Optimal. Leaf size=21 \[ e^{-4 \left (-\frac {1}{3} (1-x)^2+x\right )} (4+x) \]
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Rubi [A]
time = 0.26, antiderivative size = 41, normalized size of antiderivative = 1.95, number of steps
used = 17, number of rules used = 8, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {12, 6874,
2267, 2266, 2235, 2276, 2272, 2273} \begin {gather*} e^{\frac {4 x^2}{3}-\frac {20 x}{3}+\frac {4}{3}} x+4 e^{\frac {4 x^2}{3}-\frac {20 x}{3}+\frac {4}{3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2235
Rule 2266
Rule 2267
Rule 2272
Rule 2273
Rule 2276
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int e^{\frac {1}{3} \left (4-20 x+4 x^2\right )} \left (-77+12 x+8 x^2\right ) \, dx\\ &=\frac {1}{3} \int \left (-77 e^{\frac {1}{3} \left (4-20 x+4 x^2\right )}+12 e^{\frac {1}{3} \left (4-20 x+4 x^2\right )} x+8 e^{\frac {1}{3} \left (4-20 x+4 x^2\right )} x^2\right ) \, dx\\ &=\frac {8}{3} \int e^{\frac {1}{3} \left (4-20 x+4 x^2\right )} x^2 \, dx+4 \int e^{\frac {1}{3} \left (4-20 x+4 x^2\right )} x \, dx-\frac {77}{3} \int e^{\frac {1}{3} \left (4-20 x+4 x^2\right )} \, dx\\ &=\frac {8}{3} \int e^{\frac {4}{3}-\frac {20 x}{3}+\frac {4 x^2}{3}} x^2 \, dx+4 \int e^{\frac {4}{3}-\frac {20 x}{3}+\frac {4 x^2}{3}} x \, dx-\frac {77}{3} \int e^{\frac {4}{3}-\frac {20 x}{3}+\frac {4 x^2}{3}} \, dx\\ &=\frac {3}{2} e^{\frac {4}{3}-\frac {20 x}{3}+\frac {4 x^2}{3}}+e^{\frac {4}{3}-\frac {20 x}{3}+\frac {4 x^2}{3}} x+\frac {20}{3} \int e^{\frac {4}{3}-\frac {20 x}{3}+\frac {4 x^2}{3}} x \, dx+10 \int e^{\frac {4}{3}-\frac {20 x}{3}+\frac {4 x^2}{3}} \, dx-\frac {77 \int e^{\frac {3}{16} \left (-\frac {20}{3}+\frac {8 x}{3}\right )^2} \, dx}{3 e^7}-\int e^{\frac {4}{3}-\frac {20 x}{3}+\frac {4 x^2}{3}} \, dx\\ &=4 e^{\frac {4}{3}-\frac {20 x}{3}+\frac {4 x^2}{3}}+e^{\frac {4}{3}-\frac {20 x}{3}+\frac {4 x^2}{3}} x+\frac {77 \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {5-2 x}{\sqrt {3}}\right )}{4 e^7}+\frac {50}{3} \int e^{\frac {4}{3}-\frac {20 x}{3}+\frac {4 x^2}{3}} \, dx-\frac {\int e^{\frac {3}{16} \left (-\frac {20}{3}+\frac {8 x}{3}\right )^2} \, dx}{e^7}+\frac {10 \int e^{\frac {3}{16} \left (-\frac {20}{3}+\frac {8 x}{3}\right )^2} \, dx}{e^7}\\ &=4 e^{\frac {4}{3}-\frac {20 x}{3}+\frac {4 x^2}{3}}+e^{\frac {4}{3}-\frac {20 x}{3}+\frac {4 x^2}{3}} x+\frac {77 \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {5-2 x}{\sqrt {3}}\right )}{4 e^7}-\frac {9 \sqrt {3 \pi } \text {erfi}\left (\frac {5-2 x}{\sqrt {3}}\right )}{4 e^7}+\frac {50 \int e^{\frac {3}{16} \left (-\frac {20}{3}+\frac {8 x}{3}\right )^2} \, dx}{3 e^7}\\ &=4 e^{\frac {4}{3}-\frac {20 x}{3}+\frac {4 x^2}{3}}+e^{\frac {4}{3}-\frac {20 x}{3}+\frac {4 x^2}{3}} x\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.15, size = 18, normalized size = 0.86 \begin {gather*} e^{\frac {4}{3} \left (1-5 x+x^2\right )} (4+x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 28, normalized size = 1.33
