Optimal. Leaf size=21 \[ x (5+x) \left (-e^x+\frac {1}{4 x}+2 x\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 30, normalized size of antiderivative = 1.43, number of steps used = 10, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {12, 2196, 2194, 2176} \begin {gather*} 2 x^3-e^x x^2+10 x^2-5 e^x x+\frac {x}{4} \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 {1}{4} \int \left (1+80 x+24 x^2+e^x \left (-20-28 x-4 x^2\right )\right ) \, dx\\ &=\frac {x}{4}+10 x^2+2 x^3+\frac {1}{4} \int e^x \left (-20-28 x-4 x^2\right ) \, dx\\ &=\frac {x}{4}+10 x^2+2 x^3+\frac {1}{4} \int \left (-20 e^x-28 e^x x-4 e^x x^2\right ) \, dx\\ &=\frac {x}{4}+10 x^2+2 x^3-5 \int e^x \, dx-7 \int e^x x \, dx-\int e^x x^2 \, dx\\ &=-5 e^x+\frac {x}{4}-7 e^x x+10 x^2-e^x x^2+2 x^3+2 \int e^x x \, dx+7 \int e^x \, dx\\ &=2 e^x+\frac {x}{4}-5 e^x x+10 x^2-e^x x^2+2 x^3-2 \int e^x \, dx\\ &=\frac {x}{4}-5 e^x x+10 x^2-e^x x^2+2 x^3\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.03, size = 28, normalized size = 1.33 \begin {gather*} \frac {x}{4}+10 x^2+2 x^3-e^x \left (5 x+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 25, normalized size = 1.19 \begin {gather*} 2 \, x^{3} + 10 \, x^{2} - {\left (x^{2} + 5 \, x\right )} e^{x} + \frac {1}{4} \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.45, size = 25, normalized size = 1.19 \begin {gather*} 2 \, x^{3} + 10 \, x^{2} - {\left (x^{2} + 5 \, x\right )} e^{x} + \frac {1}{4} \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 27, normalized size = 1.29
method | result | size |
default | \(\frac {x}{4}+10 x^{2}+2 x^{3}-{\mathrm e}^{x} x^{2}-5 \,{\mathrm e}^{x} x\) | \(27\) |
norman | \(\frac {x}{4}+10 x^{2}+2 x^{3}-{\mathrm e}^{x} x^{2}-5 \,{\mathrm e}^{x} x\) | \(27\) |
risch | \(\frac {\left (-4 x^{2}-20 x \right ) {\mathrm e}^{x}}{4}+2 x^{3}+10 x^{2}+\frac {x}{4}\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 25, normalized size = 1.19 \begin {gather*} 2 \, x^{3} + 10 \, x^{2} - {\left (x^{2} + 5 \, x\right )} e^{x} + \frac {1}{4} \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.44, size = 22, normalized size = 1.05 \begin {gather*} \frac {x\,\left (40\,x-20\,{\mathrm {e}}^x-4\,x\,{\mathrm {e}}^x+8\,x^2+1\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 24, normalized size = 1.14 \begin {gather*} 2 x^{3} + 10 x^{2} + \frac {x}{4} + \left (- x^{2} - 5 x\right ) e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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