Optimal. Leaf size=15 \[ \frac{\log \left (b+c x^n\right )}{n}+\log (x) \]
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Rubi [A] time = 0.0153942, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {446, 72} \[ \frac{\log \left (b+c x^n\right )}{n}+\log (x) \]
Antiderivative was successfully verified.
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Rule 446
Rule 72
Rubi steps
\begin{align*} \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 = 0.0137174, 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 = 0.001, size = 17, normalized size = 1.1 \begin{align*}{\frac{\ln \left ({x}^{n} \left ( b+c{x}^{n} \right ) \right ) }{n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.931054, 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 = 1.09325, size = 42, normalized size = 2.8 \begin{align*} \frac{n \log \left (x\right ) + \log \left (c x^{n} + b\right )}{n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.576391, size = 29, normalized size = 1.93 \begin{align*} \begin{cases} \log{\left (x \right )} & \text{for}\: c = 0 \wedge \left (c = 0 \vee n = 0\right ) \\\frac{\left (b + 2 c\right ) \log{\left (x \right )}}{b + c} & \text{for}\: n = 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}{{\left (c x^{n} + b\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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