Optimal. Leaf size=32 \[ \frac{1}{3} e^{3 x} x^2-\frac{2}{9} e^{3 x} x+\frac{2 e^{3 x}}{27} \]
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Rubi [A] time = 0.0194959, antiderivative size = 32, normalized size of antiderivative = 1., 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 = {2176, 2194} \[ \frac{1}{3} e^{3 x} x^2-\frac{2}{9} e^{3 x} x+\frac{2 e^{3 x}}{27} \]
Antiderivative was successfully verified.
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Rule 2176
Rule 2194
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*}
Mathematica [A] time = 0.00593, size = 19, normalized size = 0.59 \[ \frac{1}{27} e^{3 x} \left (9 x^2-6 x+2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 17, normalized size = 0.5 \begin{align*}{\frac{ \left ( 9\,{x}^{2}-6\,x+2 \right ){{\rm e}^{3\,x}}}{27}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.941502, size = 22, normalized size = 0.69 \begin{align*} \frac{1}{27} \,{\left (9 \, x^{2} - 6 \, x + 2\right )} e^{\left (3 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73333, size = 43, normalized size = 1.34 \begin{align*} \frac{1}{27} \,{\left (9 \, x^{2} - 6 \, x + 2\right )} e^{\left (3 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.079068, size = 15, normalized size = 0.47 \begin{align*} \frac{\left (9 x^{2} - 6 x + 2\right ) e^{3 x}}{27} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06645, size = 22, normalized size = 0.69 \begin{align*} \frac{1}{27} \,{\left (9 \, x^{2} - 6 \, x + 2\right )} e^{\left (3 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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