Optimal. Leaf size=36 \[ \frac{b \log \left (b x^n+2\right )}{4 n}-\frac{1}{4} b \log (x)-\frac{x^{-n}}{2 n} \]
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Rubi [A] time = 0.0190837, antiderivative size = 36, 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} \[ \frac{b \log \left (b x^n+2\right )}{4 n}-\frac{1}{4} b \log (x)-\frac{x^{-n}}{2 n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{x^{-1-n}}{2+b x^n} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 (2+b x)} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{2 x^2}-\frac{b}{4 x}+\frac{b^2}{4 (2+b x)}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{x^{-n}}{2 n}-\frac{1}{4} b \log (x)+\frac{b \log \left (2+b x^n\right )}{4 n}\\ \end{align*}
Mathematica [A] time = 0.0270057, size = 31, normalized size = 0.86 \[ -\frac{-b \log \left (b x^n+2\right )+b n \log (x)+2 x^{-n}}{4 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 42, normalized size = 1.2 \begin{align*}{\frac{1}{{{\rm e}^{n\ln \left ( x \right ) }}} \left ( -{\frac{b\ln \left ( x \right ){{\rm e}^{n\ln \left ( x \right ) }}}{4}}-{\frac{1}{2\,n}} \right ) }+{\frac{b\ln \left ( 2+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{4\,n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.985828, size = 46, normalized size = 1.28 \begin{align*} -\frac{1}{4} \, b \log \left (x\right ) + \frac{b \log \left (\frac{b x^{n} + 2}{b}\right )}{4 \, n} - \frac{1}{2 \, n x^{n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.02583, size = 78, normalized size = 2.17 \begin{align*} -\frac{b n x^{n} \log \left (x\right ) - b x^{n} \log \left (b x^{n} + 2\right ) + 2}{4 \, n x^{n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 34.9272, size = 29, normalized size = 0.81 \begin{align*} \begin{cases} \frac{b \log{\left (\frac{b}{2} + x^{- n} \right )}}{4 n} - \frac{x^{- n}}{2 n} & \text{for}\: n \neq 0 \\\frac{\log{\left (x \right )}}{b + 2} & \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} + 2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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