Optimal. Leaf size=23 \[ -5+\frac {x^4 \left (x+x^3\right )}{e^4 \left (x-x^2\right )^2} \]
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Rubi [A] time = 0.05, antiderivative size = 46, normalized size of antiderivative = 2.00, number of steps used = 3, number of rules used = 2, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {12, 2074} \begin {gather*} \frac {x^3}{e^4}+\frac {2 x^2}{e^4}+\frac {4 x}{e^4}-\frac {8}{e^4 (1-x)}+\frac {2}{e^4 (1-x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2074
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {-3 x^2+x^3-5 x^4+3 x^5}{-1+3 x-3 x^2+x^3} \, dx}{e^4}\\ &=\frac {\int \left (4-\frac {4}{(-1+x)^3}-\frac {8}{(-1+x)^2}+4 x+3 x^2\right ) \, dx}{e^4}\\ &=\frac {2}{e^4 (1-x)^2}-\frac {8}{e^4 (1-x)}+\frac {4 x}{e^4}+\frac {2 x^2}{e^4}+\frac {x^3}{e^4}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 1.09 \begin {gather*} \frac {-13+26 x-13 x^2+x^3+x^5}{e^4 (-1+x)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 29, normalized size = 1.26 \begin {gather*} \frac {{\left (x^{5} + x^{3} - 6 \, x^{2} + 12 \, x - 6\right )} e^{\left (-4\right )}}{x^{2} - 2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 27, normalized size = 1.17 \begin {gather*} {\left (x^{3} + 2 \, x^{2} + 4 \, x + \frac {2 \, {\left (4 \, x - 3\right )}}{{\left (x - 1\right )}^{2}}\right )} e^{\left (-4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 24, normalized size = 1.04
method | result | size |
gosper | \(\frac {x^{3} \left (x^{2}+1\right ) {\mathrm e}^{-4}}{x^{2}-2 x +1}\) | \(24\) |
norman | \(\frac {\left ({\mathrm e}^{-1} x^{3}+{\mathrm e}^{-1} x^{5}\right ) {\mathrm e}^{-3}}{\left (x -1\right )^{2}}\) | \(28\) |
default | \({\mathrm e}^{-4} \left (x^{3}+2 x^{2}+4 x +\frac {8}{x -1}+\frac {2}{\left (x -1\right )^{2}}\right )\) | \(32\) |
risch | \({\mathrm e}^{-4} x^{3}+2 x^{2} {\mathrm e}^{-4}+4 x \,{\mathrm e}^{-4}+\frac {{\mathrm e}^{-4} \left (8 x -6\right )}{x^{2}-2 x +1}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 32, normalized size = 1.39 \begin {gather*} {\left (x^{3} + 2 \, x^{2} + 4 \, x + \frac {2 \, {\left (4 \, x - 3\right )}}{x^{2} - 2 \, x + 1}\right )} e^{\left (-4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.18, size = 41, normalized size = 1.78 \begin {gather*} 4\,x\,{\mathrm {e}}^{-4}+\frac {8\,x-6}{{\mathrm {e}}^4\,x^2-2\,{\mathrm {e}}^4\,x+{\mathrm {e}}^4}+2\,x^2\,{\mathrm {e}}^{-4}+x^3\,{\mathrm {e}}^{-4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.17, size = 42, normalized size = 1.83 \begin {gather*} \frac {x^{3}}{e^{4}} + \frac {2 x^{2}}{e^{4}} + \frac {4 x}{e^{4}} + \frac {8 x - 6}{x^{2} e^{4} - 2 x e^{4} + e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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