Optimal. Leaf size=25 \[ \frac{e \log ^q\left (c x^n\right )}{n q}+\frac{d x^m}{m} \]
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Rubi [A] time = 0.0338121, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {14, 2302, 30} \[ \frac{e \log ^q\left (c x^n\right )}{n q}+\frac{d x^m}{m} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2302
Rule 30
Rubi steps
\begin{align*} \int \frac{d x^m+e \log ^{-1+q}\left (c x^n\right )}{x} \, dx &=\int \left (d x^{-1+m}+\frac{e \log ^{-1+q}\left (c x^n\right )}{x}\right ) \, dx\\ &=\frac{d x^m}{m}+e \int \frac{\log ^{-1+q}\left (c x^n\right )}{x} \, dx\\ &=\frac{d x^m}{m}+\frac{e \operatorname{Subst}\left (\int x^{-1+q} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{d x^m}{m}+\frac{e \log ^q\left (c x^n\right )}{n q}\\ \end{align*}
Mathematica [A] time = 0.0201541, size = 25, normalized size = 1. \[ \frac{e \log ^q\left (c x^n\right )}{n q}+\frac{d x^m}{m} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 26, normalized size = 1. \begin{align*}{\frac{d{x}^{m}}{m}}+{\frac{e \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{q}}{nq}} \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 = 1.86243, size = 107, normalized size = 4.28 \begin{align*} \frac{d n q x^{m} +{\left (e m n \log \left (x\right ) + e m \log \left (c\right )\right )}{\left (n \log \left (x\right ) + \log \left (c\right )\right )}^{q - 1}}{m n q} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 60.9989, size = 53, normalized size = 2.12 \begin{align*} d \left (\begin{cases} \frac{x^{m}}{m} & \text{for}\: m \neq 0 \\\log{\left (x \right )} & \text{otherwise} \end{cases}\right ) + e \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 = 1.31321, size = 36, normalized size = 1.44 \begin{align*} \frac{d x^{m}}{m} + \frac{{\left (n \log \left (x\right ) + \log \left (c\right )\right )}^{q} e}{n q} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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