Optimal. Leaf size=44 \[ \frac {x^2 a^{3 x}}{3 \log (a)}+\frac {2 a^{3 x}}{27 \log ^3(a)}-\frac {2 x a^{3 x}}{9 \log ^2(a)} \]
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Rubi [A] time = 0.02, antiderivative size = 44, 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 = {2176, 2194} \[ \frac {x^2 a^{3 x}}{3 \log (a)}-\frac {2 x a^{3 x}}{9 \log ^2(a)}+\frac {2 a^{3 x}}{27 \log ^3(a)} \]
Antiderivative was successfully verified.
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Rule 2176
Rule 2194
Rubi steps
\begin {align*} \int a^{3 x} x^2 \, dx &=\frac {a^{3 x} x^2}{3 \log (a)}-\frac {2 \int a^{3 x} x \, dx}{3 \log (a)}\\ &=-\frac {2 a^{3 x} x}{9 \log ^2(a)}+\frac {a^{3 x} x^2}{3 \log (a)}+\frac {2 \int a^{3 x} \, dx}{9 \log ^2(a)}\\ &=\frac {2 a^{3 x}}{27 \log ^3(a)}-\frac {2 a^{3 x} x}{9 \log ^2(a)}+\frac {a^{3 x} x^2}{3 \log (a)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 29, normalized size = 0.66 \[ \frac {a^{3 x} \left (9 x^2 \log ^2(a)-6 x \log (a)+2\right )}{27 \log ^3(a)} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int a^{3 x} x^2 \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.13, size = 27, normalized size = 0.61 \[ \frac {{\left (9 \, x^{2} \log \relax (a)^{2} - 6 \, x \log \relax (a) + 2\right )} a^{3 \, x}}{27 \, \log \relax (a)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.70, size = 826, normalized size = 18.77 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 28, normalized size = 0.64
method | result | size |
gosper | \(\frac {\left (9 x^{2} \ln \relax (a )^{2}-6 x \ln \relax (a )+2\right ) a^{3 x}}{27 \ln \relax (a )^{3}}\) | \(28\) |
risch | \(\frac {\left (9 x^{2} \ln \relax (a )^{2}-6 x \ln \relax (a )+2\right ) a^{3 x}}{27 \ln \relax (a )^{3}}\) | \(28\) |
meijerg | \(-\frac {2-\frac {\left (27 x^{2} \ln \relax (a )^{2}-18 x \ln \relax (a )+6\right ) {\mathrm e}^{3 x \ln \relax (a )}}{3}}{27 \ln \relax (a )^{3}}\) | \(33\) |
norman | \(\frac {2 \,{\mathrm e}^{3 x \ln \relax (a )}}{27 \ln \relax (a )^{3}}-\frac {2 x \,{\mathrm e}^{3 x \ln \relax (a )}}{9 \ln \relax (a )^{2}}+\frac {x^{2} {\mathrm e}^{3 x \ln \relax (a )}}{3 \ln \relax (a )}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 27, normalized size = 0.61 \[ \frac {{\left (9 \, x^{2} \log \relax (a)^{2} - 6 \, x \log \relax (a) + 2\right )} a^{3 \, x}}{27 \, \log \relax (a)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 27, normalized size = 0.61 \[ \frac {a^{3\,x}\,\left (9\,x^2\,{\ln \relax (a)}^2-6\,x\,\ln \relax (a)+2\right )}{27\,{\ln \relax (a)}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 39, normalized size = 0.89 \[ \begin {cases} \frac {a^{3 x} \left (9 x^{2} \log {\relax (a )}^{2} - 6 x \log {\relax (a )} + 2\right )}{27 \log {\relax (a )}^{3}} & \text {for}\: 27 \log {\relax (a )}^{3} \neq 0 \\\frac {x^{3}}{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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