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 = 0.01, 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 = {1598, 272, 36,
29, 31} \begin {gather*} \frac {\log (x)}{b}-\frac {\log \left (b+c x^n\right )}{b n} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 272
Rule 1598
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 {\text {Subst}\left (\int \frac {1}{x (b+c x)} \, dx,x,x^n\right )}{n}\\ &=\frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,x^n\right )}{b n}-\frac {c \text {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 = 0.03, size = 25, normalized size = 1.09 \begin {gather*} \frac {\log \left (x^n\right )-\log \left (b n \left (b+c x^n\right )\right )}{b n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 26, normalized size = 1.13
method | result | size |
norman | \(\frac {\ln \left (x \right )}{b}-\frac {\ln \left (c \,{\mathrm e}^{n \ln \left (x \right )}+b \right )}{b n}\) | \(26\) |
risch | \(\frac {\ln \left (x \right )}{b}-\frac {\ln \left (x^{n}+\frac {b}{c}\right )}{b n}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 27, normalized size = 1.17 \begin {gather*} \frac {\log \left (x\right )}{b} - \frac {\log \left (\frac {c x^{n} + b}{c}\right )}{b n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 22, normalized size = 0.96 \begin {gather*} \frac {n \log \left (x\right ) - \log \left (c x^{n} + b\right )}{b n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 42 vs.
\(2 (15) = 30\).
time = 2.76, size = 42, normalized size = 1.83 \begin {gather*} \begin {cases} \frac {\log {\left (x \right )}}{c} & \text {for}\: b = 0 \wedge n = 0 \\- \frac {x^{- n}}{c n} & \text {for}\: b = 0 \\\frac {\log {\left (x \right )}}{b + c} & \text {for}\: n = 0 \\\frac {2 \log {\left (x \right )}}{b} - \frac {\log {\left (x^{n} + \frac {c x^{2 n}}{b} \right )}}{b n} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.83, size = 25, normalized size = 1.09 \begin {gather*} \frac {\log \left ({\left | x \right |}\right )}{b} - \frac {\log \left ({\left | c x^{n} + b \right |}\right )}{b n} \end {gather*}
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 \begin {gather*} -\frac {2\,\mathrm {atanh}\left (\frac {2\,c\,x^n}{b}+1\right )}{b\,n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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