Optimal. Leaf size=57 \[ -\frac {x^{-2 n}}{2 b n}+\frac {c x^{-n}}{b^2 n}+\frac {c^2 \log (x)}{b^3}-\frac {c^2 \log \left (b+c x^n\right )}{b^3 n} \]
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Rubi [A]
time = 0.03, 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 = {1598, 272, 46}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 272
Rule 1598
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 {\text {Subst}\left (\int \frac {1}{x^3 (b+c x)} \, dx,x,x^n\right )}{n}\\ &=\frac {\text {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 = 0.06, size = 48, normalized size = 0.84 \begin {gather*} -\frac {b x^{-2 n} \left (b-2 c x^n\right )-2 c^2 \log \left (x^n\right )+2 c^2 \log \left (b+c x^n\right )}{2 b^3 n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 58, normalized size = 1.02
method | result | size |
risch | \(\frac {c \,x^{-n}}{b^{2} n}-\frac {x^{-2 n}}{2 b n}+\frac {c^{2} \ln \left (x \right )}{b^{3}}-\frac {c^{2} \ln \left (x^{n}+\frac {b}{c}\right )}{b^{3} n}\) | \(58\) |
norman | \(\left (\frac {c \,{\mathrm e}^{n \ln \left (x \right )}}{b^{2} n}-\frac {1}{2 b n}+\frac {c^{2} \ln \left (x \right ) {\mathrm e}^{2 n \ln \left (x \right )}}{b^{3}}\right ) {\mathrm e}^{-2 n \ln \left (x \right )}-\frac {c^{2} \ln \left (c \,{\mathrm e}^{n \ln \left (x \right )}+b \right )}{b^{3} n}\) | \(69\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 58, normalized size = 1.02 \begin {gather*} \frac {c^{2} \log \left (x\right )}{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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 59, normalized size = 1.04 \begin {gather*} \frac {2 \, c^{2} n x^{2 \, n} \log \left (x\right ) - 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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = -0.02 \begin {gather*} \int \frac {1}{x^{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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