Integrand size = 7, antiderivative size = 14 \[ \int (a+b x)^5 \, dx=\frac {(a+b x)^6}{6 b} \]
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Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {32} \[ \int (a+b x)^5 \, dx=\frac {(a+b x)^6}{6 b} \]
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Rule 32
Rubi steps \begin{align*} \text {integral}& = \frac {(a+b x)^6}{6 b} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int (a+b x)^5 \, dx=\frac {(a+b x)^6}{6 b} \]
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Time = 0.16 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93
method | result | size |
default | \(\frac {\left (b x +a \right )^{6}}{6 b}\) | \(13\) |
gosper | \(\frac {1}{6} b^{5} x^{6}+a \,b^{4} x^{5}+\frac {5}{2} a^{2} b^{3} x^{4}+\frac {10}{3} a^{3} b^{2} x^{3}+\frac {5}{2} a^{4} b \,x^{2}+a^{5} x\) | \(54\) |
norman | \(\frac {1}{6} b^{5} x^{6}+a \,b^{4} x^{5}+\frac {5}{2} a^{2} b^{3} x^{4}+\frac {10}{3} a^{3} b^{2} x^{3}+\frac {5}{2} a^{4} b \,x^{2}+a^{5} x\) | \(54\) |
parallelrisch | \(\frac {1}{6} b^{5} x^{6}+a \,b^{4} x^{5}+\frac {5}{2} a^{2} b^{3} x^{4}+\frac {10}{3} a^{3} b^{2} x^{3}+\frac {5}{2} a^{4} b \,x^{2}+a^{5} x\) | \(54\) |
risch | \(\frac {b^{5} x^{6}}{6}+a \,b^{4} x^{5}+\frac {5 a^{2} b^{3} x^{4}}{2}+\frac {10 a^{3} b^{2} x^{3}}{3}+\frac {5 a^{4} b \,x^{2}}{2}+a^{5} x +\frac {a^{6}}{6 b}\) | \(62\) |
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Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (12) = 24\).
Time = 0.21 (sec) , antiderivative size = 53, normalized size of antiderivative = 3.79 \[ \int (a+b x)^5 \, dx=\frac {1}{6} \, b^{5} x^{6} + a b^{4} x^{5} + \frac {5}{2} \, a^{2} b^{3} x^{4} + \frac {10}{3} \, a^{3} b^{2} x^{3} + \frac {5}{2} \, a^{4} b x^{2} + a^{5} x \]
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Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (8) = 16\).
Time = 0.03 (sec) , antiderivative size = 60, normalized size of antiderivative = 4.29 \[ \int (a+b x)^5 \, dx=a^{5} x + \frac {5 a^{4} b x^{2}}{2} + \frac {10 a^{3} b^{2} x^{3}}{3} + \frac {5 a^{2} b^{3} x^{4}}{2} + a b^{4} x^{5} + \frac {b^{5} x^{6}}{6} \]
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Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (12) = 24\).
Time = 0.20 (sec) , antiderivative size = 53, normalized size of antiderivative = 3.79 \[ \int (a+b x)^5 \, dx=\frac {1}{6} \, b^{5} x^{6} + a b^{4} x^{5} + \frac {5}{2} \, a^{2} b^{3} x^{4} + \frac {10}{3} \, a^{3} b^{2} x^{3} + \frac {5}{2} \, a^{4} b x^{2} + a^{5} x \]
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none
Time = 0.30 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int (a+b x)^5 \, dx=\frac {{\left (b x + a\right )}^{6}}{6 \, b} \]
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Time = 0.01 (sec) , antiderivative size = 53, normalized size of antiderivative = 3.79 \[ \int (a+b x)^5 \, dx=a^5\,x+\frac {5\,a^4\,b\,x^2}{2}+\frac {10\,a^3\,b^2\,x^3}{3}+\frac {5\,a^2\,b^3\,x^4}{2}+a\,b^4\,x^5+\frac {b^5\,x^6}{6} \]
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