Optimal. Leaf size=28 \[ \frac {x^{n+1} \log (a x)}{n+1}-\frac {x^{n+1}}{(n+1)^2} \]
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Rubi [A] time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2304} \[ \frac {x^{n+1} \log (a x)}{n+1}-\frac {x^{n+1}}{(n+1)^2} \]
Antiderivative was successfully verified.
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Rule 2304
Rubi steps
\begin {align*} \int x^n \log (a x) \, dx &=-\frac {x^{1+n}}{(1+n)^2}+\frac {x^{1+n} \log (a x)}{1+n}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 21, normalized size = 0.75 \[ \frac {x^{n+1} ((n+1) \log (a x)-1)}{(n+1)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 32, normalized size = 1.14 \[ \frac {{\left ({\left (n + 1\right )} x \log \relax (a) + {\left (n + 1\right )} x \log \relax (x) - x\right )} x^{n}}{n^{2} + 2 \, n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 36, normalized size = 1.29 \[ \frac {x \,{\mathrm e}^{n \ln \relax (x )} \ln \left (a x \right )}{n +1}-\frac {x \,{\mathrm e}^{n \ln \relax (x )}}{n^{2}+2 n +1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 28, normalized size = 1.00 \[ \frac {x^{n + 1} \log \left (a x\right )}{n + 1} - \frac {x^{n + 1}}{{\left (n + 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 38, normalized size = 1.36 \[ \left \{\begin {array}{cl} \frac {{\ln \left (a\,x\right )}^2}{2} & \text {\ if\ \ }n=-1\\ \frac {x^{n+1}\,\left (\ln \left (a\,x\right )-\frac {1}{n+1}\right )}{n+1} & \text {\ if\ \ }n\neq -1 \end {array}\right . \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.77, size = 94, normalized size = 3.36 \[ \begin {cases} \frac {n x x^{n} \log {\relax (a )}}{n^{2} + 2 n + 1} + \frac {n x x^{n} \log {\relax (x )}}{n^{2} + 2 n + 1} + \frac {x x^{n} \log {\relax (a )}}{n^{2} + 2 n + 1} + \frac {x x^{n} \log {\relax (x )}}{n^{2} + 2 n + 1} - \frac {x x^{n}}{n^{2} + 2 n + 1} & \text {for}\: n \neq -1 \\\frac {\log {\left (a x \right )}^{2}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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