Integrand size = 17, antiderivative size = 17 \[ \int \frac {1}{5} \left (5 x^3+e^x (1+x)\right ) \, dx=2+\frac {e^x x}{5}+\frac {x^4}{4} \]
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Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.47, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {12, 2207, 2225} \[ \int \frac {1}{5} \left (5 x^3+e^x (1+x)\right ) \, dx=\frac {x^4}{4}-\frac {e^x}{5}+\frac {1}{5} e^x (x+1) \]
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Rule 12
Rule 2207
Rule 2225
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} \int \left (5 x^3+e^x (1+x)\right ) \, dx \\ & = \frac {x^4}{4}+\frac {1}{5} \int e^x (1+x) \, dx \\ & = \frac {x^4}{4}+\frac {1}{5} e^x (1+x)-\frac {\int e^x \, dx}{5} \\ & = -\frac {e^x}{5}+\frac {x^4}{4}+\frac {1}{5} e^x (1+x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94 \[ \int \frac {1}{5} \left (5 x^3+e^x (1+x)\right ) \, dx=\frac {e^x x}{5}+\frac {x^4}{4} \]
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Time = 0.15 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.71
method | result | size |
default | \(\frac {{\mathrm e}^{x} x}{5}+\frac {x^{4}}{4}\) | \(12\) |
norman | \(\frac {{\mathrm e}^{x} x}{5}+\frac {x^{4}}{4}\) | \(12\) |
risch | \(\frac {{\mathrm e}^{x} x}{5}+\frac {x^{4}}{4}\) | \(12\) |
parallelrisch | \(\frac {{\mathrm e}^{x} x}{5}+\frac {x^{4}}{4}\) | \(12\) |
parts | \(\frac {{\mathrm e}^{x} x}{5}+\frac {x^{4}}{4}\) | \(12\) |
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none
Time = 0.26 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.65 \[ \int \frac {1}{5} \left (5 x^3+e^x (1+x)\right ) \, dx=\frac {1}{4} \, x^{4} + \frac {1}{5} \, x e^{x} \]
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Time = 0.04 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.59 \[ \int \frac {1}{5} \left (5 x^3+e^x (1+x)\right ) \, dx=\frac {x^{4}}{4} + \frac {x e^{x}}{5} \]
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none
Time = 0.19 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {1}{5} \left (5 x^3+e^x (1+x)\right ) \, dx=\frac {1}{4} \, x^{4} + \frac {1}{5} \, {\left (x - 1\right )} e^{x} + \frac {1}{5} \, e^{x} \]
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none
Time = 0.25 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.65 \[ \int \frac {1}{5} \left (5 x^3+e^x (1+x)\right ) \, dx=\frac {1}{4} \, x^{4} + \frac {1}{5} \, x e^{x} \]
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Time = 9.68 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.65 \[ \int \frac {1}{5} \left (5 x^3+e^x (1+x)\right ) \, dx=\frac {x\,{\mathrm {e}}^x}{5}+\frac {x^4}{4} \]
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