Optimal. Leaf size=68 \[ \frac{1}{3} e^{3 x} x^5-\frac{5}{9} e^{3 x} x^4+\frac{38}{27} e^{3 x} x^3-\frac{38}{27} e^{3 x} x^2+\frac{76}{81} e^{3 x} x-\frac{724 e^{3 x}}{243} \]
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Rubi [A] time = 0.107394, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {2196, 2194, 2176} \[ \frac{1}{3} e^{3 x} x^5-\frac{5}{9} e^{3 x} x^4+\frac{38}{27} e^{3 x} x^3-\frac{38}{27} e^{3 x} x^2+\frac{76}{81} e^{3 x} x-\frac{724 e^{3 x}}{243} \]
Antiderivative was successfully verified.
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Rule 2196
Rule 2194
Rule 2176
Rubi steps
\begin{align*} \int e^{3 x} \left (-8+2 x^3+x^5\right ) \, dx &=\int \left (-8 e^{3 x}+2 e^{3 x} x^3+e^{3 x} x^5\right ) \, dx\\ &=2 \int e^{3 x} x^3 \, dx-8 \int e^{3 x} \, dx+\int e^{3 x} x^5 \, dx\\ &=-\frac{8 e^{3 x}}{3}+\frac{2}{3} e^{3 x} x^3+\frac{1}{3} e^{3 x} x^5-\frac{5}{3} \int e^{3 x} x^4 \, dx-2 \int e^{3 x} x^2 \, dx\\ &=-\frac{8 e^{3 x}}{3}-\frac{2}{3} e^{3 x} x^2+\frac{2}{3} e^{3 x} x^3-\frac{5}{9} e^{3 x} x^4+\frac{1}{3} e^{3 x} x^5+\frac{4}{3} \int e^{3 x} x \, dx+\frac{20}{9} \int e^{3 x} x^3 \, dx\\ &=-\frac{8 e^{3 x}}{3}+\frac{4}{9} e^{3 x} x-\frac{2}{3} e^{3 x} x^2+\frac{38}{27} e^{3 x} x^3-\frac{5}{9} e^{3 x} x^4+\frac{1}{3} e^{3 x} x^5-\frac{4}{9} \int e^{3 x} \, dx-\frac{20}{9} \int e^{3 x} x^2 \, dx\\ &=-\frac{76 e^{3 x}}{27}+\frac{4}{9} e^{3 x} x-\frac{38}{27} e^{3 x} x^2+\frac{38}{27} e^{3 x} x^3-\frac{5}{9} e^{3 x} x^4+\frac{1}{3} e^{3 x} x^5+\frac{40}{27} \int e^{3 x} x \, dx\\ &=-\frac{76 e^{3 x}}{27}+\frac{76}{81} e^{3 x} x-\frac{38}{27} e^{3 x} x^2+\frac{38}{27} e^{3 x} x^3-\frac{5}{9} e^{3 x} x^4+\frac{1}{3} e^{3 x} x^5-\frac{40}{81} \int e^{3 x} \, dx\\ &=-\frac{724 e^{3 x}}{243}+\frac{76}{81} e^{3 x} x-\frac{38}{27} e^{3 x} x^2+\frac{38}{27} e^{3 x} x^3-\frac{5}{9} e^{3 x} x^4+\frac{1}{3} e^{3 x} x^5\\ \end{align*}
Mathematica [A] time = 0.0212306, size = 34, normalized size = 0.5 \[ \frac{1}{243} e^{3 x} \left (81 x^5-135 x^4+342 x^3-342 x^2+228 x-724\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 32, normalized size = 0.5 \begin{align*}{\frac{{{\rm e}^{3\,x}} \left ( 81\,{x}^{5}-135\,{x}^{4}+342\,{x}^{3}-342\,{x}^{2}+228\,x-724 \right ) }{243}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.983796, size = 80, normalized size = 1.18 \begin{align*} \frac{1}{243} \,{\left (81 \, x^{5} - 135 \, x^{4} + 180 \, x^{3} - 180 \, x^{2} + 120 \, x - 40\right )} e^{\left (3 \, x\right )} + \frac{2}{27} \,{\left (9 \, x^{3} - 9 \, x^{2} + 6 \, x - 2\right )} e^{\left (3 \, x\right )} - \frac{8}{3} \, e^{\left (3 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.853035, size = 92, normalized size = 1.35 \begin{align*} \frac{1}{243} \,{\left (81 \, x^{5} - 135 \, x^{4} + 342 \, x^{3} - 342 \, x^{2} + 228 \, x - 724\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 = 0.101229, size = 31, normalized size = 0.46 \begin{align*} \frac{\left (81 x^{5} - 135 x^{4} + 342 x^{3} - 342 x^{2} + 228 x - 724\right ) e^{3 x}}{243} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2537, size = 42, normalized size = 0.62 \begin{align*} \frac{1}{243} \,{\left (81 \, x^{5} - 135 \, x^{4} + 342 \, x^{3} - 342 \, x^{2} + 228 \, x - 724\right )} e^{\left (3 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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