Optimal. Leaf size=21 \[ -e^{16}+x \left (x-\frac {9}{256} \left (e^x+x\right )^2\right ) \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(44\) vs. \(2(21)=42\).
time = 0.07, antiderivative size = 44, normalized size of antiderivative = 2.10, number of steps
used = 12, number of rules used = 5, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {12, 2207,
2225, 1607, 2227} \begin {gather*} -\frac {9 x^3}{256}-\frac {9 e^x x^2}{128}+x^2+\frac {9 e^{2 x}}{512}-\frac {9}{512} e^{2 x} (2 x+1) \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} &=\frac {1}{256} \int \left (e^{2 x} (-9-18 x)+512 x-27 x^2+e^x \left (-36 x-18 x^2\right )\right ) \, dx\\ &=x^2-\frac {9 x^3}{256}+\frac {1}{256} \int e^{2 x} (-9-18 x) \, dx+\frac {1}{256} \int e^x \left (-36 x-18 x^2\right ) \, dx\\ &=x^2-\frac {9 x^3}{256}-\frac {9}{512} e^{2 x} (1+2 x)+\frac {1}{256} \int e^x (-36-18 x) x \, dx+\frac {9}{256} \int e^{2 x} \, dx\\ &=\frac {9 e^{2 x}}{512}+x^2-\frac {9 x^3}{256}-\frac {9}{512} e^{2 x} (1+2 x)+\frac {1}{256} \int \left (-36 e^x x-18 e^x x^2\right ) \, dx\\ &=\frac {9 e^{2 x}}{512}+x^2-\frac {9 x^3}{256}-\frac {9}{512} e^{2 x} (1+2 x)-\frac {9}{128} \int e^x x^2 \, dx-\frac {9}{64} \int e^x x \, dx\\ &=\frac {9 e^{2 x}}{512}-\frac {9 e^x x}{64}+x^2-\frac {9 e^x x^2}{128}-\frac {9 x^3}{256}-\frac {9}{512} e^{2 x} (1+2 x)+\frac {9 \int e^x \, dx}{64}+\frac {9}{64} \int e^x x \, dx\\ &=\frac {9 e^x}{64}+\frac {9 e^{2 x}}{512}+x^2-\frac {9 e^x x^2}{128}-\frac {9 x^3}{256}-\frac {9}{512} e^{2 x} (1+2 x)-\frac {9 \int e^x \, dx}{64}\\ &=\frac {9 e^{2 x}}{512}+x^2-\frac {9 e^x x^2}{128}-\frac {9 x^3}{256}-\frac {9}{512} e^{2 x} (1+2 x)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.03, size = 31, normalized size = 1.48 \begin {gather*} \frac {1}{256} \left (-9 e^{2 x} x+256 x^2-18 e^x x^2-9 x^3\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 24, normalized size = 1.14
method | result | size |
default | \(x^{2}-\frac {9 x^{3}}{256}-\frac {9 x \,{\mathrm e}^{2 x}}{256}-\frac {9 \,{\mathrm e}^{x} x^{2}}{128}\) | \(24\) |
norman | \(x^{2}-\frac {9 x^{3}}{256}-\frac {9 x \,{\mathrm e}^{2 x}}{256}-\frac {9 \,{\mathrm e}^{x} x^{2}}{128}\) | \(24\) |
risch | \(x^{2}-\frac {9 x^{3}}{256}-\frac {9 x \,{\mathrm e}^{2 x}}{256}-\frac {9 \,{\mathrm e}^{x} x^{2}}{128}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 23, normalized size = 1.10 \begin {gather*} -\frac {9}{256} \, x^{3} - \frac {9}{128} \, x^{2} e^{x} + x^{2} - \frac {9}{256} \, x e^{\left (2 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 23, normalized size = 1.10 \begin {gather*} -\frac {9}{256} \, x^{3} - \frac {9}{128} \, x^{2} e^{x} + x^{2} - \frac {9}{256} \, x e^{\left (2 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 29, normalized size = 1.38 \begin {gather*} - \frac {9 x^{3}}{256} - \frac {9 x^{2} e^{x}}{128} + x^{2} - \frac {9 x e^{2 x}}{256} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 23, normalized size = 1.10 \begin {gather*} -\frac {9}{256} \, x^{3} - \frac {9}{128} \, x^{2} e^{x} + x^{2} - \frac {9}{256} \, x e^{\left (2 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.16, size = 23, normalized size = 1.10 \begin {gather*} -\frac {x\,\left (9\,{\mathrm {e}}^{2\,x}-256\,x+18\,x\,{\mathrm {e}}^x+9\,x^2\right )}{256} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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