Optimal. Leaf size=19 \[ -2-e^5+225 e^{-16 x} x^3+\log (4) \]
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Rubi [A] time = 0.11, antiderivative size = 10, normalized size of antiderivative = 0.53, number of steps used = 10, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {1593, 2196, 2176, 2194} \begin {gather*} 225 e^{-16 x} x^3 \end {gather*}
Antiderivative was successfully verified.
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Rule 1593
Rule 2176
Rule 2194
Rule 2196
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int e^{-16 x} (675-3600 x) x^2 \, dx\\ &=\int \left (675 e^{-16 x} x^2-3600 e^{-16 x} x^3\right ) \, dx\\ &=675 \int e^{-16 x} x^2 \, dx-3600 \int e^{-16 x} x^3 \, dx\\ &=-\frac {675}{16} e^{-16 x} x^2+225 e^{-16 x} x^3+\frac {675}{8} \int e^{-16 x} x \, dx-675 \int e^{-16 x} x^2 \, dx\\ &=-\frac {675}{128} e^{-16 x} x+225 e^{-16 x} x^3+\frac {675}{128} \int e^{-16 x} \, dx-\frac {675}{8} \int e^{-16 x} x \, dx\\ &=-\frac {675 e^{-16 x}}{2048}+225 e^{-16 x} x^3-\frac {675}{128} \int e^{-16 x} \, dx\\ &=225 e^{-16 x} x^3\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.01, size = 10, normalized size = 0.53 \begin {gather*} 225 e^{-16 x} x^3 \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 9, normalized size = 0.47 \begin {gather*} 225 \, x^{3} e^{\left (-16 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 9, normalized size = 0.47 \begin {gather*} 225 \, x^{3} e^{\left (-16 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 10, normalized size = 0.53
method | result | size |
risch | \(225 x^{3} {\mathrm e}^{-16 x}\) | \(10\) |
gosper | \(225 x^{3} {\mathrm e}^{-16 x}\) | \(12\) |
derivativedivides | \(225 x^{3} {\mathrm e}^{-16 x}\) | \(12\) |
default | \(225 x^{3} {\mathrm e}^{-16 x}\) | \(12\) |
norman | \(225 x^{3} {\mathrm e}^{-16 x}\) | \(12\) |
meijerg | \(\frac {225 \left (16384 x^{3}+3072 x^{2}+384 x +24\right ) {\mathrm e}^{-16 x}}{16384}-\frac {225 \left (768 x^{2}+96 x +6\right ) {\mathrm e}^{-16 x}}{4096}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 38, normalized size = 2.00 \begin {gather*} \frac {225}{2048} \, {\left (2048 \, x^{3} + 384 \, x^{2} + 48 \, x + 3\right )} e^{\left (-16 \, x\right )} - \frac {675}{2048} \, {\left (128 \, x^{2} + 16 \, x + 1\right )} e^{\left (-16 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.58, size = 9, normalized size = 0.47 \begin {gather*} 225\,x^3\,{\mathrm {e}}^{-16\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 8, normalized size = 0.42 \begin {gather*} 225 x^{3} e^{- 16 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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