Optimal. Leaf size=38 \[ \frac {b \log \left (a+b x^n\right )}{a^2 n}-\frac {b \log (x)}{a^2}-\frac {x^{-n}}{a n} \]
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Rubi [A] time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {266, 44} \[ \frac {b \log \left (a+b x^n\right )}{a^2 n}-\frac {b \log (x)}{a^2}-\frac {x^{-n}}{a n} \]
Antiderivative was successfully verified.
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Rule 44
Rule 266
Rubi steps
\begin {align*} \int \frac {x^{-1-n}}{a+b x^n} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x^2 (a+b x)} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{a x^2}-\frac {b}{a^2 x}+\frac {b^2}{a^2 (a+b x)}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac {x^{-n}}{a n}-\frac {b \log (x)}{a^2}+\frac {b \log \left (a+b x^n\right )}{a^2 n}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 32, normalized size = 0.84 \[ -\frac {-b \log \left (a+b x^n\right )+a x^{-n}+b n \log (x)}{a^2 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 37, normalized size = 0.97 \[ -\frac {b n x^{n} \log \relax (x) - b x^{n} \log \left (b x^{n} + a\right ) + a}{a^{2} n x^{n}} \]
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 \[ \int \frac {x^{-n - 1}}{b x^{n} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 50, normalized size = 1.32 \[ \left (-\frac {b \,{\mathrm e}^{n \ln \relax (x )} \ln \relax (x )}{a^{2}}-\frac {1}{a n}\right ) {\mathrm e}^{-n \ln \relax (x )}+\frac {b \ln \left (b \,{\mathrm e}^{n \ln \relax (x )}+a \right )}{a^{2} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.69, size = 42, normalized size = 1.11 \[ -\frac {b \log \relax (x)}{a^{2}} + \frac {b \log \left (\frac {b x^{n} + a}{b}\right )}{a^{2} n} - \frac {1}{a n x^{n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{x^{n+1}\,\left (a+b\,x^n\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 22.38, size = 48, normalized size = 1.26 \[ \begin {cases} \frac {\log {\relax (x )}}{b} & \text {for}\: a = 0 \wedge n = 0 \\- \frac {x^{- 2 n}}{2 b n} & \text {for}\: a = 0 \\\frac {\log {\relax (x )}}{a + b} & \text {for}\: n = 0 \\- \frac {x^{- n}}{a n} + \frac {b \log {\left (x^{- n} + \frac {b}{a} \right )}}{a^{2} n} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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