Optimal. Leaf size=15 \[ \frac{\log \left (b+c x^n\right )}{n}+\log (x) \]
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Rubi [A] time = 0.0253765, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {1593, 446, 72} \[ \frac{\log \left (b+c x^n\right )}{n}+\log (x) \]
Antiderivative was successfully verified.
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Rule 1593
Rule 446
Rule 72
Rubi steps
\begin{align*} \int \frac{b+2 c x^n}{b x+c x^{1+n}} \, dx &=\int \frac{b+2 c x^n}{x \left (b+c x^n\right )} \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{b+2 c x}{x (b+c x)} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{x}+\frac{c}{b+c x}\right ) \, dx,x,x^n\right )}{n}\\ &=\log (x)+\frac{\log \left (b+c x^n\right )}{n}\\ \end{align*}
Mathematica [A] time = 0.0117224, size = 15, normalized size = 1. \[ \frac{\log \left (b+c x^n\right )}{n}+\log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 18, normalized size = 1.2 \begin{align*} \ln \left ( x \right ) +{\frac{\ln \left ( c{{\rm e}^{n\ln \left ( x \right ) }}+b \right ) }{n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.05882, size = 63, normalized size = 4.2 \begin{align*} b{\left (\frac{\log \left (x\right )}{b} - \frac{\log \left (\frac{c x^{n} + b}{c}\right )}{b n}\right )} + \frac{2 \, \log \left (\frac{c x^{n} + b}{c}\right )}{n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54709, size = 61, normalized size = 4.07 \begin{align*} \frac{{\left (n - 1\right )} \log \left (x\right ) + \log \left (b x + c x^{n + 1}\right )}{n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.41776, size = 29, normalized size = 1.93 \begin{align*} \begin{cases} \log{\left (x \right )} & \text{for}\: c = 0 \wedge n = 0 \\\frac{\left (b + 2 c\right ) \log{\left (x \right )}}{b + c} & \text{for}\: n = 0 \\\log{\left (x \right )} & \text{for}\: c = 0 \\\log{\left (x \right )} + \frac{\log{\left (\frac{b}{c} + x^{n} \right )}}{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{2 \, c x^{n} + b}{b x + c x^{n + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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