Optimal. Leaf size=21 \[ \frac{x \left (a x^n\right )^{-1/n} (c x)^m}{m} \]
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Rubi [A] time = 0.0045587, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {15, 16, 32} \[ \frac{x \left (a x^n\right )^{-1/n} (c x)^m}{m} \]
Antiderivative was successfully verified.
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Rule 15
Rule 16
Rule 32
Rubi steps
\begin{align*} \int (c x)^m \left (a x^n\right )^{-1/n} \, dx &=\left (x \left (a x^n\right )^{-1/n}\right ) \int \frac{(c x)^m}{x} \, dx\\ &=\left (c x \left (a x^n\right )^{-1/n}\right ) \int (c x)^{-1+m} \, dx\\ &=\frac{x (c x)^m \left (a x^n\right )^{-1/n}}{m}\\ \end{align*}
Mathematica [A] time = 0.002479, size = 21, normalized size = 1. \[ \frac{x \left (a x^n\right )^{-1/n} (c x)^m}{m} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 22, normalized size = 1.1 \begin{align*}{\frac{x \left ( cx \right ) ^{m}}{m\sqrt [n]{a{x}^{n}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00879, size = 41, normalized size = 1.95 \begin{align*} \frac{c^{m} x e^{\left (m \log \left (x\right ) - \frac{\log \left (x^{n}\right )}{n}\right )}}{a^{\left (\frac{1}{n}\right )} m} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81905, size = 50, normalized size = 2.38 \begin{align*} \frac{e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )}}{a^{\left (\frac{1}{n}\right )} m} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10933, size = 28, normalized size = 1.33 \begin{align*} \frac{e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )}}{a^{\left (\frac{1}{n}\right )} m} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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