Optimal. Leaf size=21 \[ \frac {3+x}{e^8}-e^x \left (-4-x+x^2\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 24, normalized size of antiderivative = 1.14, number of steps used = 10, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {12, 2196, 2194, 2176} \begin {gather*} -e^x x^2+e^x x+\frac {x}{e^8}+4 e^x \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2176
Rule 2194
Rule 2196
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \left (1+e^{8+x} \left (5-x-x^2\right )\right ) \, dx}{e^8}\\ &=\frac {x}{e^8}+\frac {\int e^{8+x} \left (5-x-x^2\right ) \, dx}{e^8}\\ &=\frac {x}{e^8}+\frac {\int \left (5 e^{8+x}-e^{8+x} x-e^{8+x} x^2\right ) \, dx}{e^8}\\ &=\frac {x}{e^8}-\frac {\int e^{8+x} x \, dx}{e^8}-\frac {\int e^{8+x} x^2 \, dx}{e^8}+\frac {5 \int e^{8+x} \, dx}{e^8}\\ &=5 e^x+\frac {x}{e^8}-e^x x-e^x x^2+\frac {\int e^{8+x} \, dx}{e^8}+\frac {2 \int e^{8+x} x \, dx}{e^8}\\ &=6 e^x+\frac {x}{e^8}+e^x x-e^x x^2-\frac {2 \int e^{8+x} \, dx}{e^8}\\ &=4 e^x+\frac {x}{e^8}+e^x x-e^x x^2\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.03, size = 19, normalized size = 0.90 \begin {gather*} \frac {x}{e^8}-e^x \left (-4-x+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.27, size = 21, normalized size = 1.00 \begin {gather*} -{\left ({\left (x^{2} - x - 4\right )} e^{\left (x + 8\right )} - x\right )} e^{\left (-8\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 21, normalized size = 1.00 \begin {gather*} -{\left ({\left (x^{2} - x - 4\right )} e^{\left (x + 8\right )} - x\right )} e^{\left (-8\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 27, normalized size = 1.29
method | result | size |
risch | \({\mathrm e}^{-8} x +\left (-x^{2} {\mathrm e}^{8}+x \,{\mathrm e}^{8}+4 \,{\mathrm e}^{8}\right ) {\mathrm e}^{-8+x}\) | \(27\) |
default | \({\mathrm e}^{-8} \left (x +{\mathrm e}^{8} \left ({\mathrm e}^{x} x +4 \,{\mathrm e}^{x}-{\mathrm e}^{x} x^{2}\right )\right )\) | \(29\) |
norman | \(\left ({\mathrm e}^{-1} x +{\mathrm e}^{7} x \,{\mathrm e}^{x}+4 \,{\mathrm e}^{7} {\mathrm e}^{x}-{\mathrm e}^{7} x^{2} {\mathrm e}^{x}\right ) {\mathrm e}^{-7}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.35, size = 45, normalized size = 2.14 \begin {gather*} -{\left ({\left (x^{2} e^{8} - 2 \, x e^{8} + 2 \, e^{8}\right )} e^{x} + {\left (x e^{8} - e^{8}\right )} e^{x} - x - 5 \, e^{\left (x + 8\right )}\right )} e^{\left (-8\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 20, normalized size = 0.95 \begin {gather*} 4\,{\mathrm {e}}^x-x^2\,{\mathrm {e}}^x+x\,{\mathrm {e}}^{-8}+x\,{\mathrm {e}}^x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 14, normalized size = 0.67 \begin {gather*} \frac {x}{e^{8}} + \left (- x^{2} + x + 4\right ) e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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