Integrand size = 17, antiderivative size = 28 \[ \int \frac {(c x)^{-1+n}}{a+b x^n} \, dx=\frac {x^{-n} (c x)^n \log \left (a+b x^n\right )}{b c n} \]
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Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {274, 266} \[ \int \frac {(c x)^{-1+n}}{a+b x^n} \, dx=\frac {x^{-n} (c x)^n \log \left (a+b x^n\right )}{b c n} \]
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Rule 266
Rule 274
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{-n} (c x)^n\right ) \int \frac {x^{-1+n}}{a+b x^n} \, dx}{c} \\ & = \frac {x^{-n} (c x)^n \log \left (a+b x^n\right )}{b c n} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {(c x)^{-1+n}}{a+b x^n} \, dx=\frac {x^{1-n} (c x)^{-1+n} \log \left (a+b x^n\right )}{b n} \]
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\[\int \frac {\left (c x \right )^{-1+n}}{a +b \,x^{n}}d x\]
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none
Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \frac {(c x)^{-1+n}}{a+b x^n} \, dx=\frac {c^{n - 1} \log \left (b x^{n} + a\right )}{b n} \]
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Time = 0.46 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.61 \[ \int \frac {(c x)^{-1+n}}{a+b x^n} \, dx=\frac {c^{n - 1} \log {\left (1 + \frac {b x^{n}}{a} \right )}}{b n} \]
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none
Time = 0.22 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {(c x)^{-1+n}}{a+b x^n} \, dx=\frac {c^{n - 1} \log \left (\frac {b x^{n} + a}{b}\right )}{b n} \]
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\[ \int \frac {(c x)^{-1+n}}{a+b x^n} \, dx=\int { \frac {\left (c x\right )^{n - 1}}{b x^{n} + a} \,d x } \]
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Timed out. \[ \int \frac {(c x)^{-1+n}}{a+b x^n} \, dx=\int \frac {{\left (c\,x\right )}^{n-1}}{a+b\,x^n} \,d x \]
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