Integrand size = 34, antiderivative size = 22 \[ \int \left (2 e^{5+2 x}+e^x \left (-1-2 x-7 x^2-6 x^3-x^4\right )\right ) \, dx=-6+e^x \left (-1+e^{5+x}-\left (x+x^2\right )^2\right ) \]
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Time = 0.07 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.68, number of steps used = 19, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {2225, 2227, 2207} \[ \int \left (2 e^{5+2 x}+e^x \left (-1-2 x-7 x^2-6 x^3-x^4\right )\right ) \, dx=-e^x x^4-2 e^x x^3-e^x x^2-e^x+e^{2 x+5} \]
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Rule 2207
Rule 2225
Rule 2227
Rubi steps \begin{align*} \text {integral}& = 2 \int e^{5+2 x} \, dx+\int e^x \left (-1-2 x-7 x^2-6 x^3-x^4\right ) \, dx \\ & = e^{5+2 x}+\int \left (-e^x-2 e^x x-7 e^x x^2-6 e^x x^3-e^x x^4\right ) \, dx \\ & = e^{5+2 x}-2 \int e^x x \, dx-6 \int e^x x^3 \, dx-7 \int e^x x^2 \, dx-\int e^x \, dx-\int e^x x^4 \, dx \\ & = -e^x+e^{5+2 x}-2 e^x x-7 e^x x^2-6 e^x x^3-e^x x^4+2 \int e^x \, dx+4 \int e^x x^3 \, dx+14 \int e^x x \, dx+18 \int e^x x^2 \, dx \\ & = e^x+e^{5+2 x}+12 e^x x+11 e^x x^2-2 e^x x^3-e^x x^4-12 \int e^x x^2 \, dx-14 \int e^x \, dx-36 \int e^x x \, dx \\ & = -13 e^x+e^{5+2 x}-24 e^x x-e^x x^2-2 e^x x^3-e^x x^4+24 \int e^x x \, dx+36 \int e^x \, dx \\ & = 23 e^x+e^{5+2 x}-e^x x^2-2 e^x x^3-e^x x^4-24 \int e^x \, dx \\ & = -e^x+e^{5+2 x}-e^x x^2-2 e^x x^3-e^x x^4 \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \left (2 e^{5+2 x}+e^x \left (-1-2 x-7 x^2-6 x^3-x^4\right )\right ) \, dx=e^x \left (-1+e^{5+x}-x^2-2 x^3-x^4\right ) \]
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Time = 0.07 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27
method | result | size |
risch | \({\mathrm e}^{5+2 x}+\left (-x^{4}-2 x^{3}-x^{2}-1\right ) {\mathrm e}^{x}\) | \(28\) |
default | \(-{\mathrm e}^{x} x^{4}-2 \,{\mathrm e}^{x} x^{3}-{\mathrm e}^{x} x^{2}-{\mathrm e}^{x}+{\mathrm e}^{5} {\mathrm e}^{2 x}\) | \(34\) |
norman | \(-{\mathrm e}^{x} x^{4}-2 \,{\mathrm e}^{x} x^{3}-{\mathrm e}^{x} x^{2}-{\mathrm e}^{x}+{\mathrm e}^{5} {\mathrm e}^{2 x}\) | \(34\) |
parallelrisch | \(-{\mathrm e}^{x} x^{4}-2 \,{\mathrm e}^{x} x^{3}-{\mathrm e}^{x} x^{2}+{\mathrm e}^{x} {\mathrm e}^{5+x}-{\mathrm e}^{x}\) | \(34\) |
meijerg | \(-{\mathrm e}^{5} \left (1-{\mathrm e}^{2 x}\right )-\frac {\left (5 x^{4}-20 x^{3}+60 x^{2}-120 x +120\right ) {\mathrm e}^{x}}{5}+\frac {3 \left (-4 x^{3}+12 x^{2}-24 x +24\right ) {\mathrm e}^{x}}{2}-\frac {7 \left (3 x^{2}-6 x +6\right ) {\mathrm e}^{x}}{3}+\left (2-2 x \right ) {\mathrm e}^{x}+1-{\mathrm e}^{x}\) | \(84\) |
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Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \left (2 e^{5+2 x}+e^x \left (-1-2 x-7 x^2-6 x^3-x^4\right )\right ) \, dx=-{\left (x^{4} + 2 \, x^{3} + x^{2} + 1\right )} e^{x} + e^{\left (2 \, x + 5\right )} \]
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Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \left (2 e^{5+2 x}+e^x \left (-1-2 x-7 x^2-6 x^3-x^4\right )\right ) \, dx=\left (- x^{4} - 2 x^{3} - x^{2} - 1\right ) e^{x} + e^{5} e^{2 x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (21) = 42\).
Time = 0.20 (sec) , antiderivative size = 69, normalized size of antiderivative = 3.14 \[ \int \left (2 e^{5+2 x}+e^x \left (-1-2 x-7 x^2-6 x^3-x^4\right )\right ) \, dx=-{\left (x^{4} - 4 \, x^{3} + 12 \, x^{2} - 24 \, x + 24\right )} e^{x} - 6 \, {\left (x^{3} - 3 \, x^{2} + 6 \, x - 6\right )} e^{x} - 7 \, {\left (x^{2} - 2 \, x + 2\right )} e^{x} - 2 \, {\left (x - 1\right )} e^{x} + e^{\left (2 \, x + 5\right )} - e^{x} \]
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Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \left (2 e^{5+2 x}+e^x \left (-1-2 x-7 x^2-6 x^3-x^4\right )\right ) \, dx=-{\left (x^{4} + 2 \, x^{3} + x^{2} + 1\right )} e^{x} + e^{\left (2 \, x + 5\right )} \]
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Time = 0.06 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \left (2 e^{5+2 x}+e^x \left (-1-2 x-7 x^2-6 x^3-x^4\right )\right ) \, dx=-{\mathrm {e}}^x\,\left (x^2-{\mathrm {e}}^{x+5}+2\,x^3+x^4+1\right ) \]
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