Optimal. Leaf size=28 \[ \frac{x^{-n} (c x)^n \log \left (a+b x^n\right )}{b c n} \]
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Rubi [A] time = 0.0109541, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {268, 260} \[ \frac{x^{-n} (c x)^n \log \left (a+b x^n\right )}{b c n} \]
Antiderivative was successfully verified.
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Rule 268
Rule 260
Rubi steps
\begin{align*} \int \frac{(c x)^{-1+n}}{a+b x^n} \, dx &=\frac{\left (x^{-n} (c x)^n\right ) \int \frac{x^{-1+n}}{a+b x^n} \, dx}{c}\\ &=\frac{x^{-n} (c x)^n \log \left (a+b x^n\right )}{b c n}\\ \end{align*}
Mathematica [A] time = 0.0055718, size = 29, normalized size = 1.04 \[ \frac{x^{1-n} (c x)^{n-1} \log \left (a+b x^n\right )}{b n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.066, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( cx \right ) ^{-1+n}}{a+b{x}^{n}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.96295, size = 32, normalized size = 1.14 \begin{align*} \frac{c^{n - 1} \log \left (\frac{b x^{n} + a}{b}\right )}{b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.32587, size = 43, normalized size = 1.54 \begin{align*} \frac{c^{n - 1} \log \left (b x^{n} + a\right )}{b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.32101, size = 17, normalized size = 0.61 \begin{align*} \frac{c^{n} \log{\left (1 + \frac{b x^{n}}{a} \right )}}{b c n} \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 (c x\right )^{n - 1}}{b x^{n} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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