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 = 0.02, antiderivative size = 23, normalized size of antiderivative = 1.00, 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 29
Rule 31
Rule 36
Rule 266
Rule 1584
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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.93, size = 22, normalized size = 0.96 \[ \frac {n \log \relax (x) - \log \left (c x^{n} + b\right )}{b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 25, normalized size = 1.09 \[ \frac {\log \left ({\left | x \right |}\right )}{b} - \frac {\log \left ({\left | c x^{n} + b \right |}\right )}{b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 26, normalized size = 1.13 \[ \frac {\ln \relax (x )}{b}-\frac {\ln \left (c \,{\mathrm e}^{n \ln \relax (x )}+b \right )}{b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.90, size = 27, normalized size = 1.17 \[ \frac {\log \relax (x)}{b} - \frac {\log \left (\frac {c x^{n} + b}{c}\right )}{b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.37, size = 20, normalized size = 0.87 \[ -\frac {2\,\mathrm {atanh}\left (\frac {2\,c\,x^n}{b}+1\right )}{b\,n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 16.49, size = 66, normalized size = 2.87 \[ \begin {cases} \tilde {\infty } \log {\relax (x )} & \text {for}\: b = 0 \wedge c = 0 \wedge n = 0 \\- \frac {x^{- n}}{c n} & \text {for}\: b = 0 \\\frac {\log {\relax (x )}}{b + c} & \text {for}\: n = 0 \\\frac {\frac {n^{2} \log {\relax (x )}}{n^{2} - n} - \frac {n \log {\relax (x )}}{n^{2} - n}}{b} & \text {for}\: c = 0 \\\frac {2 \log {\relax (x )}}{b} - \frac {\log {\left (\frac {b x^{n}}{c} + x^{2 n} \right )}}{b n} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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