Optimal. Leaf size=42 \[ \frac {2 x^{m+1}}{(m+1)^3}+\frac {x^{m+1} \log ^2(x)}{m+1}-\frac {2 x^{m+1} \log (x)}{(m+1)^2} \]
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Rubi [A] time = 0.02, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2305, 2304} \[ \frac {2 x^{m+1}}{(m+1)^3}+\frac {x^{m+1} \log ^2(x)}{m+1}-\frac {2 x^{m+1} \log (x)}{(m+1)^2} \]
Antiderivative was successfully verified.
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Rule 2304
Rule 2305
Rubi steps
\begin {align*} \int x^m \log ^2(x) \, dx &=\frac {x^{1+m} \log ^2(x)}{1+m}-\frac {2 \int x^m \log (x) \, dx}{1+m}\\ &=\frac {2 x^{1+m}}{(1+m)^3}-\frac {2 x^{1+m} \log (x)}{(1+m)^2}+\frac {x^{1+m} \log ^2(x)}{1+m}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 30, normalized size = 0.71 \[ \frac {x^{m+1} \left ((m+1)^2 \log ^2(x)-2 (m+1) \log (x)+2\right )}{(m+1)^3} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^m \log ^2(x) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.92, size = 45, normalized size = 1.07 \[ \frac {{\left ({\left (m^{2} + 2 \, m + 1\right )} x \log \relax (x)^{2} - 2 \, {\left (m + 1\right )} x \log \relax (x) + 2 \, x\right )} x^{m}}{m^{3} + 3 \, m^{2} + 3 \, m + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.64, size = 84, normalized size = 2.00 \[ -\frac {2 \, m x x^{m} \log \relax (x)}{{\left (m^{2} + 2 \, m + 1\right )} {\left (m + 1\right )}} + \frac {x^{m + 1} \log \relax (x)^{2}}{m + 1} - \frac {2 \, x x^{m} \log \relax (x)}{{\left (m^{2} + 2 \, m + 1\right )} {\left (m + 1\right )}} + \frac {2 \, x x^{m}}{{\left (m^{2} + 2 \, m + 1\right )} {\left (m + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 41, normalized size = 0.98
method | result | size |
risch | \(\frac {x \left (m^{2} \ln \relax (x )^{2}+2 m \ln \relax (x )^{2}-2 m \ln \relax (x )+\ln \relax (x )^{2}-2 \ln \relax (x )+2\right ) x^{m}}{\left (1+m \right )^{3}}\) | \(41\) |
norman | \(\frac {x \ln \relax (x )^{2} {\mathrm e}^{m \ln \relax (x )}}{1+m}+\frac {2 x \,{\mathrm e}^{m \ln \relax (x )}}{m^{3}+3 m^{2}+3 m +1}-\frac {2 x \ln \relax (x ) {\mathrm e}^{m \ln \relax (x )}}{m^{2}+2 m +1}\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 42, normalized size = 1.00 \[ \frac {x^{m + 1} \log \relax (x)^{2}}{m + 1} - \frac {2 \, x^{m + 1} \log \relax (x)}{{\left (m + 1\right )}^{2}} + \frac {2 \, x^{m + 1}}{{\left (m + 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.35, size = 43, normalized size = 1.02 \[ \left \{\begin {array}{cl} \frac {{\ln \relax (x)}^3}{3} & \text {\ if\ \ }m=-1\\ \frac {x^{m+1}\,\left ({\ln \relax (x)}^2\,{\left (m+1\right )}^2-2\,\ln \relax (x)\,\left (m+1\right )+2\right )}{{\left (m+1\right )}^3} & \text {\ if\ \ }m\neq -1 \end {array}\right . \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.57, size = 155, normalized size = 3.69 \[ \begin {cases} \frac {m^{2} x x^{m} \log {\relax (x )}^{2}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {2 m x x^{m} \log {\relax (x )}^{2}}{m^{3} + 3 m^{2} + 3 m + 1} - \frac {2 m x x^{m} \log {\relax (x )}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {x x^{m} \log {\relax (x )}^{2}}{m^{3} + 3 m^{2} + 3 m + 1} - \frac {2 x x^{m} \log {\relax (x )}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {2 x x^{m}}{m^{3} + 3 m^{2} + 3 m + 1} & \text {for}\: m \neq -1 \\\frac {\log {\relax (x )}^{3}}{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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