Optimal. Leaf size=26 \[ -\frac {x^{1+m}}{(1+m)^2}+\frac {x^{1+m} \log (x)}{1+m} \]
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Rubi [A]
time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2341}
\begin {gather*} \frac {x^{m+1} \log (x)}{m+1}-\frac {x^{m+1}}{(m+1)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2341
Rubi steps
\begin {align*} \int x^m \log (x) \, dx &=-\frac {x^{1+m}}{(1+m)^2}+\frac {x^{1+m} \log (x)}{1+m}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 19, normalized size = 0.73 \begin {gather*} \frac {x^{1+m} (-1+(1+m) \log (x))}{(1+m)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 19, normalized size = 0.73
method | result | size |
risch | \(\frac {x \left (m \ln \left (x \right )+\ln \left (x \right )-1\right ) x^{m}}{\left (1+m \right )^{2}}\) | \(19\) |
norman | \(\frac {x \ln \left (x \right ) {\mathrm e}^{m \ln \left (x \right )}}{1+m}-\frac {x \,{\mathrm e}^{m \ln \left (x \right )}}{m^{2}+2 m +1}\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.40, size = 26, normalized size = 1.00 \begin {gather*} \frac {x^{m + 1} \log \left (x\right )}{m + 1} - \frac {x^{m + 1}}{{\left (m + 1\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.77, size = 25, normalized size = 0.96 \begin {gather*} \frac {{\left ({\left (m + 1\right )} x \log \left (x\right ) - x\right )} x^{m}}{m^{2} + 2 \, m + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 56 vs.
\(2 (20) = 40\).
time = 0.20, size = 56, normalized size = 2.15 \begin {gather*} \begin {cases} \frac {m x x^{m} \log {\left (x \right )}}{m^{2} + 2 m + 1} + \frac {x x^{m} \log {\left (x \right )}}{m^{2} + 2 m + 1} - \frac {x x^{m}}{m^{2} + 2 m + 1} & \text {for}\: m \neq -1 \\\frac {\log {\left (x \right )}^{2}}{2} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.40, size = 32, normalized size = 1.23 \begin {gather*} \left \{\begin {array}{cl} \frac {{\ln \left (x\right )}^2}{2} & \text {\ if\ \ }m=-1\\ \frac {x^{m+1}\,\left (\ln \left (x\right )\,\left (m+1\right )-1\right )}{{\left (m+1\right )}^2} & \text {\ if\ \ }m\neq -1 \end {array}\right . \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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