Optimal. Leaf size=15 \[ \frac{\log \left (a x^n+b\right )}{a n} \]
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Rubi [A] time = 0.0076386, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {1593, 260} \[ \frac{\log \left (a x^n+b\right )}{a n} \]
Antiderivative was successfully verified.
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Rule 1593
Rule 260
Rubi steps
\begin{align*} \int \frac{1}{a x+b x^{1-n}} \, dx &=\int \frac{x^{-1+n}}{b+a x^n} \, dx\\ &=\frac{\log \left (b+a x^n\right )}{a n}\\ \end{align*}
Mathematica [A] time = 0.0044175, size = 15, normalized size = 1. \[ \frac{\log \left (a x^n+b\right )}{a n} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.01, size = 41, normalized size = 2.7 \begin{align*} -{\frac{\ln \left ( x \right ) }{an}}+{\frac{\ln \left ( x \right ) }{a}}+{\frac{\ln \left ( ax+b{{\rm e}^{ \left ( 1-n \right ) \ln \left ( x \right ) }} \right ) }{an}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02379, size = 26, normalized size = 1.73 \begin{align*} \frac{\log \left (\frac{a x^{n} + b}{a}\right )}{a n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.90674, size = 68, normalized size = 4.53 \begin{align*} \frac{{\left (n - 1\right )} \log \left (x\right ) + \log \left (a x + b x^{-n + 1}\right )}{a n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.99376, size = 39, normalized size = 2.6 \begin{align*} \begin{cases} \tilde{\infty } \log{\left (x \right )} & \text{for}\: a = 0 \wedge b = 0 \wedge n = 0 \\\frac{\log{\left (x \right )}}{a} & \text{for}\: b = 0 \\\frac{\log{\left (x \right )}}{a + b} & \text{for}\: n = 0 \\\frac{x^{n}}{b n} & \text{for}\: a = 0 \\\frac{\log{\left (x \right )}}{a} + \frac{\log{\left (\frac{a}{b} + x^{- n} \right )}}{a n} & \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{1}{a x + b x^{-n + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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