Optimal. Leaf size=15 \[ \frac {\log \left (b+c x^n\right )}{n}+\log (x) \]
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Rubi [A] time = 0.04, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {1584, 446, 72} \[ \frac {\log \left (b+c x^n\right )}{n}+\log (x) \]
Antiderivative was successfully verified.
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Rule 72
Rule 446
Rule 1584
Rubi steps
\begin {align*} \int \frac {x^{-1+n} \left (b+2 c x^n\right )}{b x^n+c x^{2 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*}
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Mathematica [A] time = 0.01, size = 15, normalized size = 1.00 \[ \frac {\log \left (b+c x^n\right )}{n}+\log (x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 17, normalized size = 1.13 \[ \frac {n \log \relax (x) + \log \left (c x^{n} + b\right )}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 17, normalized size = 1.13 \[ \frac {\log \left ({\left | c x^{n} + b \right |}\right )}{n} + \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 18, normalized size = 1.20 \[ \ln \relax (x )+\frac {\ln \left (c \,{\mathrm e}^{n \ln \relax (x )}+b \right )}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 47, normalized size = 3.13 \[ b {\left (\frac {\log \relax (x)}{b} - \frac {\log \left (\frac {c x^{n} + b}{c}\right )}{b n}\right )} + \frac {2 \, \log \left (\frac {c x^{n} + b}{c}\right )}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.23, size = 28, normalized size = 1.87 \[ \frac {2\,\left (\ln \left (b+c\,x^n\right )-\mathrm {atanh}\left (\frac {2\,c\,x^n}{b}+1\right )\right )}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 31.23, size = 48, normalized size = 3.20 \[ \begin {cases} \log {\relax (x )} & \text {for}\: c = 0 \wedge n = 0 \\\frac {\left (b + 2 c\right ) \log {\relax (x )}}{b + c} & \text {for}\: n = 0 \\\frac {n^{2} \log {\relax (x )}}{n^{2} - n} - \frac {n \log {\relax (x )}}{n^{2} - n} & \text {for}\: c = 0 \\\log {\relax (x )} + \frac {\log {\left (\frac {b}{c} + x^{n} \right )}}{n} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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