Optimal. Leaf size=21 \[ \frac {2 x}{5 e^2 \left (x-x \left (1+x+x^2\right )\right )} \]
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Rubi [A] time = 0.02, antiderivative size = 15, normalized size of antiderivative = 0.71, number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {12, 1594, 27, 74} \begin {gather*} -\frac {2}{5 e^2 x (x+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 74
Rule 1594
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {2+4 x}{5 x^2+10 x^3+5 x^4} \, dx}{e^2}\\ &=\frac {\int \frac {2+4 x}{x^2 \left (5+10 x+5 x^2\right )} \, dx}{e^2}\\ &=\frac {\int \frac {2+4 x}{5 x^2 (1+x)^2} \, dx}{e^2}\\ &=\frac {\int \frac {2+4 x}{x^2 (1+x)^2} \, dx}{5 e^2}\\ &=-\frac {2}{5 e^2 x (1+x)}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.01, size = 18, normalized size = 0.86 \begin {gather*} \frac {2 \left (-\frac {1}{x}+\frac {1}{1+x}\right )}{5 e^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 11, normalized size = 0.52 \begin {gather*} -\frac {2 \, e^{\left (-2\right )}}{5 \, {\left (x^{2} + x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 11, normalized size = 0.52 \begin {gather*} -\frac {2 \, e^{\left (-2\right )}}{5 \, {\left (x^{2} + x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 13, normalized size = 0.62
method | result | size |
risch | \(-\frac {2 \,{\mathrm e}^{-2}}{5 x \left (x +1\right )}\) | \(13\) |
gosper | \(-\frac {2 \,{\mathrm e}^{-2}}{5 x \left (x +1\right )}\) | \(15\) |
norman | \(-\frac {2 \,{\mathrm e}^{-2}}{5 x \left (x +1\right )}\) | \(15\) |
default | \(\frac {2 \,{\mathrm e}^{-2} \left (\frac {1}{x +1}-\frac {1}{x}\right )}{5}\) | \(18\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 11, normalized size = 0.52 \begin {gather*} -\frac {2 \, e^{\left (-2\right )}}{5 \, {\left (x^{2} + x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 12, normalized size = 0.57 \begin {gather*} -\frac {2\,{\mathrm {e}}^{-2}}{5\,x\,\left (x+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 17, normalized size = 0.81 \begin {gather*} - \frac {2}{5 x^{2} e^{2} + 5 x e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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