Optimal. Leaf size=32 \[ \frac {1}{3} e^{3 x} x^2-\frac {5}{9} e^{3 x} x+\frac {5 e^{3 x}}{27} \]
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Rubi [A] time = 0.05, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {1593, 2196, 2176, 2194} \[ \frac {1}{3} e^{3 x} x^2-\frac {5}{9} e^{3 x} x+\frac {5 e^{3 x}}{27} \]
Antiderivative was successfully verified.
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Rule 1593
Rule 2176
Rule 2194
Rule 2196
Rubi steps
\begin {align*} \int e^{3 x} \left (-x+x^2\right ) \, dx &=\int e^{3 x} (-1+x) x \, dx\\ &=\int \left (-e^{3 x} x+e^{3 x} x^2\right ) \, dx\\ &=-\int e^{3 x} x \, dx+\int e^{3 x} x^2 \, dx\\ &=-\frac {1}{3} e^{3 x} x+\frac {1}{3} e^{3 x} x^2+\frac {1}{3} \int e^{3 x} \, dx-\frac {2}{3} \int e^{3 x} x \, dx\\ &=\frac {e^{3 x}}{9}-\frac {5}{9} e^{3 x} x+\frac {1}{3} e^{3 x} x^2+\frac {2}{9} \int e^{3 x} \, dx\\ &=\frac {5 e^{3 x}}{27}-\frac {5}{9} e^{3 x} x+\frac {1}{3} e^{3 x} x^2\\ \end {align*}
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Mathematica [A] time = 0.03, size = 19, normalized size = 0.59 \[ \frac {1}{27} e^{3 x} \left (9 x^2-15 x+5\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.38, size = 16, normalized size = 0.50 \[ \frac {1}{27} \, {\left (9 \, x^{2} - 15 \, x + 5\right )} e^{\left (3 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 16, normalized size = 0.50 \[ \frac {1}{27} \, {\left (9 \, x^{2} - 15 \, x + 5\right )} e^{\left (3 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 17, normalized size = 0.53 \[ \frac {\left (9 x^{2}-15 x +5\right ) {\mathrm e}^{3 x}}{27} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.95, size = 28, normalized size = 0.88 \[ \frac {1}{27} \, {\left (9 \, x^{2} - 6 \, x + 2\right )} e^{\left (3 \, x\right )} - \frac {1}{9} \, {\left (3 \, x - 1\right )} e^{\left (3 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.53, size = 16, normalized size = 0.50 \[ \frac {{\mathrm {e}}^{3\,x}\,\left (9\,x^2-15\,x+5\right )}{27} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 15, normalized size = 0.47 \[ \frac {\left (9 x^{2} - 15 x + 5\right ) e^{3 x}}{27} \]
Verification of antiderivative is not currently implemented for this CAS.
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