Integrand size = 18, antiderivative size = 5 \[ \int \frac {(a+b x)^5}{(a c+b c x)^5} \, dx=\frac {x}{c^5} \]
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Time = 0.00 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {21, 8} \[ \int \frac {(a+b x)^5}{(a c+b c x)^5} \, dx=\frac {x}{c^5} \]
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Rule 8
Rule 21
Rubi steps \begin{align*} \text {integral}& = \frac {\int 1 \, dx}{c^5} \\ & = \frac {x}{c^5} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^5}{(a c+b c x)^5} \, dx=\frac {x}{c^5} \]
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Time = 0.15 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.20
method | result | size |
default | \(\frac {x}{c^{5}}\) | \(6\) |
risch | \(\frac {x}{c^{5}}\) | \(6\) |
norman | \(\frac {\frac {b^{4} x^{5}}{c}+\frac {a^{4} x}{c}+\frac {4 a \,b^{3} x^{4}}{c}+\frac {4 a^{3} b \,x^{2}}{c}+\frac {6 a^{2} b^{2} x^{3}}{c}}{c^{4} \left (b x +a \right )^{4}}\) | \(69\) |
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none
Time = 0.22 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^5}{(a c+b c x)^5} \, dx=\frac {x}{c^{5}} \]
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Time = 0.09 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.60 \[ \int \frac {(a+b x)^5}{(a c+b c x)^5} \, dx=\frac {x}{c^{5}} \]
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none
Time = 0.21 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^5}{(a c+b c x)^5} \, dx=\frac {x}{c^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 15 vs. \(2 (5) = 10\).
Time = 0.28 (sec) , antiderivative size = 15, normalized size of antiderivative = 3.00 \[ \int \frac {(a+b x)^5}{(a c+b c x)^5} \, dx=\frac {b c x + a c}{b c^{6}} \]
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Time = 0.01 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^5}{(a c+b c x)^5} \, dx=\frac {x}{c^5} \]
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