^{l x} + {\left (2 \, k^{2} l + 5 \, k l^{2} + 2 \, l^{3}\right )} a^{3 \, k x} + {\left (2 \, k^{3} + 5 \, k^{2} l + 2 \, k l^{2}\right )} a^{3 \, l x}}{3 \, {\left (2 \, k^{3} l + 5 \, k^{2} l^{2} + 2 \, k l^{3}\right )} \log \left (a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 665 vs.
\(2 (63) = 126\).
time = 2.52, size = 665, normalized size = 8.42 \begin {gather*} \begin {cases} 8 x & \text {for}\: a = 1 \wedge \left (a = 1 \vee k = 0\right ) \wedge \left (a = 1 \vee l = 0\right ) \\\frac {a^{3 l x}}{3 l \log {\left (a \right )}} + \frac {3 a^{2 l x}}{2 l \log {\left (a \right )}} + \frac {3 a^{l x}}{l \log {\left (a \right )}} + x & \text {for}\: k = 0 \\\frac {a^{3 l x}}{3 l \log {\left (a \right )}} + 3 x - \frac {a^{- 3 l x}}{l \log {\left (a \right )}} - \frac {a^{- 6 l x}}{6 l \log {\left (a \right )}} & \text {for}\: k = - 2 l \\\frac {2 a^{\frac {3 l x}{2}}}{l \log {\left (a \right )}} + \frac {a^{3 l x}}{3 l \log {\left (a \right )}} + 3 x - \frac {2 a^{- \frac {3 l x}{2}}}{3 l \log {\left (a \right )}} & \text {for}\: k = - \frac {l}{2} \\\frac {a^{3 k x}}{3 k \log {\left (a \right )}} + \frac {3 a^{2 k x}}{2 k \log {\left (a \right )}} + \frac {3 a^{k x}}{k \log {\left (a \right )}} + x & \text {for}\: l = 0 \\\frac {2 a^{3 k x} k^{2} l}{6 k^{3} l \log {\left (a \right )} + 15 k^{2} l^{2} \log {\left (a \right )} + 6 k l^{3} \log {\left (a \right )}} + \frac {5 a^{3 k x} k l^{2}}{6 k^{3} l \log {\left (a \right )} + 15 k^{2} l^{2} \log {\left (a \right )} + 6 k l^{3} \log {\left (a \right )}} + \frac {2 a^{3 k x} l^{3}}{6 k^{3} l \log {\left (a \right )} + 15 k^{2} l^{2} \log {\left (a \right )} + 6 k l^{3} \log {\left (a \right )}} + \frac {9 a^{2 k x} a^{l x} k^{2} l}{6 k^{3} l \log {\left (a \right )} + 15 k^{2} l^{2} \log {\left (a \right )} + 6 k l^{3} \log {\left (a \right )}} + \frac {18 a^{2 k x} a^{l x} k l^{2}}{6 k^{3} l \log {\left (a \right )} + 15 k^{2} l^{2} \log {\left (a \right )} + 6 k l^{3} \log {\left (a \right )}} + \frac {18 a^{k x} a^{2 l x} k^{2} l}{6 k^{3} l \log {\left (a \right )} + 15 k^{2} l^{2} \log {\left (a \right )} + 6 k l^{3} \log {\left (a \right )}} + \frac {9 a^{k x} a^{2 l x} k l^{2}}{6 k^{3} l \log {\left (a \right )} + 15 k^{2} l^{2} \log {\left (a \right )} + 6 k l^{3} \log {\left (a \right )}} + \frac {2 a^{3 l x} k^{3}}{6 k^{3} l \log {\left (a \right )} + 15 k^{2} l^{2} \log {\left (a \right )} + 6 k l^{3} \log {\left (a \right )}} + \frac {5 a^{3 l x} k^{2} l}{6 k^{3} l \log {\left (a \right )} + 15 k^{2} l^{2} \log {\left (a \right )} + 6 k l^{3} \log {\left (a \right )}} + \frac {2 a^{3 l x} k l^{2}}{6 k^{3} l \log {\left (a \right )} + 15 k^{2} l^{2} \log {\left (a \right )} + 6 k l^{3} \log {\left (a \right )}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 0.86, size = 1033, normalized size = 13.08 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.39, size = 81, normalized size = 1.03 \begin {gather*} \frac {3\,a^{k\,x}\,a^{2\,l\,x}}{k\,\ln \left (a\right )+2\,l\,\ln \left (a\right )}+\frac {3\,a^{2\,k\,x}\,a^{l\,x}}{2\,k\,\ln \left (a\right )+l\,\ln \left (a\right )}+\frac {a^{3\,k\,x}}{3\,k\,\ln \left (a\right )}+\frac {a^{3\,l\,x}}{3\,l\,\ln \left (a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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