Integrand size = 17, antiderivative size = 19 \[ \int e^{-16 x} \left (675 x^2-3600 x^3\right ) \, dx=-2-e^5+225 e^{-16 x} x^3+\log (4) \]
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Time = 0.07 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.53, number of steps used = 10, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {1607, 2227, 2207, 2225} \[ \int e^{-16 x} \left (675 x^2-3600 x^3\right ) \, dx=225 e^{-16 x} x^3 \]
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Rule 1607
Rule 2207
Rule 2225
Rule 2227
Rubi steps \begin{align*} \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{align*}
Time = 0.05 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.53 \[ \int e^{-16 x} \left (675 x^2-3600 x^3\right ) \, dx=225 e^{-16 x} x^3 \]
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Time = 0.21 (sec) , antiderivative size = 10, normalized size of antiderivative = 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 | \(225 x^{3} {\mathrm e}^{-16 x}\) | \(12\) |
parallelrisch | \(225 x^{3} {\mathrm e}^{-16 x}\) | \(12\) |
meijerg | \(\frac {225 \left (16384 x^{3}+3072 x^{2}+384 x +24\right ) {\mathrm e}^{-16 x}}{16384}-\frac {225 \left (768 x^{2}+96 x +6\right ) {\mathrm e}^{-16 x}}{4096}\) | \(39\) |
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Time = 0.24 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.47 \[ \int e^{-16 x} \left (675 x^2-3600 x^3\right ) \, dx=225 \, x^{3} e^{\left (-16 \, x\right )} \]
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Time = 0.05 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.42 \[ \int e^{-16 x} \left (675 x^2-3600 x^3\right ) \, dx=225 x^{3} e^{- 16 x} \]
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Time = 0.20 (sec) , antiderivative size = 38, normalized size of antiderivative = 2.00 \[ \int e^{-16 x} \left (675 x^2-3600 x^3\right ) \, dx=\frac {225}{2048} \, {\left (2048 \, x^{3} + 384 \, x^{2} + 48 \, x + 3\right )} e^{\left (-16 \, x\right )} - \frac {675}{2048} \, {\left (128 \, x^{2} + 16 \, x + 1\right )} e^{\left (-16 \, x\right )} \]
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Time = 0.28 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.47 \[ \int e^{-16 x} \left (675 x^2-3600 x^3\right ) \, dx=225 \, x^{3} e^{\left (-16 \, x\right )} \]
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Time = 16.49 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.47 \[ \int e^{-16 x} \left (675 x^2-3600 x^3\right ) \, dx=225\,x^3\,{\mathrm {e}}^{-16\,x} \]
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