Optimal. Leaf size=21 \[ 2 x^2 \left (2+x+x \left (-e^{2 x}+x+\log (4)\right )\right ) \]
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Rubi [A] time = 0.10, antiderivative size = 30, normalized size of antiderivative = 1.43, number of steps used = 12, number of rules used = 5, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {6, 1593, 2196, 2176, 2194} \begin {gather*} 2 x^4-2 e^{2 x} x^3+2 x^3 (1+\log (4))+4 x^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 1593
Rule 2176
Rule 2194
Rule 2196
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (8 x+8 x^3+e^{2 x} \left (-6 x^2-4 x^3\right )+x^2 (6+6 \log (4))\right ) \, dx\\ &=4 x^2+2 x^4+2 x^3 (1+\log (4))+\int e^{2 x} \left (-6 x^2-4 x^3\right ) \, dx\\ &=4 x^2+2 x^4+2 x^3 (1+\log (4))+\int e^{2 x} (-6-4 x) x^2 \, dx\\ &=4 x^2+2 x^4+2 x^3 (1+\log (4))+\int \left (-6 e^{2 x} x^2-4 e^{2 x} x^3\right ) \, dx\\ &=4 x^2+2 x^4+2 x^3 (1+\log (4))-4 \int e^{2 x} x^3 \, dx-6 \int e^{2 x} x^2 \, dx\\ &=4 x^2-3 e^{2 x} x^2-2 e^{2 x} x^3+2 x^4+2 x^3 (1+\log (4))+6 \int e^{2 x} x \, dx+6 \int e^{2 x} x^2 \, dx\\ &=3 e^{2 x} x+4 x^2-2 e^{2 x} x^3+2 x^4+2 x^3 (1+\log (4))-3 \int e^{2 x} \, dx-6 \int e^{2 x} x \, dx\\ &=-\frac {3 e^{2 x}}{2}+4 x^2-2 e^{2 x} x^3+2 x^4+2 x^3 (1+\log (4))+3 \int e^{2 x} \, dx\\ &=4 x^2-2 e^{2 x} x^3+2 x^4+2 x^3 (1+\log (4))\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.05, size = 23, normalized size = 1.10 \begin {gather*} 2 x^2 \left (2+x^2+x \left (1-e^{2 x}+\log (4)\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 32, normalized size = 1.52 \begin {gather*} 2 \, x^{4} - 2 \, x^{3} e^{\left (2 \, x\right )} + 4 \, x^{3} \log \relax (2) + 2 \, x^{3} + 4 \, x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 32, normalized size = 1.52 \begin {gather*} 2 \, x^{4} - 2 \, x^{3} e^{\left (2 \, x\right )} + 4 \, x^{3} \log \relax (2) + 2 \, x^{3} + 4 \, x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 31, normalized size = 1.48
method | result | size |
norman | \(\left (4 \ln \relax (2)+2\right ) x^{3}+4 x^{2}+2 x^{4}-2 \,{\mathrm e}^{2 x} x^{3}\) | \(31\) |
derivativedivides | \(4 x^{2}+2 x^{3}+2 x^{4}-2 \,{\mathrm e}^{2 x} x^{3}+4 x^{3} \ln \relax (2)\) | \(33\) |
default | \(4 x^{2}+2 x^{3}+2 x^{4}-2 \,{\mathrm e}^{2 x} x^{3}+4 x^{3} \ln \relax (2)\) | \(33\) |
risch | \(4 x^{2}+2 x^{3}+2 x^{4}-2 \,{\mathrm e}^{2 x} x^{3}+4 x^{3} \ln \relax (2)\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 32, normalized size = 1.52 \begin {gather*} 2 \, x^{4} - 2 \, x^{3} e^{\left (2 \, x\right )} + 4 \, x^{3} \log \relax (2) + 2 \, x^{3} + 4 \, x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.56, size = 28, normalized size = 1.33 \begin {gather*} x^3\,\left (\ln \left (16\right )+2\right )-2\,x^3\,{\mathrm {e}}^{2\,x}+4\,x^2+2\,x^4 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 29, normalized size = 1.38 \begin {gather*} 2 x^{4} - 2 x^{3} e^{2 x} + x^{3} \left (2 + 4 \log {\relax (2 )}\right ) + 4 x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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