Optimal. Leaf size=14 \[ \frac{\left (b x^n\right )^p}{n p} \]
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Rubi [A] time = 0.0038183, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {15, 30} \[ \frac{\left (b x^n\right )^p}{n p} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rubi steps
\begin{align*} \int \frac{\left (b x^n\right )^p}{x} \, dx &=\left (x^{-n p} \left (b x^n\right )^p\right ) \int x^{-1+n p} \, dx\\ &=\frac{\left (b x^n\right )^p}{n p}\\ \end{align*}
Mathematica [A] time = 0.0011997, size = 14, normalized size = 1. \[ \frac{\left (b x^n\right )^p}{n p} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 15, normalized size = 1.1 \begin{align*}{\frac{ \left ( b{x}^{n} \right ) ^{p}}{np}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.959177, size = 20, normalized size = 1.43 \begin{align*} \frac{b^{p}{\left (x^{n}\right )}^{p}}{n p} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.34491, size = 45, normalized size = 3.21 \begin{align*} \frac{e^{\left (n p \log \left (x\right ) + p \log \left (b\right )\right )}}{n p} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.27713, size = 22, normalized size = 1.57 \begin{align*} \begin{cases} \log{\left (x \right )} & \text{for}\: n = 0 \wedge p = 0 \\b^{p} \log{\left (x \right )} & \text{for}\: n = 0 \\\log{\left (x \right )} & \text{for}\: p = 0 \\\frac{b^{p} \left (x^{n}\right )^{p}}{n p} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (b x^{n}\right )^{p}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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