Optimal. Leaf size=17 \[ -\left (6+e-3 e^{2 x} x^2\right )^2 \]
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Rubi [A] time = 0.22, antiderivative size = 33, normalized size of antiderivative = 1.94, number of steps used = 27, number of rules used = 4, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1593, 2196, 2176, 2194} \begin {gather*} -9 e^{4 x} x^4+36 e^{2 x} x^2+6 e^{2 x+1} x^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 1593
Rule 2176
Rule 2194
Rule 2196
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int e^{4 x} \left (-36 x^3-36 x^4\right ) \, dx+\int e^{2 x} \left (72 x+72 x^2+e \left (12 x+12 x^2\right )\right ) \, dx\\ &=\int e^{4 x} (-36-36 x) x^3 \, dx+\int \left (72 e^{2 x} x+72 e^{2 x} x^2+12 e^{1+2 x} x (1+x)\right ) \, dx\\ &=12 \int e^{1+2 x} x (1+x) \, dx+72 \int e^{2 x} x \, dx+72 \int e^{2 x} x^2 \, dx+\int \left (-36 e^{4 x} x^3-36 e^{4 x} x^4\right ) \, dx\\ &=36 e^{2 x} x+36 e^{2 x} x^2+12 \int \left (e^{1+2 x} x+e^{1+2 x} x^2\right ) \, dx-36 \int e^{2 x} \, dx-36 \int e^{4 x} x^3 \, dx-36 \int e^{4 x} x^4 \, dx-72 \int e^{2 x} x \, dx\\ &=-18 e^{2 x}+36 e^{2 x} x^2-9 e^{4 x} x^3-9 e^{4 x} x^4+12 \int e^{1+2 x} x \, dx+12 \int e^{1+2 x} x^2 \, dx+27 \int e^{4 x} x^2 \, dx+36 \int e^{2 x} \, dx+36 \int e^{4 x} x^3 \, dx\\ &=6 e^{1+2 x} x+36 e^{2 x} x^2+\frac {27}{4} e^{4 x} x^2+6 e^{1+2 x} x^2-9 e^{4 x} x^4-6 \int e^{1+2 x} \, dx-12 \int e^{1+2 x} x \, dx-\frac {27}{2} \int e^{4 x} x \, dx-27 \int e^{4 x} x^2 \, dx\\ &=-3 e^{1+2 x}-\frac {27}{8} e^{4 x} x+36 e^{2 x} x^2+6 e^{1+2 x} x^2-9 e^{4 x} x^4+\frac {27}{8} \int e^{4 x} \, dx+6 \int e^{1+2 x} \, dx+\frac {27}{2} \int e^{4 x} x \, dx\\ &=\frac {27 e^{4 x}}{32}+36 e^{2 x} x^2+6 e^{1+2 x} x^2-9 e^{4 x} x^4-\frac {27}{8} \int e^{4 x} \, dx\\ &=36 e^{2 x} x^2+6 e^{1+2 x} x^2-9 e^{4 x} x^4\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.03, size = 19, normalized size = 1.12 \begin {gather*} -\left (-6-e+3 e^{2 x} x^2\right )^2 \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 28, normalized size = 1.65 \begin {gather*} -9 \, x^{4} e^{\left (4 \, x\right )} + 6 \, {\left (x^{2} e + 6 \, x^{2}\right )} e^{\left (2 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 30, normalized size = 1.76 \begin {gather*} -9 \, x^{4} e^{\left (4 \, x\right )} + 36 \, x^{2} e^{\left (2 \, x\right )} + 6 \, x^{2} e^{\left (2 \, x + 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 24, normalized size = 1.41
method | result | size |
risch | \(-9 x^{4} {\mathrm e}^{4 x}+6 \left ({\mathrm e}+6\right ) x^{2} {\mathrm e}^{2 x}\) | \(24\) |
norman | \(\left (6 \,{\mathrm e}+36\right ) x^{2} {\mathrm e}^{2 x}-9 x^{4} {\mathrm e}^{4 x}\) | \(25\) |
default | \(36 \,{\mathrm e}^{2 x} x^{2}+12 \,{\mathrm e} \left (\frac {{\mathrm e}^{2 x} x^{2}}{2}-\frac {x \,{\mathrm e}^{2 x}}{2}+\frac {{\mathrm e}^{2 x}}{4}\right )+12 \,{\mathrm e} \left (\frac {x \,{\mathrm e}^{2 x}}{2}-\frac {{\mathrm e}^{2 x}}{4}\right )-9 x^{4} {\mathrm e}^{4 x}\) | \(65\) |
meijerg | \(-\frac {9 \left (1280 x^{4}-1280 x^{3}+960 x^{2}-480 x +120\right ) {\mathrm e}^{4 x}}{1280}+\frac {9 \left (-256 x^{3}+192 x^{2}-96 x +24\right ) {\mathrm e}^{4 x}}{256}-\frac {3 \left ({\mathrm e}+6\right ) \left (2-\frac {\left (12 x^{2}-12 x +6\right ) {\mathrm e}^{2 x}}{3}\right )}{2}+3 \left ({\mathrm e}+6\right ) \left (1-\frac {\left (-4 x +2\right ) {\mathrm e}^{2 x}}{2}\right )\) | \(92\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.37, size = 74, normalized size = 4.35 \begin {gather*} -9 \, x^{4} e^{\left (4 \, x\right )} + 3 \, {\left (2 \, x^{2} e - 2 \, x e + e\right )} e^{\left (2 \, x\right )} + 18 \, {\left (2 \, x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x\right )} + 3 \, {\left (2 \, x e - e\right )} e^{\left (2 \, x\right )} + 18 \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 24, normalized size = 1.41 \begin {gather*} 3\,x^2\,{\mathrm {e}}^{2\,x}\,\left (2\,\mathrm {e}-3\,x^2\,{\mathrm {e}}^{2\,x}+12\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 27, normalized size = 1.59 \begin {gather*} - 9 x^{4} e^{4 x} + \left (6 e x^{2} + 36 x^{2}\right ) e^{2 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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