Optimal. Leaf size=14 \[ 15+27 e^{-2 x} x^2 \log (5) \]
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Rubi [A]
time = 0.06, antiderivative size = 12, normalized size of antiderivative = 0.86, number of steps
used = 9, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {12, 1607, 2227,
2207, 2225} \begin {gather*} 27 e^{-2 x} x^2 \log (5) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 1607
Rule 2207
Rule 2225
Rule 2227
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\log (5) \int e^{-2 x} \left (54 x-54 x^2\right ) \, dx\\ &=\log (5) \int e^{-2 x} (54-54 x) x \, dx\\ &=\log (5) \int \left (54 e^{-2 x} x-54 e^{-2 x} x^2\right ) \, dx\\ &=(54 \log (5)) \int e^{-2 x} x \, dx-(54 \log (5)) \int e^{-2 x} x^2 \, dx\\ &=-27 e^{-2 x} x \log (5)+27 e^{-2 x} x^2 \log (5)+(27 \log (5)) \int e^{-2 x} \, dx-(54 \log (5)) \int e^{-2 x} x \, dx\\ &=-\frac {27}{2} e^{-2 x} \log (5)+27 e^{-2 x} x^2 \log (5)-(27 \log (5)) \int e^{-2 x} \, dx\\ &=27 e^{-2 x} x^2 \log (5)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 12, normalized size = 0.86 \begin {gather*} 27 e^{-2 x} x^2 \log (5) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 12, normalized size = 0.86
method | result | size |
gosper | \(27 \ln \left (5\right ) x^{2} {\mathrm e}^{-2 x}\) | \(12\) |
default | \(27 \ln \left (5\right ) x^{2} {\mathrm e}^{-2 x}\) | \(12\) |
norman | \(27 \ln \left (5\right ) x^{2} {\mathrm e}^{-2 x}\) | \(12\) |
risch | \(27 \ln \left (5\right ) x^{2} {\mathrm e}^{-2 x}\) | \(12\) |
meijerg | \(-\frac {27 \ln \left (5\right ) \left (2-\frac {\left (12 x^{2}+12 x +6\right ) {\mathrm e}^{-2 x}}{3}\right )}{4}+\frac {27 \ln \left (5\right ) \left (1-\frac {\left (4 x +2\right ) {\mathrm e}^{-2 x}}{2}\right )}{2}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 31 vs.
\(2 (13) = 26\).
time = 0.26, size = 31, normalized size = 2.21 \begin {gather*} \frac {27}{2} \, {\left ({\left (2 \, x^{2} + 2 \, x + 1\right )} e^{\left (-2 \, x\right )} - {\left (2 \, x + 1\right )} e^{\left (-2 \, x\right )}\right )} \log \left (5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 11, normalized size = 0.79 \begin {gather*} 27 \, x^{2} e^{\left (-2 \, x\right )} \log \left (5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 12, normalized size = 0.86 \begin {gather*} 27 x^{2} e^{- 2 x} \log {\left (5 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 11, normalized size = 0.79 \begin {gather*} 27 \, x^{2} e^{\left (-2 \, x\right )} \log \left (5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 11, normalized size = 0.79 \begin {gather*} 27\,x^2\,{\mathrm {e}}^{-2\,x}\,\ln \left (5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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