Optimal. Leaf size=15 \[ \frac {2 e^x x^4}{27 (-1+x)} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(50\) vs. \(2(15)=30\).
time = 0.13, antiderivative size = 50, normalized size of antiderivative = 3.33, number of steps
used = 18, number of rules used = 8, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {27, 12, 1608,
2230, 2225, 2208, 2209, 2207} \begin {gather*} \frac {2 e^x x^3}{27}+\frac {2 e^x x^2}{27}+\frac {2 e^x x}{27}+\frac {2 e^x}{27}-\frac {2 e^x}{27 (1-x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 1608
Rule 2207
Rule 2208
Rule 2209
Rule 2225
Rule 2230
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^x \left (-8 x^3+4 x^4+2 x^5\right )}{27 (-1+x)^2} \, dx\\ &=\frac {1}{27} \int \frac {e^x \left (-8 x^3+4 x^4+2 x^5\right )}{(-1+x)^2} \, dx\\ &=\frac {1}{27} \int \frac {e^x x^3 \left (-8+4 x+2 x^2\right )}{(-1+x)^2} \, dx\\ &=\frac {1}{27} \int \left (4 e^x-\frac {2 e^x}{(-1+x)^2}+\frac {2 e^x}{-1+x}+6 e^x x+8 e^x x^2+2 e^x x^3\right ) \, dx\\ &=-\left (\frac {2}{27} \int \frac {e^x}{(-1+x)^2} \, dx\right )+\frac {2}{27} \int \frac {e^x}{-1+x} \, dx+\frac {2}{27} \int e^x x^3 \, dx+\frac {4 \int e^x \, dx}{27}+\frac {2}{9} \int e^x x \, dx+\frac {8}{27} \int e^x x^2 \, dx\\ &=\frac {4 e^x}{27}-\frac {2 e^x}{27 (1-x)}+\frac {2 e^x x}{9}+\frac {8 e^x x^2}{27}+\frac {2 e^x x^3}{27}+\frac {2}{27} e \text {Ei}(-1+x)-\frac {2}{27} \int \frac {e^x}{-1+x} \, dx-\frac {2 \int e^x \, dx}{9}-\frac {2}{9} \int e^x x^2 \, dx-\frac {16}{27} \int e^x x \, dx\\ &=-\frac {2 e^x}{27}-\frac {2 e^x}{27 (1-x)}-\frac {10 e^x x}{27}+\frac {2 e^x x^2}{27}+\frac {2 e^x x^3}{27}+\frac {4}{9} \int e^x x \, dx+\frac {16 \int e^x \, dx}{27}\\ &=\frac {14 e^x}{27}-\frac {2 e^x}{27 (1-x)}+\frac {2 e^x x}{27}+\frac {2 e^x x^2}{27}+\frac {2 e^x x^3}{27}-\frac {4 \int e^x \, dx}{9}\\ &=\frac {2 e^x}{27}-\frac {2 e^x}{27 (1-x)}+\frac {2 e^x x}{27}+\frac {2 e^x x^2}{27}+\frac {2 e^x x^3}{27}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.08, size = 15, normalized size = 1.00 \begin {gather*} \frac {2 e^x x^4}{27 (-1+x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(33\) vs.
\(2(12)=24\).
time = 1.77, size = 34, normalized size = 2.27
method | result | size |
gosper | \(\frac {2 x^{4} {\mathrm e}^{x}}{27 \left (x -1\right )}\) | \(13\) |
norman | \(\frac {2 x^{4} {\mathrm e}^{x}}{27 \left (x -1\right )}\) | \(13\) |
risch | \(\frac {2 x^{4} {\mathrm e}^{x}}{27 \left (x -1\right )}\) | \(13\) |
default | \(\frac {2 \,{\mathrm e}^{x} x^{3}}{27}+\frac {2 \,{\mathrm e}^{x} x^{2}}{27}+\frac {2 \,{\mathrm e}^{x} x}{27}+\frac {2 \,{\mathrm e}^{x}}{27}+\frac {2 \,{\mathrm e}^{x}}{27 \left (x -1\right )}\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 12, normalized size = 0.80 \begin {gather*} \frac {2 \, x^{4} e^{x}}{27 \, {\left (x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 12, normalized size = 0.80 \begin {gather*} \frac {2 \, x^{4} e^{x}}{27 \, {\left (x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 12, normalized size = 0.80 \begin {gather*} \frac {2 x^{4} e^{x}}{27 x - 27} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 12, normalized size = 0.80 \begin {gather*} \frac {2 \, x^{4} e^{x}}{27 \, {\left (x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.01, size = 14, normalized size = 0.93 \begin {gather*} \frac {2\,x^4\,{\mathrm {e}}^x}{27\,\left (x-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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