Integrand size = 44, antiderivative size = 19 \[ \int \left (-32 x+12 x^2+16 x^3+e^{2 x} \left (2 x+2 x^2\right )+e^x \left (-12 x^2-4 x^3\right )\right ) \, dx=4 x^2 \left (-4+x+\left (-\frac {e^x}{2}+x\right )^2\right ) \]
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Time = 0.09 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.74, number of steps used = 19, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1607, 2227, 2207, 2225} \[ \int \left (-32 x+12 x^2+16 x^3+e^{2 x} \left (2 x+2 x^2\right )+e^x \left (-12 x^2-4 x^3\right )\right ) \, dx=4 x^4-4 e^x x^3+4 x^3+e^{2 x} x^2-16 x^2 \]
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Rule 1607
Rule 2207
Rule 2225
Rule 2227
Rubi steps \begin{align*} \text {integral}& = -16 x^2+4 x^3+4 x^4+\int e^{2 x} \left (2 x+2 x^2\right ) \, dx+\int e^x \left (-12 x^2-4 x^3\right ) \, dx \\ & = -16 x^2+4 x^3+4 x^4+\int e^x (-12-4 x) x^2 \, dx+\int e^{2 x} x (2+2 x) \, dx \\ & = -16 x^2+4 x^3+4 x^4+\int \left (2 e^{2 x} x+2 e^{2 x} x^2\right ) \, dx+\int \left (-12 e^x x^2-4 e^x x^3\right ) \, dx \\ & = -16 x^2+4 x^3+4 x^4+2 \int e^{2 x} x \, dx+2 \int e^{2 x} x^2 \, dx-4 \int e^x x^3 \, dx-12 \int e^x x^2 \, dx \\ & = e^{2 x} x-16 x^2-12 e^x x^2+e^{2 x} x^2+4 x^3-4 e^x x^3+4 x^4-2 \int e^{2 x} x \, dx+12 \int e^x x^2 \, dx+24 \int e^x x \, dx-\int e^{2 x} \, dx \\ & = -\frac {e^{2 x}}{2}+24 e^x x-16 x^2+e^{2 x} x^2+4 x^3-4 e^x x^3+4 x^4-24 \int e^x \, dx-24 \int e^x x \, dx+\int e^{2 x} \, dx \\ & = -24 e^x-16 x^2+e^{2 x} x^2+4 x^3-4 e^x x^3+4 x^4+24 \int e^x \, dx \\ & = -16 x^2+e^{2 x} x^2+4 x^3-4 e^x x^3+4 x^4 \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.26 \[ \int \left (-32 x+12 x^2+16 x^3+e^{2 x} \left (2 x+2 x^2\right )+e^x \left (-12 x^2-4 x^3\right )\right ) \, dx=x^2 \left (e^{2 x}-4 e^x x+4 \left (-4+x+x^2\right )\right ) \]
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Time = 0.04 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.68
method | result | size |
default | \({\mathrm e}^{2 x} x^{2}-4 \,{\mathrm e}^{x} x^{3}-16 x^{2}+4 x^{3}+4 x^{4}\) | \(32\) |
norman | \({\mathrm e}^{2 x} x^{2}-4 \,{\mathrm e}^{x} x^{3}-16 x^{2}+4 x^{3}+4 x^{4}\) | \(32\) |
risch | \({\mathrm e}^{2 x} x^{2}-4 \,{\mathrm e}^{x} x^{3}-16 x^{2}+4 x^{3}+4 x^{4}\) | \(32\) |
parallelrisch | \({\mathrm e}^{2 x} x^{2}-4 \,{\mathrm e}^{x} x^{3}-16 x^{2}+4 x^{3}+4 x^{4}\) | \(32\) |
parts | \({\mathrm e}^{2 x} x^{2}-4 \,{\mathrm e}^{x} x^{3}-16 x^{2}+4 x^{3}+4 x^{4}\) | \(32\) |
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Time = 0.24 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.63 \[ \int \left (-32 x+12 x^2+16 x^3+e^{2 x} \left (2 x+2 x^2\right )+e^x \left (-12 x^2-4 x^3\right )\right ) \, dx=4 \, x^{4} - 4 \, x^{3} e^{x} + 4 \, x^{3} + x^{2} e^{\left (2 \, x\right )} - 16 \, x^{2} \]
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Time = 0.05 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.63 \[ \int \left (-32 x+12 x^2+16 x^3+e^{2 x} \left (2 x+2 x^2\right )+e^x \left (-12 x^2-4 x^3\right )\right ) \, dx=4 x^{4} - 4 x^{3} e^{x} + 4 x^{3} + x^{2} e^{2 x} - 16 x^{2} \]
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Time = 0.20 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.63 \[ \int \left (-32 x+12 x^2+16 x^3+e^{2 x} \left (2 x+2 x^2\right )+e^x \left (-12 x^2-4 x^3\right )\right ) \, dx=4 \, x^{4} - 4 \, x^{3} e^{x} + 4 \, x^{3} + x^{2} e^{\left (2 \, x\right )} - 16 \, x^{2} \]
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Time = 0.28 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.63 \[ \int \left (-32 x+12 x^2+16 x^3+e^{2 x} \left (2 x+2 x^2\right )+e^x \left (-12 x^2-4 x^3\right )\right ) \, dx=4 \, x^{4} - 4 \, x^{3} e^{x} + 4 \, x^{3} + x^{2} e^{\left (2 \, x\right )} - 16 \, x^{2} \]
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Time = 13.80 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21 \[ \int \left (-32 x+12 x^2+16 x^3+e^{2 x} \left (2 x+2 x^2\right )+e^x \left (-12 x^2-4 x^3\right )\right ) \, dx=x^2\,\left (4\,x+{\mathrm {e}}^{2\,x}-4\,x\,{\mathrm {e}}^x+4\,x^2-16\right ) \]
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