Optimal. Leaf size=27 \[ \frac {3+\frac {1}{3} \left (4+x+2 x^2\right )}{5-e^3+x^2} \]
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Rubi [A] time = 0.08, antiderivative size = 46, normalized size of antiderivative = 1.70, number of steps used = 4, number of rules used = 4, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1994, 28, 1814, 8} \begin {gather*} \frac {\left (5-e^3\right ) x-2 e^6+7 e^3+15}{3 \left (5-e^3\right ) \left (x^2-e^3+5\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 28
Rule 1814
Rule 1994
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5+e^3 (-1-4 x)-6 x-x^2}{3 \left (5-e^3\right )^2+6 \left (5-e^3\right ) x^2+3 x^4} \, dx\\ &=3 \int \frac {5+e^3 (-1-4 x)-6 x-x^2}{\left (3 \left (5-e^3\right )+3 x^2\right )^2} \, dx\\ &=\frac {15+7 e^3-2 e^6+\left (5-e^3\right ) x}{3 \left (5-e^3\right ) \left (5-e^3+x^2\right )}-\frac {\int 0 \, dx}{2 \left (5-e^3\right )}\\ &=\frac {15+7 e^3-2 e^6+\left (5-e^3\right ) x}{3 \left (5-e^3\right ) \left (5-e^3+x^2\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 0.96 \begin {gather*} -\frac {-3-2 e^3-x}{3 \left (5-e^3+x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 20, normalized size = 0.74 \begin {gather*} \frac {x + 2 \, e^{3} + 3}{3 \, {\left (x^{2} - e^{3} + 5\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 21, normalized size = 0.78
method | result | size |
gosper | \(-\frac {3+x +2 \,{\mathrm e}^{3}}{3 \left (-x^{2}+{\mathrm e}^{3}-5\right )}\) | \(21\) |
default | \(-\frac {3+x +2 \,{\mathrm e}^{3}}{3 \left (-x^{2}+{\mathrm e}^{3}-5\right )}\) | \(21\) |
norman | \(\frac {-\frac {x}{3}-1-\frac {2 \,{\mathrm e}^{3}}{3}}{-x^{2}+{\mathrm e}^{3}-5}\) | \(22\) |
risch | \(\frac {-\frac {x}{3}-1-\frac {2 \,{\mathrm e}^{3}}{3}}{-x^{2}+{\mathrm e}^{3}-5}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 20, normalized size = 0.74 \begin {gather*} \frac {x + 2 \, e^{3} + 3}{3 \, {\left (x^{2} - e^{3} + 5\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 21, normalized size = 0.78 \begin {gather*} \frac {\frac {x}{3}+\frac {2\,{\mathrm {e}}^3}{3}+1}{x^2-{\mathrm {e}}^3+5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.40, size = 22, normalized size = 0.81 \begin {gather*} - \frac {- x - 2 e^{3} - 3}{3 x^{2} - 3 e^{3} + 15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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