Optimal. Leaf size=33 \[ \frac{\left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^{p+1}}{b m n (p+1)} \]
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Rubi [A] time = 0.0939415, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {2302, 30, 2445} \[ \frac{\left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^{p+1}}{b m n (p+1)} \]
Antiderivative was successfully verified.
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Rule 2302
Rule 30
Rule 2445
Rubi steps
\begin{align*} \int \frac{\left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^p}{x} \, dx &=\operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c d^n x^{m n}\right )\right )^p}{x} \, dx,c d^n x^{m n},c \left (d x^m\right )^n\right )\\ &=\operatorname{Subst}\left (\frac{\operatorname{Subst}\left (\int x^p \, dx,x,a+b \log \left (c d^n x^{m n}\right )\right )}{b m n},c d^n x^{m n},c \left (d x^m\right )^n\right )\\ &=\frac{\left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^{1+p}}{b m n (1+p)}\\ \end{align*}
Mathematica [A] time = 0.008967, size = 33, normalized size = 1. \[ \frac{\left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^{p+1}}{b m n (p+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 34, normalized size = 1. \begin{align*}{\frac{ \left ( a+b\ln \left ( c \left ( d{x}^{m} \right ) ^{n} \right ) \right ) ^{1+p}}{mnb \left ( 1+p \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.89625, size = 144, normalized size = 4.36 \begin{align*} \frac{{\left (b m n \log \left (x\right ) + b n \log \left (d\right ) + b \log \left (c\right ) + a\right )}{\left (b m n \log \left (x\right ) + b n \log \left (d\right ) + b \log \left (c\right ) + a\right )}^{p}}{b m n p + b m n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.96086, size = 80, normalized size = 2.42 \begin{align*} - \begin{cases} - a^{p} \log{\left (x \right )} & \text{for}\: b = 0 \\- \left (a + b \log{\left (c d^{n} \right )}\right )^{p} \log{\left (x \right )} & \text{for}\: m = 0 \\- \left (a + b \log{\left (c \right )}\right )^{p} \log{\left (x \right )} & \text{for}\: n = 0 \\- \frac{\begin{cases} \frac{\left (a + b \log{\left (c \left (d x^{m}\right )^{n} \right )}\right )^{p + 1}}{p + 1} & \text{for}\: p \neq -1 \\\log{\left (a + b \log{\left (c \left (d x^{m}\right )^{n} \right )} \right )} & \text{otherwise} \end{cases}}{b m n} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31088, size = 49, normalized size = 1.48 \begin{align*} \frac{{\left (b m n \log \left (x\right ) + b n \log \left (d\right ) + b \log \left (c\right ) + a\right )}^{p + 1}}{b m n{\left (p + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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