Integrand size = 15, antiderivative size = 19 \[ \int x^{-1+n} \left (a+b x^n\right )^5 \, dx=\frac {\left (a+b x^n\right )^6}{6 b n} \]
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Time = 0.00 (sec) , antiderivative size = 19, 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 = {267} \[ \int x^{-1+n} \left (a+b x^n\right )^5 \, dx=\frac {\left (a+b x^n\right )^6}{6 b n} \]
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Rule 267
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a+b x^n\right )^6}{6 b n} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int x^{-1+n} \left (a+b x^n\right )^5 \, dx=\frac {\left (a+b x^n\right )^6}{6 b n} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(83\) vs. \(2(17)=34\).
Time = 4.41 (sec) , antiderivative size = 84, normalized size of antiderivative = 4.42
method | result | size |
risch | \(\frac {b^{5} x^{6 n}}{6 n}+\frac {a \,b^{4} x^{5 n}}{n}+\frac {5 a^{2} b^{3} x^{4 n}}{2 n}+\frac {10 a^{3} b^{2} x^{3 n}}{3 n}+\frac {5 a^{4} b \,x^{2 n}}{2 n}+\frac {a^{5} x^{n}}{n}\) | \(84\) |
parallelrisch | \(\frac {x \,x^{5 n} x^{-1+n} b^{5}+6 x \,x^{4 n} x^{-1+n} a \,b^{4}+15 x \,x^{3 n} x^{-1+n} a^{2} b^{3}+20 x \,x^{2 n} x^{-1+n} a^{3} b^{2}+15 x \,x^{n} x^{-1+n} a^{4} b +6 x \,x^{-1+n} a^{5}}{6 n}\) | \(103\) |
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Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (17) = 34\).
Time = 0.27 (sec) , antiderivative size = 71, normalized size of antiderivative = 3.74 \[ \int x^{-1+n} \left (a+b x^n\right )^5 \, dx=\frac {b^{5} x^{6 \, n} + 6 \, a b^{4} x^{5 \, n} + 15 \, a^{2} b^{3} x^{4 \, n} + 20 \, a^{3} b^{2} x^{3 \, n} + 15 \, a^{4} b x^{2 \, n} + 6 \, a^{5} x^{n}}{6 \, n} \]
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Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (12) = 24\).
Time = 0.75 (sec) , antiderivative size = 124, normalized size of antiderivative = 6.53 \[ \int x^{-1+n} \left (a+b x^n\right )^5 \, dx=\begin {cases} \frac {a^{5} x x^{n - 1}}{n} + \frac {5 a^{4} b x x^{n} x^{n - 1}}{2 n} + \frac {10 a^{3} b^{2} x x^{2 n} x^{n - 1}}{3 n} + \frac {5 a^{2} b^{3} x x^{3 n} x^{n - 1}}{2 n} + \frac {a b^{4} x x^{4 n} x^{n - 1}}{n} + \frac {b^{5} x x^{5 n} x^{n - 1}}{6 n} & \text {for}\: n \neq 0 \\\left (a + b\right )^{5} \log {\left (x \right )} & \text {otherwise} \end {cases} \]
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none
Time = 0.21 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int x^{-1+n} \left (a+b x^n\right )^5 \, dx=\frac {{\left (b x^{n} + a\right )}^{6}}{6 \, b n} \]
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Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (17) = 34\).
Time = 0.28 (sec) , antiderivative size = 71, normalized size of antiderivative = 3.74 \[ \int x^{-1+n} \left (a+b x^n\right )^5 \, dx=\frac {b^{5} x^{6 \, n} + 6 \, a b^{4} x^{5 \, n} + 15 \, a^{2} b^{3} x^{4 \, n} + 20 \, a^{3} b^{2} x^{3 \, n} + 15 \, a^{4} b x^{2 \, n} + 6 \, a^{5} x^{n}}{6 \, n} \]
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Time = 5.76 (sec) , antiderivative size = 83, normalized size of antiderivative = 4.37 \[ \int x^{-1+n} \left (a+b x^n\right )^5 \, dx=\frac {a^5\,x^n}{n}+\frac {b^5\,x^{6\,n}}{6\,n}+\frac {10\,a^3\,b^2\,x^{3\,n}}{3\,n}+\frac {5\,a^2\,b^3\,x^{4\,n}}{2\,n}+\frac {5\,a^4\,b\,x^{2\,n}}{2\,n}+\frac {a\,b^4\,x^{5\,n}}{n} \]
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