Optimal. Leaf size=43 \[ x \left (a+b \log \left (c x^n\right )\right )^2-2 a b n x-2 b^2 n x \log \left (c x^n\right )+2 b^2 n^2 x \]
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Rubi [A] time = 0.0128483, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2296, 2295} \[ x \left (a+b \log \left (c x^n\right )\right )^2-2 a b n x-2 b^2 n x \log \left (c x^n\right )+2 b^2 n^2 x \]
Antiderivative was successfully verified.
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Rule 2296
Rule 2295
Rubi steps
\begin{align*} \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx &=x \left (a+b \log \left (c x^n\right )\right )^2-(2 b n) \int \left (a+b \log \left (c x^n\right )\right ) \, dx\\ &=-2 a b n x+x \left (a+b \log \left (c x^n\right )\right )^2-\left (2 b^2 n\right ) \int \log \left (c x^n\right ) \, dx\\ &=-2 a b n x+2 b^2 n^2 x-2 b^2 n x \log \left (c x^n\right )+x \left (a+b \log \left (c x^n\right )\right )^2\\ \end{align*}
Mathematica [A] time = 0.008462, size = 33, normalized size = 0.77 \[ x \left (\left (a+b \log \left (c x^n\right )\right )^2-2 b n \left (a+b \log \left (c x^n\right )-b n\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 63, normalized size = 1.5 \begin{align*} x{a}^{2}+{b}^{2}x \left ( \ln \left ( c{{\rm e}^{n\ln \left ( x \right ) }} \right ) \right ) ^{2}+2\,{b}^{2}{n}^{2}x-2\,{b}^{2}nx\ln \left ( c{{\rm e}^{n\ln \left ( x \right ) }} \right ) +2\,xab\ln \left ( c{x}^{n} \right ) -2\,abnx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08919, size = 77, normalized size = 1.79 \begin{align*} b^{2} x \log \left (c x^{n}\right )^{2} - 2 \, a b n x + 2 \, a b x \log \left (c x^{n}\right ) + 2 \,{\left (n^{2} x - n x \log \left (c x^{n}\right )\right )} b^{2} + a^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.792463, size = 197, normalized size = 4.58 \begin{align*} b^{2} n^{2} x \log \left (x\right )^{2} + b^{2} x \log \left (c\right )^{2} - 2 \,{\left (b^{2} n - a b\right )} x \log \left (c\right ) +{\left (2 \, b^{2} n^{2} - 2 \, a b n + a^{2}\right )} x + 2 \,{\left (b^{2} n x \log \left (c\right ) -{\left (b^{2} n^{2} - a b n\right )} x\right )} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.607413, size = 109, normalized size = 2.53 \begin{align*} a^{2} x + 2 a b n x \log{\left (x \right )} - 2 a b n x + 2 a b x \log{\left (c \right )} + b^{2} n^{2} x \log{\left (x \right )}^{2} - 2 b^{2} n^{2} x \log{\left (x \right )} + 2 b^{2} n^{2} x + 2 b^{2} n x \log{\left (c \right )} \log{\left (x \right )} - 2 b^{2} n x \log{\left (c \right )} + b^{2} x \log{\left (c \right )}^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1966, size = 119, normalized size = 2.77 \begin{align*} b^{2} n^{2} x \log \left (x\right )^{2} - 2 \, b^{2} n^{2} x \log \left (x\right ) + 2 \, b^{2} n x \log \left (c\right ) \log \left (x\right ) + 2 \, b^{2} n^{2} x - 2 \, b^{2} n x \log \left (c\right ) + b^{2} x \log \left (c\right )^{2} + 2 \, a b n x \log \left (x\right ) - 2 \, a b n x + 2 \, a b x \log \left (c\right ) + a^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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