Optimal. Leaf size=15 \[ x \left (e^{1+\frac {20}{9 x^2}}+x\right ) \]
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Rubi [A]
time = 0.06, antiderivative size = 17, normalized size of antiderivative = 1.13, number of steps
used = 4, number of rules used = 3, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {12, 14, 2326}
\begin {gather*} x^2+e^{\frac {20}{9 x^2}+1} x \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2326
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{9} \int \frac {18 x^3+e^{\frac {20+9 x^2}{9 x^2}} \left (-40+9 x^2\right )}{x^2} \, dx\\ &=\frac {1}{9} \int \left (18 x+\frac {e^{1+\frac {20}{9 x^2}} \left (-40+9 x^2\right )}{x^2}\right ) \, dx\\ &=x^2+\frac {1}{9} \int \frac {e^{1+\frac {20}{9 x^2}} \left (-40+9 x^2\right )}{x^2} \, dx\\ &=e^{1+\frac {20}{9 x^2}} x+x^2\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.02, size = 17, normalized size = 1.13 \begin {gather*} e^{1+\frac {20}{9 x^2}} x+x^2 \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.04, size = 59, normalized size = 3.93
method | result | size |
risch | \(x^{2}+x \,{\mathrm e}^{\frac {9 x^{2}+20}{9 x^{2}}}\) | \(20\) |
norman | \(\frac {x^{3}+{\mathrm e}^{\frac {9 x^{2}+20}{9 x^{2}}} x^{2}}{x}\) | \(26\) |
derivativedivides | \(x^{2}-\frac {2 i {\mathrm e} \sqrt {\pi }\, \sqrt {5}\, \erf \left (\frac {2 i \sqrt {5}}{3 x}\right )}{3}-{\mathrm e} \left (-{\mathrm e}^{\frac {20}{9 x^{2}}} x -\frac {2 i \sqrt {\pi }\, \sqrt {5}\, \erf \left (\frac {2 i \sqrt {5}}{3 x}\right )}{3}\right )\) | \(59\) |
default | \(x^{2}-\frac {2 i {\mathrm e} \sqrt {\pi }\, \sqrt {5}\, \erf \left (\frac {2 i \sqrt {5}}{3 x}\right )}{3}-{\mathrm e} \left (-{\mathrm e}^{\frac {20}{9 x^{2}}} x -\frac {2 i \sqrt {\pi }\, \sqrt {5}\, \erf \left (\frac {2 i \sqrt {5}}{3 x}\right )}{3}\right )\) | \(59\) |
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 = 61, normalized size = 4.07 \begin {gather*} \frac {1}{3} \, \sqrt {5} x \sqrt {-\frac {1}{x^{2}}} e \Gamma \left (-\frac {1}{2}, -\frac {20}{9 \, x^{2}}\right ) + x^{2} + \frac {2 \, \sqrt {5} \sqrt {\pi } {\left (\operatorname {erf}\left (\frac {2}{3} \, \sqrt {5} \sqrt {-\frac {1}{x^{2}}}\right ) - 1\right )} e}{3 \, x \sqrt {-\frac {1}{x^{2}}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 19, normalized size = 1.27 \begin {gather*} x^{2} + x e^{\left (\frac {9 \, x^{2} + 20}{9 \, x^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 15, normalized size = 1.00 \begin {gather*} x^{2} + x e^{\frac {x^{2} + \frac {20}{9}}{x^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 19, normalized size = 1.27 \begin {gather*} x^{2} + x e^{\left (\frac {9 \, x^{2} + 20}{9 \, x^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 8.07, size = 12, normalized size = 0.80 \begin {gather*} x\,\left (x+{\mathrm {e}}^{\frac {20}{9\,x^2}+1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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