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 = 0.0072916, antiderivative size = 27, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.0010958, 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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Maple [A] time = 0.052, size = 29, normalized size = 1.1 \begin{align*}{\frac{a{x}^{2}}{2}}+{\frac{{x}^{2}b\ln \left ( c{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{2}}-{\frac{bn{x}^{2}}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1695, size = 35, normalized size = 1.3 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.803031, size = 82, normalized size = 3.04 \begin{align*} \frac{1}{2} \, b n x^{2} \log \left (x\right ) + \frac{1}{2} \, b x^{2} \log \left (c\right ) - \frac{1}{4} \,{\left (b n - 2 \, a\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.460422, size = 36, normalized size = 1.33 \begin{align*} \frac{a x^{2}}{2} + \frac{b n x^{2} \log{\left (x \right )}}{2} - \frac{b n x^{2}}{4} + \frac{b x^{2} \log{\left (c \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17518, size = 42, normalized size = 1.56 \begin{align*} \frac{1}{2} \, b n x^{2} \log \left (x\right ) - \frac{1}{4} \, b n x^{2} + \frac{1}{2} \, b x^{2} \log \left (c\right ) + \frac{1}{2} \, a x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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