Optimal. Leaf size=20 \[ -x+e^{3 x} x \left (4+\frac {x^3}{4}\right ) \]
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Rubi [A] time = 0.16, antiderivative size = 24, normalized size of antiderivative = 1.20, number of steps used = 16, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {12, 2196, 2194, 2176} \begin {gather*} \frac {1}{4} e^{3 x} x^4+4 e^{3 x} x-x \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2176
Rule 2194
Rule 2196
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int \left (-4+e^{3 x} \left (16+48 x+4 x^3+3 x^4\right )\right ) \, dx\\ &=-x+\frac {1}{4} \int e^{3 x} \left (16+48 x+4 x^3+3 x^4\right ) \, dx\\ &=-x+\frac {1}{4} \int \left (16 e^{3 x}+48 e^{3 x} x+4 e^{3 x} x^3+3 e^{3 x} x^4\right ) \, dx\\ &=-x+\frac {3}{4} \int e^{3 x} x^4 \, dx+4 \int e^{3 x} \, dx+12 \int e^{3 x} x \, dx+\int e^{3 x} x^3 \, dx\\ &=\frac {4 e^{3 x}}{3}-x+4 e^{3 x} x+\frac {1}{3} e^{3 x} x^3+\frac {1}{4} e^{3 x} x^4-4 \int e^{3 x} \, dx-\int e^{3 x} x^2 \, dx-\int e^{3 x} x^3 \, dx\\ &=-x+4 e^{3 x} x-\frac {1}{3} e^{3 x} x^2+\frac {1}{4} e^{3 x} x^4+\frac {2}{3} \int e^{3 x} x \, dx+\int e^{3 x} x^2 \, dx\\ &=-x+\frac {38}{9} e^{3 x} x+\frac {1}{4} e^{3 x} x^4-\frac {2}{9} \int e^{3 x} \, dx-\frac {2}{3} \int e^{3 x} x \, dx\\ &=-\frac {2 e^{3 x}}{27}-x+4 e^{3 x} x+\frac {1}{4} e^{3 x} x^4+\frac {2}{9} \int e^{3 x} \, dx\\ &=-x+4 e^{3 x} x+\frac {1}{4} e^{3 x} x^4\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.02, size = 24, normalized size = 1.20 \begin {gather*} -x+\frac {1}{3} e^{3 x} \left (12 x+\frac {3 x^4}{4}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 17, normalized size = 0.85 \begin {gather*} \frac {1}{4} \, {\left (x^{4} + 16 \, x\right )} e^{\left (3 \, x\right )} - x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 17, normalized size = 0.85 \begin {gather*} \frac {1}{4} \, {\left (x^{4} + 16 \, x\right )} e^{\left (3 \, x\right )} - x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 18, normalized size = 0.90
method | result | size |
risch | \(\frac {\left (x^{4}+16 x \right ) {\mathrm e}^{3 x}}{4}-x\) | \(18\) |
derivativedivides | \(-x +\frac {{\mathrm e}^{3 x} x^{4}}{4}+4 x \,{\mathrm e}^{3 x}\) | \(21\) |
default | \(-x +\frac {{\mathrm e}^{3 x} x^{4}}{4}+4 x \,{\mathrm e}^{3 x}\) | \(21\) |
norman | \(-x +\frac {{\mathrm e}^{3 x} x^{4}}{4}+4 x \,{\mathrm e}^{3 x}\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.39, size = 68, normalized size = 3.40 \begin {gather*} \frac {1}{108} \, {\left (27 \, x^{4} - 36 \, x^{3} + 36 \, x^{2} - 24 \, x + 8\right )} e^{\left (3 \, x\right )} + \frac {1}{27} \, {\left (9 \, x^{3} - 9 \, x^{2} + 6 \, x - 2\right )} e^{\left (3 \, x\right )} + \frac {4}{3} \, {\left (3 \, x - 1\right )} e^{\left (3 \, x\right )} - x + \frac {4}{3} \, e^{\left (3 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 19, normalized size = 0.95 \begin {gather*} \frac {x\,\left (16\,{\mathrm {e}}^{3\,x}+x^3\,{\mathrm {e}}^{3\,x}-4\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 14, normalized size = 0.70 \begin {gather*} - x + \frac {\left (x^{4} + 16 x\right ) e^{3 x}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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