Optimal. Leaf size=15 \[ \frac {\log \left (b+c x^n\right )}{n}+\log (x) \]
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Rubi [A] time = 0.02, antiderivative size = 15, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {\log \left (b+c x^n\right )}{n}+\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 72
Rule 446
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*}
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Mathematica [A] time = 0.02, size = 15, normalized size = 1.00 \begin {gather*} \frac {\log \left (b+c x^n\right )}{n}+\log (x) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 24, normalized size = 1.60 \begin {gather*} \frac {\log \left (b n+c n x^n\right )}{n}+\frac {\log \left (x^n\right )}{n} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 17, normalized size = 1.13 \begin {gather*} \frac {n \log \relax (x) + \log \left (c x^{n} + b\right )}{n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, c x^{n} + b}{{\left (c x^{n} + b\right )} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 17, normalized size = 1.13 \begin {gather*} \frac {\ln \left (\left (c \,x^{n}+b \right ) x^{n}\right )}{n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.49, size = 47, normalized size = 3.13 \begin {gather*} b {\left (\frac {\log \relax (x)}{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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.72, size = 15, normalized size = 1.00 \begin {gather*} \ln \relax (x)+\frac {\ln \left (b+c\,x^n\right )}{n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.62, size = 29, normalized size = 1.93 \begin {gather*} \begin {cases} \log {\relax (x )} & \text {for}\: c = 0 \wedge n = 0 \\\frac {\left (b + 2 c\right ) \log {\relax (x )}}{b + c} & \text {for}\: n = 0 \\\log {\relax (x )} & \text {for}\: c = 0 \\\log {\relax (x )} + \frac {\log {\left (\frac {b}{c} + x^{n} \right )}}{n} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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