Integrand size = 42, antiderivative size = 17 \[ \int \frac {e^{\frac {x^2}{1+x+x^2}} \left (2 x+x^2\right )}{1+2 x+3 x^2+2 x^3+x^4} \, dx=e^{\frac {x}{x+\frac {x+x^2}{x^2}}} \]
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Time = 0.21 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1607, 6820, 6838} \[ \int \frac {e^{\frac {x^2}{1+x+x^2}} \left (2 x+x^2\right )}{1+2 x+3 x^2+2 x^3+x^4} \, dx=e^{\frac {x^2}{x^2+x+1}} \]
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Rule 1607
Rule 6820
Rule 6838
Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{\frac {x^2}{1+x+x^2}} x (2+x)}{1+2 x+3 x^2+2 x^3+x^4} \, dx \\ & = \int \frac {e^{\frac {x^2}{1+x+x^2}} x (2+x)}{\left (1+x+x^2\right )^2} \, dx \\ & = e^{\frac {x^2}{1+x+x^2}} \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \frac {e^{\frac {x^2}{1+x+x^2}} \left (2 x+x^2\right )}{1+2 x+3 x^2+2 x^3+x^4} \, dx=e^{\frac {x^2}{1+x+x^2}} \]
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Time = 0.68 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82
method | result | size |
gosper | \({\mathrm e}^{\frac {x^{2}}{x^{2}+x +1}}\) | \(14\) |
risch | \({\mathrm e}^{\frac {x^{2}}{x^{2}+x +1}}\) | \(14\) |
parallelrisch | \({\mathrm e}^{\frac {x^{2}}{x^{2}+x +1}}\) | \(14\) |
norman | \(\frac {x \,{\mathrm e}^{\frac {x^{2}}{x^{2}+x +1}}+{\mathrm e}^{\frac {x^{2}}{x^{2}+x +1}} x^{2}+{\mathrm e}^{\frac {x^{2}}{x^{2}+x +1}}}{x^{2}+x +1}\) | \(56\) |
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none
Time = 0.23 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {e^{\frac {x^2}{1+x+x^2}} \left (2 x+x^2\right )}{1+2 x+3 x^2+2 x^3+x^4} \, dx=e^{\left (\frac {x^{2}}{x^{2} + x + 1}\right )} \]
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Time = 0.09 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.59 \[ \int \frac {e^{\frac {x^2}{1+x+x^2}} \left (2 x+x^2\right )}{1+2 x+3 x^2+2 x^3+x^4} \, dx=e^{\frac {x^{2}}{x^{2} + x + 1}} \]
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none
Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.41 \[ \int \frac {e^{\frac {x^2}{1+x+x^2}} \left (2 x+x^2\right )}{1+2 x+3 x^2+2 x^3+x^4} \, dx=e^{\left (-\frac {x}{x^{2} + x + 1} - \frac {1}{x^{2} + x + 1} + 1\right )} \]
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none
Time = 0.26 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {e^{\frac {x^2}{1+x+x^2}} \left (2 x+x^2\right )}{1+2 x+3 x^2+2 x^3+x^4} \, dx=e^{\left (\frac {x^{2}}{x^{2} + x + 1}\right )} \]
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Time = 0.18 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {e^{\frac {x^2}{1+x+x^2}} \left (2 x+x^2\right )}{1+2 x+3 x^2+2 x^3+x^4} \, dx={\mathrm {e}}^{\frac {x^2}{x^2+x+1}} \]
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