Optimal. Leaf size=23 \[ \frac{\log (x)}{b}-\frac{\log \left (b+c x^n\right )}{b n} \]
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Rubi [A] time = 0.0185387, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {1584, 266, 36, 29, 31} \[ \frac{\log (x)}{b}-\frac{\log \left (b+c x^n\right )}{b n} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 266
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{x^{-1+n}}{b x^n+c x^{2 n}} \, dx &=\int \frac{1}{x \left (b+c x^n\right )} \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x (b+c x)} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^n\right )}{b n}-\frac{c \operatorname{Subst}\left (\int \frac{1}{b+c x} \, dx,x,x^n\right )}{b n}\\ &=\frac{\log (x)}{b}-\frac{\log \left (b+c x^n\right )}{b n}\\ \end{align*}
Mathematica [A] time = 0.006655, size = 22, normalized size = 0.96 \[ \frac{n \log (x)-\log \left (b+c x^n\right )}{b n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 26, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( x \right ) }{b}}-{\frac{\ln \left ( c{{\rm e}^{n\ln \left ( x \right ) }}+b \right ) }{bn}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.989826, size = 36, normalized size = 1.57 \begin{align*} \frac{\log \left (x\right )}{b} - \frac{\log \left (\frac{c x^{n} + b}{c}\right )}{b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.85103, size = 47, normalized size = 2.04 \begin{align*} \frac{n \log \left (x\right ) - \log \left (c x^{n} + b\right )}{b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 14.6952, size = 42, normalized size = 1.83 \begin{align*} \begin{cases} \frac{\log{\left (x \right )}}{c} & \text{for}\: b = 0 \wedge n = 0 \\\frac{\log{\left (x \right )}}{b + c} & \text{for}\: n = 0 \\- \frac{x^{- n}}{c n} & \text{for}\: b = 0 \\\frac{2 \log{\left (x \right )}}{b} - \frac{\log{\left (x^{n} + \frac{c x^{2 n}}{b} \right )}}{b n} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10034, size = 34, normalized size = 1.48 \begin{align*} \frac{\log \left ({\left | x \right |}\right )}{b} - \frac{\log \left ({\left | c x^{n} + b \right |}\right )}{b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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