Integrand size = 23, antiderivative size = 76 \[ \int \frac {x^{-1-2 n}}{b x^n+c x^{2 n}} \, dx=-\frac {x^{-3 n}}{3 b n}+\frac {c x^{-2 n}}{2 b^2 n}-\frac {c^2 x^{-n}}{b^3 n}-\frac {c^3 \log (x)}{b^4}+\frac {c^3 \log \left (b+c x^n\right )}{b^4 n} \]
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Time = 0.03 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {1598, 272, 46} \[ \int \frac {x^{-1-2 n}}{b x^n+c x^{2 n}} \, dx=\frac {c^3 \log \left (b+c x^n\right )}{b^4 n}-\frac {c^3 \log (x)}{b^4}-\frac {c^2 x^{-n}}{b^3 n}+\frac {c x^{-2 n}}{2 b^2 n}-\frac {x^{-3 n}}{3 b n} \]
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Rule 46
Rule 272
Rule 1598
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^{-1-3 n}}{b+c x^n} \, dx \\ & = \frac {\text {Subst}\left (\int \frac {1}{x^4 (b+c x)} \, dx,x,x^n\right )}{n} \\ & = \frac {\text {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*}
Time = 0.09 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.83 \[ \int \frac {x^{-1-2 n}}{b x^n+c x^{2 n}} \, dx=-\frac {b x^{-3 n} \left (2 b^2-3 b c x^n+6 c^2 x^{2 n}\right )+6 c^3 \log \left (x^n\right )-6 c^3 \log \left (b+c x^n\right )}{6 b^4 n} \]
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Time = 0.62 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.99
method | result | size |
risch | \(-\frac {c^{2} x^{-n}}{b^{3} n}+\frac {c \,x^{-2 n}}{2 b^{2} n}-\frac {x^{-3 n}}{3 b n}-\frac {c^{3} \ln \left (x \right )}{b^{4}}+\frac {c^{3} \ln \left (x^{n}+\frac {b}{c}\right )}{b^{4} n}\) | \(75\) |
norman | \(\left (-\frac {1}{3 b n}+\frac {c \,{\mathrm e}^{n \ln \left (x \right )}}{2 b^{2} n}-\frac {c^{2} {\mathrm e}^{2 n \ln \left (x \right )}}{b^{3} n}-\frac {c^{3} \ln \left (x \right ) {\mathrm e}^{3 n \ln \left (x \right )}}{b^{4}}\right ) {\mathrm e}^{-3 n \ln \left (x \right )}+\frac {c^{3} \ln \left (c \,{\mathrm e}^{n \ln \left (x \right )}+b \right )}{b^{4} n}\) | \(88\) |
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Time = 0.28 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.95 \[ \int \frac {x^{-1-2 n}}{b x^n+c x^{2 n}} \, dx=-\frac {6 \, c^{3} n x^{3 \, n} \log \left (x\right ) - 6 \, c^{3} x^{3 \, n} \log \left (c x^{n} + b\right ) + 6 \, b c^{2} x^{2 \, n} - 3 \, b^{2} c x^{n} + 2 \, b^{3}}{6 \, b^{4} n x^{3 \, n}} \]
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Exception generated. \[ \int \frac {x^{-1-2 n}}{b x^n+c x^{2 n}} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
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Time = 0.17 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.93 \[ \int \frac {x^{-1-2 n}}{b x^n+c x^{2 n}} \, dx=-\frac {c^{3} \log \left (x\right )}{b^{4}} + \frac {c^{3} \log \left (\frac {c x^{n} + b}{c}\right )}{b^{4} n} - \frac {6 \, c^{2} x^{2 \, n} - 3 \, b c x^{n} + 2 \, b^{2}}{6 \, b^{3} n x^{3 \, n}} \]
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\[ \int \frac {x^{-1-2 n}}{b x^n+c x^{2 n}} \, dx=\int { \frac {x^{-2 \, n - 1}}{c x^{2 \, n} + b x^{n}} \,d x } \]
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Timed out. \[ \int \frac {x^{-1-2 n}}{b x^n+c x^{2 n}} \, dx=\int \frac {1}{x^{2\,n+1}\,\left (b\,x^n+c\,x^{2\,n}\right )} \,d x \]
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