Optimal. Leaf size=52 \[ \frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac{1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{4} b^2 n^2 x^2 \]
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Rubi [A] time = 0.0230684, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2305, 2304} \[ \frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac{1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{4} b^2 n^2 x^2 \]
Antiderivative was successfully verified.
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Rule 2305
Rule 2304
Rubi steps
\begin{align*} \int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx &=\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2-(b n) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx\\ &=\frac{1}{4} b^2 n^2 x^2-\frac{1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2\\ \end{align*}
Mathematica [A] time = 0.0130538, size = 41, normalized size = 0.79 \[ \frac{1}{4} x^2 \left (2 \left (a+b \log \left (c x^n\right )\right )^2+b n \left (-2 a-2 b \log \left (c x^n\right )+b n\right )\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.213, size = 692, normalized size = 13.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11882, size = 95, normalized size = 1.83 \begin{align*} \frac{1}{2} \, b^{2} x^{2} \log \left (c x^{n}\right )^{2} - \frac{1}{2} \, a b n x^{2} + a b x^{2} \log \left (c x^{n}\right ) + \frac{1}{2} \, a^{2} x^{2} + \frac{1}{4} \,{\left (n^{2} x^{2} - 2 \, n x^{2} \log \left (c x^{n}\right )\right )} b^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.87917, size = 243, normalized size = 4.67 \begin{align*} \frac{1}{2} \, b^{2} n^{2} x^{2} \log \left (x\right )^{2} + \frac{1}{2} \, b^{2} x^{2} \log \left (c\right )^{2} - \frac{1}{2} \,{\left (b^{2} n - 2 \, a b\right )} x^{2} \log \left (c\right ) + \frac{1}{4} \,{\left (b^{2} n^{2} - 2 \, a b n + 2 \, a^{2}\right )} x^{2} + \frac{1}{2} \,{\left (2 \, b^{2} n x^{2} \log \left (c\right ) -{\left (b^{2} n^{2} - 2 \, a b n\right )} x^{2}\right )} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.02012, size = 126, normalized size = 2.42 \begin{align*} \frac{a^{2} x^{2}}{2} + a b n x^{2} \log{\left (x \right )} - \frac{a b n x^{2}}{2} + a b x^{2} \log{\left (c \right )} + \frac{b^{2} n^{2} x^{2} \log{\left (x \right )}^{2}}{2} - \frac{b^{2} n^{2} x^{2} \log{\left (x \right )}}{2} + \frac{b^{2} n^{2} x^{2}}{4} + b^{2} n x^{2} \log{\left (c \right )} \log{\left (x \right )} - \frac{b^{2} n x^{2} \log{\left (c \right )}}{2} + \frac{b^{2} x^{2} \log{\left (c \right )}^{2}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15036, size = 146, normalized size = 2.81 \begin{align*} \frac{1}{2} \, b^{2} n^{2} x^{2} \log \left (x\right )^{2} - \frac{1}{2} \, b^{2} n^{2} x^{2} \log \left (x\right ) + b^{2} n x^{2} \log \left (c\right ) \log \left (x\right ) + \frac{1}{4} \, b^{2} n^{2} x^{2} - \frac{1}{2} \, b^{2} n x^{2} \log \left (c\right ) + \frac{1}{2} \, b^{2} x^{2} \log \left (c\right )^{2} + a b n x^{2} \log \left (x\right ) - \frac{1}{2} \, a b n x^{2} + a b x^{2} \log \left (c\right ) + \frac{1}{2} \, a^{2} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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