Optimal. Leaf size=20 \[ \frac{1}{2} \left (a x+b \log ^2\left (c x^n\right )\right )^2 \]
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Rubi [A] time = 0.0929345, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054, Rules used = {2561, 2544} \[ \frac{1}{2} \left (a x+b \log ^2\left (c x^n\right )\right )^2 \]
Antiderivative was successfully verified.
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Rule 2561
Rule 2544
Rubi steps
\begin{align*} \int \left (\frac{a}{x}+\frac{2 b n \log \left (c x^n\right )}{x^2}\right ) \left (a x^2+b x \log ^2\left (c x^n\right )\right ) \, dx &=\int \frac{\left (a x+2 b n \log \left (c x^n\right )\right ) \left (a x^2+b x \log ^2\left (c x^n\right )\right )}{x^2} \, dx\\ &=\int \frac{\left (a x+2 b n \log \left (c x^n\right )\right ) \left (a x+b \log ^2\left (c x^n\right )\right )}{x} \, dx\\ &=\frac{1}{2} \left (a x+b \log ^2\left (c x^n\right )\right )^2\\ \end{align*}
Mathematica [A] time = 0.0040864, size = 38, normalized size = 1.9 \[ \frac{a^2 x^2}{2}+a b x \log ^2\left (c x^n\right )+\frac{1}{2} b^2 \log ^4\left (c x^n\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.018, size = 63, normalized size = 3.2 \begin{align*}{\frac{{a}^{2}{x}^{2}}{2}}+abx \left ( \ln \left ( c{{\rm e}^{n\ln \left ( x \right ) }} \right ) \right ) ^{2}-2\,bnax\ln \left ( c{{\rm e}^{n\ln \left ( x \right ) }} \right ) +{\frac{{b}^{2} \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{4}}{2}}+2\,\ln \left ( c{x}^{n} \right ) abnx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.04964, size = 100, normalized size = 5. \begin{align*} \frac{1}{2} \, b^{2} \log \left (c x^{n}\right )^{4} - 2 \, a b n^{2} x + 2 \, a b n x \log \left (c x^{n}\right ) + a b x \log \left (c x^{n}\right )^{2} + \frac{1}{2} \, a^{2} x^{2} + 2 \,{\left (n^{2} x - n x \log \left (c x^{n}\right )\right )} a b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.96197, size = 230, normalized size = 11.5 \begin{align*} \frac{1}{2} \, b^{2} n^{4} \log \left (x\right )^{4} + 2 \, b^{2} n^{3} \log \left (c\right ) \log \left (x\right )^{3} + a b x \log \left (c\right )^{2} + \frac{1}{2} \, a^{2} x^{2} +{\left (3 \, b^{2} n^{2} \log \left (c\right )^{2} + a b n^{2} x\right )} \log \left (x\right )^{2} + 2 \,{\left (b^{2} n \log \left (c\right )^{3} + a b n x \log \left (c\right )\right )} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.31994, size = 117, normalized size = 5.85 \begin{align*} \frac{a^{2} x^{2}}{2} + a b n^{2} x \log{\left (x \right )}^{2} - 2 a b n^{2} x \log{\left (x \right )} + 2 a b n x \log{\left (c \right )} \log{\left (x \right )} - 2 a b n x \log{\left (c \right )} + 2 a b n x \log{\left (c x^{n} \right )} + a b x \log{\left (c \right )}^{2} - 2 b^{2} n \left (\begin{cases} - \log{\left (c \right )}^{3} \log{\left (x \right )} & \text{for}\: n = 0 \\- \frac{\log{\left (c x^{n} \right )}^{4}}{4 n} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.23081, size = 122, normalized size = 6.1 \begin{align*} \frac{1}{2} \, b^{2} n^{4} \log \left (x\right )^{4} + 2 \, b^{2} n^{3} \log \left (c\right ) \log \left (x\right )^{3} + 2 \, b^{2} n \log \left (c\right )^{3} \log \left (x\right ) + 2 \, a b n x \log \left (c\right ) \log \left (x\right ) + a b x \log \left (c\right )^{2} + \frac{1}{2} \, a^{2} x^{2} +{\left (3 \, b^{2} n^{2} \log \left (c\right )^{2} + a b n^{2} x\right )} \log \left (x\right )^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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