Integrand size = 15, antiderivative size = 18 \[ \int \left (a+b x^n\right )^{-1-\frac {1}{n}} \, dx=\frac {x \left (a+b x^n\right )^{-1/n}}{a} \]
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Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {197} \[ \int \left (a+b x^n\right )^{-1-\frac {1}{n}} \, dx=\frac {x \left (a+b x^n\right )^{-1/n}}{a} \]
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Rule 197
Rubi steps \begin{align*} \text {integral}& = \frac {x \left (a+b x^n\right )^{-1/n}}{a} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^n\right )^{-1-\frac {1}{n}} \, dx=\frac {x \left (a+b x^n\right )^{-1/n}}{a} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(46\) vs. \(2(18)=36\).
Time = 3.91 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.61
method | result | size |
parallelrisch | \(\frac {x \,x^{n} \left (a +b \,x^{n}\right )^{-\frac {1+n}{n}} b +x \left (a +b \,x^{n}\right )^{-\frac {1+n}{n}} a}{a}\) | \(47\) |
norman | \(x \,{\mathrm e}^{\left (-1-\frac {1}{n}\right ) \ln \left (a +b \,{\mathrm e}^{n \ln \left (x \right )}\right )}+\frac {b x \,{\mathrm e}^{n \ln \left (x \right )} {\mathrm e}^{\left (-1-\frac {1}{n}\right ) \ln \left (a +b \,{\mathrm e}^{n \ln \left (x \right )}\right )}}{a}\) | \(53\) |
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none
Time = 0.29 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.72 \[ \int \left (a+b x^n\right )^{-1-\frac {1}{n}} \, dx=\frac {b x x^{n} + a x}{{\left (b x^{n} + a\right )}^{\frac {n + 1}{n}} a} \]
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Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (12) = 24\).
Time = 0.69 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.33 \[ \int \left (a+b x^n\right )^{-1-\frac {1}{n}} \, dx=\frac {a^{\frac {1}{n}} a^{-1 - \frac {1}{n}} b^{- \frac {1}{n}} \left (\frac {a x^{- n}}{b} + 1\right )^{- \frac {1}{n}} \Gamma \left (\frac {1}{n}\right )}{n \Gamma \left (1 + \frac {1}{n}\right )} \]
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\[ \int \left (a+b x^n\right )^{-1-\frac {1}{n}} \, dx=\int { {\left (b x^{n} + a\right )}^{-\frac {1}{n} - 1} \,d x } \]
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\[ \int \left (a+b x^n\right )^{-1-\frac {1}{n}} \, dx=\int { {\left (b x^{n} + a\right )}^{-\frac {1}{n} - 1} \,d x } \]
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Time = 5.90 (sec) , antiderivative size = 75, normalized size of antiderivative = 4.17 \[ \int \left (a+b x^n\right )^{-1-\frac {1}{n}} \, dx=\frac {b\,x^{n+1}\,\left (\frac {a}{b\,x^n}-{\left (\frac {a}{b\,x^n}+1\right )}^{\frac {n+1}{n}}+1\right )}{a\,n\,\left (\frac {n+1}{n}-1\right )\,{\left (a+b\,x^n\right )}^{\frac {n+1}{n}}} \]
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