Optimal. Leaf size=93 \[ -\frac {x^{-4 n}}{4 b n}+\frac {c x^{-3 n}}{3 b^2 n}-\frac {c^2 x^{-2 n}}{2 b^3 n}+\frac {c^3 x^{-n}}{b^4 n}+\frac {c^4 \log (x)}{b^5}-\frac {c^4 \log \left (b+c x^n\right )}{b^5 n} \]
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Rubi [A]
time = 0.04, antiderivative size = 93, 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 = {1598, 272, 46}
\begin {gather*} -\frac {c^4 \log \left (b+c x^n\right )}{b^5 n}+\frac {c^4 \log (x)}{b^5}+\frac {c^3 x^{-n}}{b^4 n}-\frac {c^2 x^{-2 n}}{2 b^3 n}+\frac {c x^{-3 n}}{3 b^2 n}-\frac {x^{-4 n}}{4 b n} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 272
Rule 1598
Rubi steps
\begin {align*} \int \frac {x^{-1-3 n}}{b x^n+c x^{2 n}} \, dx &=\int \frac {x^{-1-4 n}}{b+c x^n} \, dx\\ &=\frac {\text {Subst}\left (\int \frac {1}{x^5 (b+c x)} \, dx,x,x^n\right )}{n}\\ &=\frac {\text {Subst}\left (\int \left (\frac {1}{b x^5}-\frac {c}{b^2 x^4}+\frac {c^2}{b^3 x^3}-\frac {c^3}{b^4 x^2}+\frac {c^4}{b^5 x}-\frac {c^5}{b^5 (b+c x)}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac {x^{-4 n}}{4 b n}+\frac {c x^{-3 n}}{3 b^2 n}-\frac {c^2 x^{-2 n}}{2 b^3 n}+\frac {c^3 x^{-n}}{b^4 n}+\frac {c^4 \log (x)}{b^5}-\frac {c^4 \log \left (b+c x^n\right )}{b^5 n}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 76, normalized size = 0.82 \begin {gather*} -\frac {b x^{-4 n} \left (3 b^3-4 b^2 c x^n+6 b c^2 x^{2 n}-12 c^3 x^{3 n}\right )-12 c^4 \log \left (x^n\right )+12 c^4 \log \left (b+c x^n\right )}{12 b^5 n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 90, normalized size = 0.97
method | result | size |
risch | \(\frac {c^{3} x^{-n}}{b^{4} n}-\frac {c^{2} x^{-2 n}}{2 b^{3} n}+\frac {c \,x^{-3 n}}{3 b^{2} n}-\frac {x^{-4 n}}{4 b n}+\frac {c^{4} \ln \left (x \right )}{b^{5}}-\frac {c^{4} \ln \left (x^{n}+\frac {b}{c}\right )}{b^{5} n}\) | \(90\) |
norman | \(\left (\frac {c^{3} {\mathrm e}^{3 n \ln \left (x \right )}}{b^{4} n}-\frac {1}{4 b n}+\frac {c \,{\mathrm e}^{n \ln \left (x \right )}}{3 b^{2} n}-\frac {c^{2} {\mathrm e}^{2 n \ln \left (x \right )}}{2 b^{3} n}+\frac {c^{4} \ln \left (x \right ) {\mathrm e}^{4 n \ln \left (x \right )}}{b^{5}}\right ) {\mathrm e}^{-4 n \ln \left (x \right )}-\frac {c^{4} \ln \left (c \,{\mathrm e}^{n \ln \left (x \right )}+b \right )}{b^{5} n}\) | \(105\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 84, normalized size = 0.90 \begin {gather*} \frac {c^{4} \log \left (x\right )}{b^{5}} - \frac {c^{4} \log \left (\frac {c x^{n} + b}{c}\right )}{b^{5} n} + \frac {12 \, c^{3} x^{3 \, n} - 6 \, b c^{2} x^{2 \, n} + 4 \, b^{2} c x^{n} - 3 \, b^{3}}{12 \, b^{4} n x^{4 \, n}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 85, normalized size = 0.91 \begin {gather*} \frac {12 \, c^{4} n x^{4 \, n} \log \left (x\right ) - 12 \, c^{4} x^{4 \, n} \log \left (c x^{n} + b\right ) + 12 \, b c^{3} x^{3 \, n} - 6 \, b^{2} c^{2} x^{2 \, n} + 4 \, b^{3} c x^{n} - 3 \, b^{4}}{12 \, b^{5} n x^{4 \, n}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 40.51, size = 88, normalized size = 0.95 \begin {gather*} - \frac {x^{- 4 n}}{4 b n} + \frac {c x^{- 3 n}}{3 b^{2} n} - \frac {c^{2} x^{- 2 n}}{2 b^{3} n} + \frac {c^{3} x^{- n}}{b^{4} n} - \frac {c^{5} \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^{5} n} + \frac {c^{4} \log {\left (x^{n} \right )}}{b^{5} n} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{x^{3\,n+1}\,\left (b\,x^n+c\,x^{2\,n}\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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