Optimal. Leaf size=27 \[ \frac{1}{2} x^2 \log \left (c \left (b x^n\right )^p\right )-\frac{1}{4} n p x^2 \]
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Rubi [A] time = 0.016509, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2304, 2445} \[ \frac{1}{2} x^2 \log \left (c \left (b x^n\right )^p\right )-\frac{1}{4} n p x^2 \]
Antiderivative was successfully verified.
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Rule 2304
Rule 2445
Rubi steps
\begin{align*} \int x \log \left (c \left (b x^n\right )^p\right ) \, dx &=\operatorname{Subst}\left (\int x \log \left (b^p c x^{n p}\right ) \, dx,b^p c x^{n p},c \left (b x^n\right )^p\right )\\ &=-\frac{1}{4} n p x^2+\frac{1}{2} x^2 \log \left (c \left (b x^n\right )^p\right )\\ \end{align*}
Mathematica [A] time = 0.0009574, size = 27, normalized size = 1. \[ \frac{1}{2} x^2 \log \left (c \left (b x^n\right )^p\right )-\frac{1}{4} n p x^2 \]
Antiderivative was successfully verified.
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Maple [F] time = 0.026, size = 0, normalized size = 0. \begin{align*} \int x\ln \left ( c \left ( b{x}^{n} \right ) ^{p} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16532, size = 31, normalized size = 1.15 \begin{align*} -\frac{1}{4} \, n p x^{2} + \frac{1}{2} \, x^{2} \log \left (\left (b x^{n}\right )^{p} c\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.824537, size = 95, normalized size = 3.52 \begin{align*} \frac{1}{2} \, n p x^{2} \log \left (x\right ) - \frac{1}{4} \, n p x^{2} + \frac{1}{2} \, p x^{2} \log \left (b\right ) + \frac{1}{2} \, x^{2} \log \left (c\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.05761, size = 37, normalized size = 1.37 \begin{align*} \frac{n p x^{2} \log{\left (x \right )}}{2} - \frac{n p x^{2}}{4} + \frac{p x^{2} \log{\left (b \right )}}{2} + \frac{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 = 1.30976, size = 43, normalized size = 1.59 \begin{align*} \frac{1}{2} \, n p x^{2} \log \left (x\right ) - \frac{1}{4} \, n p x^{2} + \frac{1}{2} \, p x^{2} \log \left (b\right ) + \frac{1}{2} \, x^{2} \log \left (c\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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