Optimal. Leaf size=16 \[ a x^m+b \log ^q\left (c x^n\right ) \]
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Rubi [A] time = 0.034799, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {14, 2302, 30} \[ a x^m+b \log ^q\left (c x^n\right ) \]
Antiderivative was successfully verified.
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Rule 14
Rule 2302
Rule 30
Rubi steps
\begin{align*} \int \frac{a m x^m+b n q \log ^{-1+q}\left (c x^n\right )}{x} \, dx &=\int \left (a m x^{-1+m}+\frac{b n q \log ^{-1+q}\left (c x^n\right )}{x}\right ) \, dx\\ &=a x^m+(b n q) \int \frac{\log ^{-1+q}\left (c x^n\right )}{x} \, dx\\ &=a x^m+(b q) \operatorname{Subst}\left (\int x^{-1+q} \, dx,x,\log \left (c x^n\right )\right )\\ &=a x^m+b \log ^q\left (c x^n\right )\\ \end{align*}
Mathematica [A] time = 0.0146627, size = 16, normalized size = 1. \[ a x^m+b \log ^q\left (c x^n\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 17, normalized size = 1.1 \begin{align*} a{x}^{m}+b \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{q} \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 = 1.9989, size = 82, normalized size = 5.12 \begin{align*}{\left (b n \log \left (x\right ) + b \log \left (c\right )\right )}{\left (n \log \left (x\right ) + \log \left (c\right )\right )}^{q - 1} + a x^{m} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 61.9015, size = 58, normalized size = 3.62 \begin{align*} a m \left (\begin{cases} \frac{x^{m}}{m} & \text{for}\: m \neq 0 \\\log{\left (x \right )} & \text{otherwise} \end{cases}\right ) + b n q \left (\begin{cases} \frac{\log{\left (x \right )}}{\log{\left (c \right )}} & \text{for}\: n = 0 \wedge q = 0 \\\frac{\log{\left (c \right )}^{q} \log{\left (x \right )}}{\log{\left (c \right )}} & \text{for}\: n = 0 \\\frac{\log{\left (n \log{\left (x \right )} + \log{\left (c \right )} \right )}}{n} & \text{for}\: q = 0 \\\frac{\left (n \log{\left (x \right )} + \log{\left (c \right )}\right )^{q}}{n q} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31518, size = 23, normalized size = 1.44 \begin{align*}{\left (n \log \left (x\right ) + \log \left (c\right )\right )}^{q} b + a x^{m} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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