Optimal. Leaf size=25 \[ e^{\frac {x}{2 (1+2 x) \left (x+\frac {3+x}{x}\right )}} \]
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Rubi [A] time = 0.48, antiderivative size = 26, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 4, integrand size = 70, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {1594, 6688, 12, 6706} \begin {gather*} e^{\frac {x^2}{2 \left (2 x^3+3 x^2+7 x+3\right )}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 1594
Rule 6688
Rule 6706
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {x^2}{6+14 x+6 x^2+4 x^3}} x \left (6+7 x-2 x^3\right )}{18+84 x+134 x^2+108 x^3+74 x^4+24 x^5+8 x^6} \, dx\\ &=\int \frac {e^{\frac {x^2}{6+14 x+6 x^2+4 x^3}} x \left (6+7 x-2 x^3\right )}{2 \left (3+7 x+3 x^2+2 x^3\right )^2} \, dx\\ &=\frac {1}{2} \int \frac {e^{\frac {x^2}{6+14 x+6 x^2+4 x^3}} x \left (6+7 x-2 x^3\right )}{\left (3+7 x+3 x^2+2 x^3\right )^2} \, dx\\ &=e^{\frac {x^2}{2 \left (3+7 x+3 x^2+2 x^3\right )}}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.39, size = 23, normalized size = 0.92 \begin {gather*} e^{\frac {x^2}{6+14 x+6 x^2+4 x^3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 23, normalized size = 0.92 \begin {gather*} e^{\left (\frac {x^{2}}{2 \, {\left (2 \, x^{3} + 3 \, x^{2} + 7 \, x + 3\right )}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 23, normalized size = 0.92 \begin {gather*} e^{\left (\frac {x^{2}}{2 \, {\left (2 \, x^{3} + 3 \, x^{2} + 7 \, x + 3\right )}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 22, normalized size = 0.88
method | result | size |
risch | \({\mathrm e}^{\frac {x^{2}}{2 \left (2 x +1\right ) \left (x^{2}+x +3\right )}}\) | \(22\) |
gosper | \({\mathrm e}^{\frac {x^{2}}{4 x^{3}+6 x^{2}+14 x +6}}\) | \(24\) |
norman | \(\frac {7 x \,{\mathrm e}^{\frac {x^{2}}{4 x^{3}+6 x^{2}+14 x +6}}+3 x^{2} {\mathrm e}^{\frac {x^{2}}{4 x^{3}+6 x^{2}+14 x +6}}+2 x^{3} {\mathrm e}^{\frac {x^{2}}{4 x^{3}+6 x^{2}+14 x +6}}+3 \,{\mathrm e}^{\frac {x^{2}}{4 x^{3}+6 x^{2}+14 x +6}}}{2 x^{3}+3 x^{2}+7 x +3}\) | \(123\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 32, normalized size = 1.28 \begin {gather*} e^{\left (\frac {5 \, x}{22 \, {\left (x^{2} + x + 3\right )}} - \frac {3}{22 \, {\left (x^{2} + x + 3\right )}} + \frac {1}{22 \, {\left (2 \, x + 1\right )}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.02, size = 23, normalized size = 0.92 \begin {gather*} {\mathrm {e}}^{\frac {1}{22\,\left (2\,x+1\right )}+\frac {\frac {5\,x}{22}-\frac {3}{22}}{x^2+x+3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.25, size = 19, normalized size = 0.76 \begin {gather*} e^{\frac {x^{2}}{4 x^{3} + 6 x^{2} + 14 x + 6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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