Integrand size = 37, antiderivative size = 15 \[ \int \frac {18 x^3+e^{\frac {20+9 x^2}{9 x^2}} \left (-40+9 x^2\right )}{9 x^2} \, dx=x \left (e^{1+\frac {20}{9 x^2}}+x\right ) \]
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Time = 0.04 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {12, 14, 2326} \[ \int \frac {18 x^3+e^{\frac {20+9 x^2}{9 x^2}} \left (-40+9 x^2\right )}{9 x^2} \, dx=x^2+e^{\frac {20}{9 x^2}+1} x \]
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Rule 12
Rule 14
Rule 2326
Rubi steps \begin{align*} \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{align*}
Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \frac {18 x^3+e^{\frac {20+9 x^2}{9 x^2}} \left (-40+9 x^2\right )}{9 x^2} \, dx=e^{1+\frac {20}{9 x^2}} x+x^2 \]
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Time = 0.66 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.33
method | result | size |
risch | \(x^{2}+x \,{\mathrm e}^{\frac {9 x^{2}+20}{9 x^{2}}}\) | \(20\) |
parallelrisch | \(x^{2}+x \,{\mathrm e}^{\frac {9 x^{2}+20}{9 x^{2}}}\) | \(20\) |
norman | \(\frac {x^{3}+{\mathrm e}^{\frac {9 x^{2}+20}{9 x^{2}}} x^{2}}{x}\) | \(26\) |
derivativedivides | \(x^{2}-\frac {2 i {\mathrm e} \sqrt {\pi }\, \sqrt {5}\, \operatorname {erf}\left (\frac {2 i \sqrt {5}}{3 x}\right )}{3}-{\mathrm e} \left (-{\mathrm e}^{\frac {20}{9 x^{2}}} x -\frac {2 i \sqrt {\pi }\, \sqrt {5}\, \operatorname {erf}\left (\frac {2 i \sqrt {5}}{3 x}\right )}{3}\right )\) | \(59\) |
default | \(x^{2}-\frac {2 i {\mathrm e} \sqrt {\pi }\, \sqrt {5}\, \operatorname {erf}\left (\frac {2 i \sqrt {5}}{3 x}\right )}{3}-{\mathrm e} \left (-{\mathrm e}^{\frac {20}{9 x^{2}}} x -\frac {2 i \sqrt {\pi }\, \sqrt {5}\, \operatorname {erf}\left (\frac {2 i \sqrt {5}}{3 x}\right )}{3}\right )\) | \(59\) |
parts | \(x^{2}-\frac {2 i {\mathrm e} \sqrt {\pi }\, \sqrt {5}\, \operatorname {erf}\left (\frac {2 i \sqrt {5}}{3 x}\right )}{3}-{\mathrm e} \left (-{\mathrm e}^{\frac {20}{9 x^{2}}} x -\frac {2 i \sqrt {\pi }\, \sqrt {5}\, \operatorname {erf}\left (\frac {2 i \sqrt {5}}{3 x}\right )}{3}\right )\) | \(59\) |
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Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.27 \[ \int \frac {18 x^3+e^{\frac {20+9 x^2}{9 x^2}} \left (-40+9 x^2\right )}{9 x^2} \, dx=x^{2} + x e^{\left (\frac {9 \, x^{2} + 20}{9 \, x^{2}}\right )} \]
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Time = 0.07 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {18 x^3+e^{\frac {20+9 x^2}{9 x^2}} \left (-40+9 x^2\right )}{9 x^2} \, dx=x^{2} + x e^{\frac {x^{2} + \frac {20}{9}}{x^{2}}} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.22 (sec) , antiderivative size = 61, normalized size of antiderivative = 4.07 \[ \int \frac {18 x^3+e^{\frac {20+9 x^2}{9 x^2}} \left (-40+9 x^2\right )}{9 x^2} \, dx=\frac {1}{3} \, \sqrt {5} x \sqrt {-\frac {1}{x^{2}}} e \Gamma \left (-\frac {1}{2}, -\frac {20}{9 \, x^{2}}\right ) + x^{2} + \frac {2 \, \sqrt {5} \sqrt {\pi } {\left (\operatorname {erf}\left (\frac {2}{3} \, \sqrt {5} \sqrt {-\frac {1}{x^{2}}}\right ) - 1\right )} e}{3 \, x \sqrt {-\frac {1}{x^{2}}}} \]
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Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.27 \[ \int \frac {18 x^3+e^{\frac {20+9 x^2}{9 x^2}} \left (-40+9 x^2\right )}{9 x^2} \, dx=x^{2} + x e^{\left (\frac {9 \, x^{2} + 20}{9 \, x^{2}}\right )} \]
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Time = 15.31 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80 \[ \int \frac {18 x^3+e^{\frac {20+9 x^2}{9 x^2}} \left (-40+9 x^2\right )}{9 x^2} \, dx=x\,\left (x+{\mathrm {e}}^{\frac {20}{9\,x^2}+1}\right ) \]
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