Optimal. Leaf size=32 \[ \frac {2 e^{3 x}}{27}-\frac {2}{9} e^{3 x} x+\frac {1}{3} e^{3 x} x^2 \]
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Rubi [A]
time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2207, 2225}
\begin {gather*} \frac {1}{3} e^{3 x} x^2-\frac {2}{9} e^{3 x} x+\frac {2 e^{3 x}}{27} \end {gather*}
Antiderivative was successfully verified.
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Rule 2207
Rule 2225
Rubi steps
\begin {align*} \int e^{3 x} x^2 \, dx &=\frac {1}{3} e^{3 x} x^2-\frac {2}{3} \int e^{3 x} x \, dx\\ &=-\frac {2}{9} e^{3 x} x+\frac {1}{3} e^{3 x} x^2+\frac {2}{9} \int e^{3 x} \, dx\\ &=\frac {2 e^{3 x}}{27}-\frac {2}{9} e^{3 x} x+\frac {1}{3} e^{3 x} x^2\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 19, normalized size = 0.59 \begin {gather*} \frac {1}{27} e^{3 x} \left (2-6 x+9 x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.74, size = 17, normalized size = 0.53 \begin {gather*} \frac {\left (2-6 x+9 x^2\right ) E^{3 x}}{27} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 24, normalized size = 0.75
method | result | size |
risch | \(\left (\frac {1}{3} x^{2}-\frac {2}{9} x +\frac {2}{27}\right ) {\mathrm e}^{3 x}\) | \(16\) |
gosper | \(\frac {\left (9 x^{2}-6 x +2\right ) {\mathrm e}^{3 x}}{27}\) | \(17\) |
meijerg | \(-\frac {2}{27}+\frac {\left (27 x^{2}-18 x +6\right ) {\mathrm e}^{3 x}}{81}\) | \(19\) |
derivativedivides | \(\frac {2 \,{\mathrm e}^{3 x}}{27}-\frac {2 \,{\mathrm e}^{3 x} x}{9}+\frac {{\mathrm e}^{3 x} x^{2}}{3}\) | \(24\) |
default | \(\frac {2 \,{\mathrm e}^{3 x}}{27}-\frac {2 \,{\mathrm e}^{3 x} x}{9}+\frac {{\mathrm e}^{3 x} x^{2}}{3}\) | \(24\) |
norman | \(\frac {2 \,{\mathrm e}^{3 x}}{27}-\frac {2 \,{\mathrm e}^{3 x} x}{9}+\frac {{\mathrm e}^{3 x} x^{2}}{3}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 16, normalized size = 0.50 \begin {gather*} \frac {1}{27} \, {\left (9 \, x^{2} - 6 \, x + 2\right )} e^{\left (3 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 16, normalized size = 0.50 \begin {gather*} \frac {1}{27} \, {\left (9 \, x^{2} - 6 \, x + 2\right )} e^{\left (3 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 15, normalized size = 0.47 \begin {gather*} \frac {\left (9 x^{2} - 6 x + 2\right ) e^{3 x}}{27} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 18, normalized size = 0.56 \begin {gather*} \frac {1}{27} \left (9 x^{2}-6 x+2\right ) \mathrm {e}^{3 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 16, normalized size = 0.50 \begin {gather*} \frac {{\mathrm {e}}^{3\,x}\,\left (9\,x^2-6\,x+2\right )}{27} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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