Integrand size = 17, antiderivative size = 21 \[ \int e^{1+x} \left (1+2 x+x^2+x^3\right ) \, dx=1+e^{1+x} (1-x) \left (-5+x-x^2\right ) \]
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Time = 0.05 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.67, number of steps used = 12, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2227, 2225, 2207} \[ \int e^{1+x} \left (1+2 x+x^2+x^3\right ) \, dx=e^{x+1} x^3-2 e^{x+1} x^2+6 e^{x+1} x-5 e^{x+1} \]
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Rule 2207
Rule 2225
Rule 2227
Rubi steps \begin{align*} \text {integral}& = \int \left (e^{1+x}+2 e^{1+x} x+e^{1+x} x^2+e^{1+x} x^3\right ) \, dx \\ & = 2 \int e^{1+x} x \, dx+\int e^{1+x} \, dx+\int e^{1+x} x^2 \, dx+\int e^{1+x} x^3 \, dx \\ & = e^{1+x}+2 e^{1+x} x+e^{1+x} x^2+e^{1+x} x^3-2 \int e^{1+x} \, dx-2 \int e^{1+x} x \, dx-3 \int e^{1+x} x^2 \, dx \\ & = -e^{1+x}-2 e^{1+x} x^2+e^{1+x} x^3+2 \int e^{1+x} \, dx+6 \int e^{1+x} x \, dx \\ & = e^{1+x}+6 e^{1+x} x-2 e^{1+x} x^2+e^{1+x} x^3-6 \int e^{1+x} \, dx \\ & = -5 e^{1+x}+6 e^{1+x} x-2 e^{1+x} x^2+e^{1+x} x^3 \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int e^{1+x} \left (1+2 x+x^2+x^3\right ) \, dx=e^{1+x} \left (-5+6 x-2 x^2+x^3\right ) \]
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Time = 0.12 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90
method | result | size |
gosper | \(\left (x^{3}-2 x^{2}+6 x -5\right ) {\mathrm e}^{1+x}\) | \(19\) |
risch | \(\left (x^{3}-2 x^{2}+6 x -5\right ) {\mathrm e}^{1+x}\) | \(19\) |
norman | \(x^{3} {\mathrm e}^{1+x}+6 x \,{\mathrm e}^{1+x}-2 x^{2} {\mathrm e}^{1+x}-5 \,{\mathrm e}^{1+x}\) | \(32\) |
parallelrisch | \(x^{3} {\mathrm e}^{1+x}+6 x \,{\mathrm e}^{1+x}-2 x^{2} {\mathrm e}^{1+x}-5 \,{\mathrm e}^{1+x}\) | \(32\) |
derivativedivides | \({\mathrm e}^{1+x} \left (1+x \right )^{3}-5 \,{\mathrm e}^{1+x} \left (1+x \right )^{2}+13 \,{\mathrm e}^{1+x} \left (1+x \right )-14 \,{\mathrm e}^{1+x}\) | \(38\) |
default | \({\mathrm e}^{1+x} \left (1+x \right )^{3}-5 \,{\mathrm e}^{1+x} \left (1+x \right )^{2}+13 \,{\mathrm e}^{1+x} \left (1+x \right )-14 \,{\mathrm e}^{1+x}\) | \(38\) |
parts | \(x^{3} {\mathrm e}^{1+x}+x^{2} {\mathrm e}^{1+x}+2 x \,{\mathrm e}^{1+x}-12 \,{\mathrm e}^{1+x}+10 \,{\mathrm e}^{1+x} \left (1+x \right )-3 \,{\mathrm e}^{1+x} \left (1+x \right )^{2}\) | \(51\) |
meijerg | \({\mathrm e} \left (6-\frac {\left (-4 x^{3}+12 x^{2}-24 x +24\right ) {\mathrm e}^{x}}{4}\right )-{\mathrm e} \left (2-\frac {\left (3 x^{2}-6 x +6\right ) {\mathrm e}^{x}}{3}\right )+2 \,{\mathrm e} \left (1-\frac {\left (2-2 x \right ) {\mathrm e}^{x}}{2}\right )-\left (1-{\mathrm e}^{x}\right ) {\mathrm e}\) | \(71\) |
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Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86 \[ \int e^{1+x} \left (1+2 x+x^2+x^3\right ) \, dx={\left (x^{3} - 2 \, x^{2} + 6 \, x - 5\right )} e^{\left (x + 1\right )} \]
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Time = 0.04 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int e^{1+x} \left (1+2 x+x^2+x^3\right ) \, dx=\left (x^{3} - 2 x^{2} + 6 x - 5\right ) e^{x + 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (18) = 36\).
Time = 0.20 (sec) , antiderivative size = 63, normalized size of antiderivative = 3.00 \[ \int e^{1+x} \left (1+2 x+x^2+x^3\right ) \, dx={\left (x^{3} e - 3 \, x^{2} e + 6 \, x e - 6 \, e\right )} e^{x} + {\left (x^{2} e - 2 \, x e + 2 \, e\right )} e^{x} + 2 \, {\left (x e - e\right )} e^{x} + e^{\left (x + 1\right )} \]
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Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86 \[ \int e^{1+x} \left (1+2 x+x^2+x^3\right ) \, dx={\left (x^{3} - 2 \, x^{2} + 6 \, x - 5\right )} e^{\left (x + 1\right )} \]
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Time = 0.05 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86 \[ \int e^{1+x} \left (1+2 x+x^2+x^3\right ) \, dx={\mathrm {e}}^{x+1}\,\left (x^3-2\,x^2+6\,x-5\right ) \]
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