Optimal. Leaf size=27 \[ \frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} b n x^2 \]
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Rubi [A] time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2304} \[ \frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} b n x^2 \]
Antiderivative was successfully verified.
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Rule 2304
Rubi steps
\begin {align*} \int x \left (a+b \log \left (c x^n\right )\right ) \, dx &=-\frac {1}{4} b n x^2+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 32, normalized size = 1.19 \[ \frac {a x^2}{2}+\frac {1}{2} b x^2 \log \left (c x^n\right )-\frac {1}{4} b n x^2 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 30, normalized size = 1.11 \[ \frac {1}{2} \, b n x^{2} \log \relax (x) + \frac {1}{2} \, b x^{2} \log \relax (c) - \frac {1}{4} \, {\left (b n - 2 \, a\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 31, normalized size = 1.15 \[ \frac {1}{2} \, b n x^{2} \log \relax (x) - \frac {1}{4} \, b n x^{2} + \frac {1}{2} \, b x^{2} \log \relax (c) + \frac {1}{2} \, a x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 29, normalized size = 1.07 \[ -\frac {b n \,x^{2}}{4}+\frac {b \,x^{2} \ln \left (c \,{\mathrm e}^{n \ln \relax (x )}\right )}{2}+\frac {a \,x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 26, normalized size = 0.96 \[ -\frac {1}{4} \, b n x^{2} + \frac {1}{2} \, b x^{2} \log \left (c x^{n}\right ) + \frac {1}{2} \, a x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.26, size = 25, normalized size = 0.93 \[ x^2\,\left (\frac {a}{2}-\frac {b\,n}{4}\right )+\frac {b\,x^2\,\ln \left (c\,x^n\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.46, size = 36, normalized size = 1.33 \[ \frac {a x^{2}}{2} + \frac {b n x^{2} \log {\relax (x )}}{2} - \frac {b n x^{2}}{4} + \frac {b x^{2} \log {\relax (c )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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