method | result | size |
gosper | \(\left (4+x \right ) {\mathrm e}^{\frac {4}{3} x^{2}-\frac {20}{3} x +\frac {4}{3}}\) | \(18\) |
norman | \(\left (4+x \right ) {\mathrm e}^{\frac {4}{3} x^{2}-\frac {20}{3} x +\frac {4}{3}}\) | \(18\) |
risch | \(\frac {\left (3 x +12\right ) {\mathrm e}^{\frac {4}{3} x^{2}-\frac {20}{3} x +\frac {4}{3}}}{3}\) | \(19\) |
default | \(4 \,{\mathrm e}^{\frac {4}{3} x^{2}-\frac {20}{3} x +\frac {4}{3}}+x \,{\mathrm e}^{\frac {4}{3} x^{2}-\frac {20}{3} x +\frac {4}{3}}\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.52, size = 196, normalized size = 9.33 \begin {gather*} \frac {77}{12} i \, \sqrt {3} \sqrt {\pi } \operatorname {erf}\left (\frac {2}{3} i \, \sqrt {3} x - \frac {5}{3} i \, \sqrt {3}\right ) e^{\left (-7\right )} - \frac {1}{6} \, \sqrt {3} {\left (\frac {3 \, \sqrt {3} \sqrt {\frac {1}{3}} {\left (2 \, x - 5\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {1}{3} \, {\left (2 \, x - 5\right )}^{2}\right )}{\left (-{\left (2 \, x - 5\right )}^{2}\right )^{\frac {3}{2}}} - \frac {25 \, \sqrt {3} \sqrt {\frac {1}{3}} \sqrt {\pi } {\left (2 \, x - 5\right )} {\left (\operatorname {erf}\left (\sqrt {\frac {1}{3}} \sqrt {-{\left (2 \, x - 5\right )}^{2}}\right ) - 1\right )}}{\sqrt {-{\left (2 \, x - 5\right )}^{2}}} - 10 \, \sqrt {3} e^{\left (\frac {1}{3} \, {\left (2 \, x - 5\right )}^{2}\right )}\right )} e^{\left (-7\right )} + \frac {1}{2} \, \sqrt {3} {\left (\frac {5 \, \sqrt {3} \sqrt {\frac {1}{3}} \sqrt {\pi } {\left (2 \, x - 5\right )} {\left (\operatorname {erf}\left (\sqrt {\frac {1}{3}} \sqrt {-{\left (2 \, x - 5\right )}^{2}}\right ) - 1\right )}}{\sqrt {-{\left (2 \, x - 5\right )}^{2}}} + \sqrt {3} e^{\left (\frac {1}{3} \, {\left (2 \, x - 5\right )}^{2}\right )}\right )} e^{\left (-7\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 15, normalized size = 0.71 \begin {gather*} {\left (x + 4\right )} e^{\left (\frac {4}{3} \, x^{2} - \frac {20}{3} \, x + \frac {4}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 19, normalized size = 0.90 \begin {gather*} \left (x + 4\right ) e^{\frac {4 x^{2}}{3} - \frac {20 x}{3} + \frac {4}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 15, normalized size = 0.71 \begin {gather*} {\left (x + 4\right )} e^{\left (\frac {4}{3} \, x^{2} - \frac {20}{3} \, x + \frac {4}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.41, size = 15, normalized size = 0.71 \begin {gather*} {\mathrm {e}}^{\frac {4\,x^2}{3}-\frac {20\,x}{3}+\frac {4}{3}}\,\left (x+4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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