Optimal. Leaf size=38 \[ \frac{b \log \left (a+b x^n\right )}{a^2 n}-\frac{b \log (x)}{a^2}-\frac{x^{-n}}{a n} \]
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Rubi [A] time = 0.0215037, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {266, 44} \[ \frac{b \log \left (a+b x^n\right )}{a^2 n}-\frac{b \log (x)}{a^2}-\frac{x^{-n}}{a n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{x^{-1-n}}{a+b x^n} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 (a+b x)} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a x^2}-\frac{b}{a^2 x}+\frac{b^2}{a^2 (a+b x)}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{x^{-n}}{a n}-\frac{b \log (x)}{a^2}+\frac{b \log \left (a+b x^n\right )}{a^2 n}\\ \end{align*}
Mathematica [A] time = 0.0271071, size = 32, normalized size = 0.84 \[ -\frac{-b \log \left (a+b x^n\right )+a x^{-n}+b n \log (x)}{a^2 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 50, normalized size = 1.3 \begin{align*}{\frac{1}{{{\rm e}^{n\ln \left ( x \right ) }}} \left ( -{\frac{1}{an}}-{\frac{b\ln \left ( x \right ){{\rm e}^{n\ln \left ( x \right ) }}}{{a}^{2}}} \right ) }+{\frac{b\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{a}^{2}n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.975883, size = 57, normalized size = 1.5 \begin{align*} -\frac{b \log \left (x\right )}{a^{2}} + \frac{b \log \left (\frac{b x^{n} + a}{b}\right )}{a^{2} n} - \frac{1}{a n x^{n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.07272, size = 78, normalized size = 2.05 \begin{align*} -\frac{b n x^{n} \log \left (x\right ) - b x^{n} \log \left (b x^{n} + a\right ) + a}{a^{2} n x^{n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 57.1692, size = 48, normalized size = 1.26 \begin{align*} \begin{cases} \frac{\log{\left (x \right )}}{b} & \text{for}\: a = 0 \wedge n = 0 \\- \frac{x^{- 2 n}}{2 b n} & \text{for}\: a = 0 \\\frac{\log{\left (x \right )}}{a + b} & \text{for}\: n = 0 \\- \frac{x^{- n}}{a n} + \frac{b \log{\left (x^{- n} + \frac{b}{a} \right )}}{a^{2} 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{x^{-n - 1}}{b x^{n} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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