Optimal. Leaf size=26 \[ -7+3 x-\frac {x-e^{\left .\frac {1}{3}\right /x} (4+x)}{x} \]
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Rubi [A]
time = 0.13, antiderivative size = 27, normalized size of antiderivative = 1.04, number of steps
used = 8, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {12, 14, 6874,
2243, 2240} \begin {gather*} 3 x+e^{\left .\frac {1}{3}\right /x}+\frac {4 e^{\left .\frac {1}{3}\right /x}}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2240
Rule 2243
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \frac {e^{\left .\frac {1}{3}\right /x} (-4-13 x)+9 x^3}{x^3} \, dx\\ &=\frac {1}{3} \int \left (9-\frac {e^{\left .\frac {1}{3}\right /x} (4+13 x)}{x^3}\right ) \, dx\\ &=3 x-\frac {1}{3} \int \frac {e^{\left .\frac {1}{3}\right /x} (4+13 x)}{x^3} \, dx\\ &=3 x-\frac {1}{3} \int \left (\frac {4 e^{\left .\frac {1}{3}\right /x}}{x^3}+\frac {13 e^{\left .\frac {1}{3}\right /x}}{x^2}\right ) \, dx\\ &=3 x-\frac {4}{3} \int \frac {e^{\left .\frac {1}{3}\right /x}}{x^3} \, dx-\frac {13}{3} \int \frac {e^{\left .\frac {1}{3}\right /x}}{x^2} \, dx\\ &=13 e^{\left .\frac {1}{3}\right /x}+\frac {4 e^{\left .\frac {1}{3}\right /x}}{x}+3 x+4 \int \frac {e^{\left .\frac {1}{3}\right /x}}{x^2} \, dx\\ &=e^{\left .\frac {1}{3}\right /x}+\frac {4 e^{\left .\frac {1}{3}\right /x}}{x}+3 x\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.03, size = 24, normalized size = 0.92 \begin {gather*} -\frac {1}{3} e^{\left .\frac {1}{3}\right /x} \left (-3-\frac {12}{x}\right )+3 x \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.79, size = 22, normalized size = 0.85
method | result | size |
risch | \(3 x +\frac {\left (4+x \right ) {\mathrm e}^{\frac {1}{3 x}}}{x}\) | \(18\) |
derivativedivides | \(3 x +\frac {4 \,{\mathrm e}^{\frac {1}{3 x}}}{x}+{\mathrm e}^{\frac {1}{3 x}}\) | \(22\) |
default | \(3 x +\frac {4 \,{\mathrm e}^{\frac {1}{3 x}}}{x}+{\mathrm e}^{\frac {1}{3 x}}\) | \(22\) |
norman | \(\frac {x^{2} {\mathrm e}^{\frac {1}{3 x}}+3 x^{3}+4 \,{\mathrm e}^{\frac {1}{3 x}} x}{x^{2}}\) | \(30\) |
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 = 0.28, size = 21, normalized size = 0.81 \begin {gather*} 3 \, x + 13 \, e^{\left (\frac {1}{3 \, x}\right )} - 12 \, \Gamma \left (2, -\frac {1}{3 \, x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 20, normalized size = 0.77 \begin {gather*} \frac {3 \, x^{2} + {\left (x + 4\right )} e^{\left (\frac {1}{3 \, x}\right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 14, normalized size = 0.54 \begin {gather*} 3 x + \frac {\left (x + 4\right ) e^{\frac {1}{3 x}}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 25, normalized size = 0.96 \begin {gather*} x {\left (\frac {e^{\left (\frac {1}{3 \, x}\right )}}{x} + \frac {4 \, e^{\left (\frac {1}{3 \, x}\right )}}{x^{2}} + 3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.50, size = 21, normalized size = 0.81 \begin {gather*} 3\,x+{\mathrm {e}}^{\frac {1}{3\,x}}+\frac {4\,{\mathrm {e}}^{\frac {1}{3\,x}}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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