Optimal. Leaf size=76 \[ \frac {c^3 \log \left (b+c x^n\right )}{b^4 n}-\frac {c^3 \log (x)}{b^4}-\frac {c^2 x^{-n}}{b^3 n}+\frac {c x^{-2 n}}{2 b^2 n}-\frac {x^{-3 n}}{3 b n} \]
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Rubi [A] time = 0.05, antiderivative size = 76, 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} \[ -\frac {c^2 x^{-n}}{b^3 n}+\frac {c^3 \log \left (b+c x^n\right )}{b^4 n}-\frac {c^3 \log (x)}{b^4}+\frac {c x^{-2 n}}{2 b^2 n}-\frac {x^{-3 n}}{3 b n} \]
Antiderivative was successfully verified.
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Rule 44
Rule 266
Rule 1584
Rubi steps
\begin {align*} \int \frac {x^{-1-2 n}}{b x^n+c x^{2 n}} \, dx &=\int \frac {x^{-1-3 n}}{b+c x^n} \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x^4 (b+c x)} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{b x^4}-\frac {c}{b^2 x^3}+\frac {c^2}{b^3 x^2}-\frac {c^3}{b^4 x}+\frac {c^4}{b^4 (b+c x)}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac {x^{-3 n}}{3 b n}+\frac {c x^{-2 n}}{2 b^2 n}-\frac {c^2 x^{-n}}{b^3 n}-\frac {c^3 \log (x)}{b^4}+\frac {c^3 \log \left (b+c x^n\right )}{b^4 n}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 62, normalized size = 0.82 \[ -\frac {b x^{-3 n} \left (2 b^2-3 b c x^n+6 c^2 x^{2 n}\right )-6 c^3 \log \left (b+c x^n\right )+6 c^3 n \log (x)}{6 b^4 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 72, normalized size = 0.95 \[ -\frac {6 \, c^{3} n x^{3 \, n} \log \relax (x) - 6 \, c^{3} x^{3 \, n} \log \left (c x^{n} + b\right ) + 6 \, b c^{2} x^{2 \, n} - 3 \, b^{2} c x^{n} + 2 \, b^{3}}{6 \, b^{4} n x^{3 \, n}} \]
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 \[ \int \frac {x^{-2 \, n - 1}}{c x^{2 \, n} + b x^{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 88, normalized size = 1.16 \[ \left (-\frac {c^{3} {\mathrm e}^{3 n \ln \relax (x )} \ln \relax (x )}{b^{4}}+\frac {c \,{\mathrm e}^{n \ln \relax (x )}}{2 b^{2} n}-\frac {c^{2} {\mathrm e}^{2 n \ln \relax (x )}}{b^{3} n}-\frac {1}{3 b n}\right ) {\mathrm e}^{-3 n \ln \relax (x )}+\frac {c^{3} \ln \left (c \,{\mathrm e}^{n \ln \relax (x )}+b \right )}{b^{4} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.91, size = 71, normalized size = 0.93 \[ -\frac {c^{3} \log \relax (x)}{b^{4}} + \frac {c^{3} \log \left (\frac {c x^{n} + b}{c}\right )}{b^{4} n} - \frac {6 \, c^{2} x^{2 \, n} - 3 \, b c x^{n} + 2 \, b^{2}}{6 \, b^{3} n x^{3 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^{2\,n+1}\,\left (b\,x^n+c\,x^{2\,n}\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 52.91, size = 73, normalized size = 0.96 \[ - \frac {x^{- 3 n}}{3 b n} + \frac {c x^{- 2 n}}{2 b^{2} n} - \frac {c^{2} x^{- n}}{b^{3} n} + \frac {c^{4} \left (\begin {cases} \frac {x^{n}}{b} & \text {for}\: c = 0 \\\frac {\log {\left (b + c x^{n} \right )}}{c} & \text {otherwise} \end {cases}\right )}{b^{4} n} - \frac {c^{3} \log {\left (x^{n} \right )}}{b^{4} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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