Optimal. Leaf size=30 \[ x \left (-5+\frac {e^x (11-x)}{2 (1-x)}-\frac {x^2}{9}\right ) \]
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Rubi [A] time = 0.18, antiderivative size = 36, normalized size of antiderivative = 1.20, number of steps used = 13, number of rules used = 8, integrand size = 52, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {27, 12, 6688, 2199, 2194, 2177, 2178, 2176} \begin {gather*} -\frac {x^3}{9}+\frac {e^x x}{2}-5 x-5 e^x+\frac {5 e^x}{1-x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 2176
Rule 2177
Rule 2178
Rule 2194
Rule 2199
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-30+60 x-32 x^2+4 x^3-2 x^4+e^x \left (33+27 x-33 x^2+3 x^3\right )}{6 (-1+x)^2} \, dx\\ &=\frac {1}{6} \int \frac {-30+60 x-32 x^2+4 x^3-2 x^4+e^x \left (33+27 x-33 x^2+3 x^3\right )}{(-1+x)^2} \, dx\\ &=\frac {1}{6} \int \left (-2 \left (15+x^2\right )+\frac {3 e^x \left (11+9 x-11 x^2+x^3\right )}{(-1+x)^2}\right ) \, dx\\ &=-\left (\frac {1}{3} \int \left (15+x^2\right ) \, dx\right )+\frac {1}{2} \int \frac {e^x \left (11+9 x-11 x^2+x^3\right )}{(-1+x)^2} \, dx\\ &=-5 x-\frac {x^3}{9}+\frac {1}{2} \int \left (-9 e^x+\frac {10 e^x}{(-1+x)^2}-\frac {10 e^x}{-1+x}+e^x x\right ) \, dx\\ &=-5 x-\frac {x^3}{9}+\frac {1}{2} \int e^x x \, dx-\frac {9 \int e^x \, dx}{2}+5 \int \frac {e^x}{(-1+x)^2} \, dx-5 \int \frac {e^x}{-1+x} \, dx\\ &=-\frac {9 e^x}{2}+\frac {5 e^x}{1-x}-5 x+\frac {e^x x}{2}-\frac {x^3}{9}-5 e \text {Ei}(-1+x)-\frac {\int e^x \, dx}{2}+5 \int \frac {e^x}{-1+x} \, dx\\ &=-5 e^x+\frac {5 e^x}{1-x}-5 x+\frac {e^x x}{2}-\frac {x^3}{9}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.08, size = 28, normalized size = 0.93 \begin {gather*} -5 x-\frac {x^3}{9}+\frac {1}{2} e^x \left (-10-\frac {10}{-1+x}+x\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 37, normalized size = 1.23 \begin {gather*} -\frac {2 \, x^{4} - 2 \, x^{3} + 90 \, x^{2} - 9 \, {\left (x^{2} - 11 \, x\right )} e^{x} - 90 \, x}{18 \, {\left (x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 38, normalized size = 1.27 \begin {gather*} -\frac {2 \, x^{4} - 2 \, x^{3} - 9 \, x^{2} e^{x} + 90 \, x^{2} + 99 \, x e^{x} - 90 \, x}{18 \, {\left (x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.42, size = 23, normalized size = 0.77
method | result | size |
risch | \(-\frac {x^{3}}{9}-5 x +\frac {x \left (x -11\right ) {\mathrm e}^{x}}{2 x -2}\) | \(23\) |
default | \(-5 x -\frac {x^{3}}{9}-\frac {5 \,{\mathrm e}^{x}}{x -1}-5 \,{\mathrm e}^{x}+\frac {{\mathrm e}^{x} x}{2}\) | \(28\) |
norman | \(\frac {-5 x^{2}+\frac {x^{3}}{9}-\frac {x^{4}}{9}-\frac {11 \,{\mathrm e}^{x} x}{2}+\frac {{\mathrm e}^{x} x^{2}}{2}+5}{x -1}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {1}{9} \, x^{3} - 5 \, x + \frac {{\left (x^{2} - 11 \, x\right )} e^{x}}{2 \, {\left (x - 1\right )}} - \frac {11 \, e E_{2}\left (-x + 1\right )}{2 \, {\left (x - 1\right )}} - \frac {11}{2} \, \int \frac {e^{x}}{x^{2} - 2 \, x + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 27, normalized size = 0.90 \begin {gather*} x\,\left (\frac {{\mathrm {e}}^x}{2}-5\right )-5\,{\mathrm {e}}^x-\frac {5\,{\mathrm {e}}^x}{x-1}-\frac {x^3}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 22, normalized size = 0.73 \begin {gather*} - \frac {x^{3}}{9} - 5 x + \frac {\left (x^{2} - 11 x\right ) e^{x}}{2 x - 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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