Integrand size = 30, antiderivative size = 19 \[ \int \frac {x^{-1+n} \left (b+2 c x^n\right )}{a+b x^n+c x^{2 n}} \, dx=\frac {\log \left (a+b x^n+c x^{2 n}\right )}{n} \]
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Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1482, 642} \[ \int \frac {x^{-1+n} \left (b+2 c x^n\right )}{a+b x^n+c x^{2 n}} \, dx=\frac {\log \left (a+b x^n+c x^{2 n}\right )}{n} \]
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Rule 642
Rule 1482
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^n\right )}{n} \\ & = \frac {\log \left (a+b x^n+c x^{2 n}\right )}{n} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.74 \[ \int \frac {x^{-1+n} \left (b+2 c x^n\right )}{a+b x^n+c x^{2 n}} \, dx=-\frac {2 \log \left (x^{-n}\right )}{n}+\frac {\log \left (c+a x^{-2 n}+b x^{-n}\right )}{n} \]
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Time = 0.39 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.26
method | result | size |
norman | \(\frac {\ln \left (a +b \,{\mathrm e}^{n \ln \left (x \right )}+c \,{\mathrm e}^{2 n \ln \left (x \right )}\right )}{n}\) | \(24\) |
risch | \(\frac {\ln \left (x^{2 n}+\frac {b \,x^{n}}{c}+\frac {a}{c}\right )}{n}\) | \(25\) |
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Time = 0.29 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {x^{-1+n} \left (b+2 c x^n\right )}{a+b x^n+c x^{2 n}} \, dx=\frac {\log \left (c x^{2 \, n} + b x^{n} + a\right )}{n} \]
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Timed out. \[ \int \frac {x^{-1+n} \left (b+2 c x^n\right )}{a+b x^n+c x^{2 n}} \, dx=\text {Timed out} \]
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Time = 0.23 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21 \[ \int \frac {x^{-1+n} \left (b+2 c x^n\right )}{a+b x^n+c x^{2 n}} \, dx=\frac {\log \left (\frac {c x^{2 \, n} + b x^{n} + a}{c}\right )}{n} \]
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Time = 0.30 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {x^{-1+n} \left (b+2 c x^n\right )}{a+b x^n+c x^{2 n}} \, dx=\frac {\log \left (c x^{2 \, n} + b x^{n} + a\right )}{n} \]
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Time = 8.89 (sec) , antiderivative size = 121, normalized size of antiderivative = 6.37 \[ \int \frac {x^{-1+n} \left (b+2 c x^n\right )}{a+b x^n+c x^{2 n}} \, dx=-\frac {2\,b\,\mathrm {atan}\left (\frac {b}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,x^n}{\sqrt {4\,a\,c-b^2}}\right )-\ln \left (a+b\,x^n+c\,x^{2\,n}\right )\,\sqrt {4\,a\,c-b^2}}{n\,\sqrt {4\,a\,c-b^2}}-\frac {2\,b\,\mathrm {atanh}\left (\frac {b+2\,c\,x^n}{\sqrt {b^2-4\,a\,c}}\right )}{n\,\sqrt {b^2-4\,a\,c}} \]
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