Optimal. Leaf size=57 \[ -\frac {c^2 \log \left (b+c x^n\right )}{b^3 n}+\frac {c^2 \log (x)}{b^3}+\frac {c x^{-n}}{b^2 n}-\frac {x^{-2 n}}{2 b n} \]
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Rubi [A] time = 0.04, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {1584, 266, 44} \[ -\frac {c^2 \log \left (b+c x^n\right )}{b^3 n}+\frac {c^2 \log (x)}{b^3}+\frac {c x^{-n}}{b^2 n}-\frac {x^{-2 n}}{2 b n} \]
Antiderivative was successfully verified.
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Rule 44
Rule 266
Rule 1584
Rubi steps
\begin {align*} \int \frac {x^{-1-n}}{b x^n+c x^{2 n}} \, dx &=\int \frac {x^{-1-2 n}}{b+c x^n} \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x^3 (b+c x)} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{b x^3}-\frac {c}{b^2 x^2}+\frac {c^2}{b^3 x}-\frac {c^3}{b^3 (b+c x)}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac {x^{-2 n}}{2 b n}+\frac {c x^{-n}}{b^2 n}+\frac {c^2 \log (x)}{b^3}-\frac {c^2 \log \left (b+c x^n\right )}{b^3 n}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 49, normalized size = 0.86 \[ \frac {-2 c^2 \log \left (b+c x^n\right )+b x^{-2 n} \left (2 c x^n-b\right )+2 c^2 n \log (x)}{2 b^3 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 59, normalized size = 1.04 \[ \frac {2 \, c^{2} n x^{2 \, n} \log \relax (x) - 2 \, c^{2} x^{2 \, n} \log \left (c x^{n} + b\right ) + 2 \, b c x^{n} - b^{2}}{2 \, b^{3} n x^{2 \, n}} \]
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 \[ \int \frac {x^{-n - 1}}{c x^{2 \, n} + b x^{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 69, normalized size = 1.21 \[ \left (\frac {c^{2} {\mathrm e}^{2 n \ln \relax (x )} \ln \relax (x )}{b^{3}}+\frac {c \,{\mathrm e}^{n \ln \relax (x )}}{b^{2} n}-\frac {1}{2 b n}\right ) {\mathrm e}^{-2 n \ln \relax (x )}-\frac {c^{2} \ln \left (c \,{\mathrm e}^{n \ln \relax (x )}+b \right )}{b^{3} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.91, size = 58, normalized size = 1.02 \[ \frac {c^{2} \log \relax (x)}{b^{3}} - \frac {c^{2} \log \left (\frac {c x^{n} + b}{c}\right )}{b^{3} n} + \frac {2 \, c x^{n} - b}{2 \, b^{2} n x^{2 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{x^{n+1}\,\left (b\,x^n+c\,x^{2\,n}\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
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