Integrand size = 20, antiderivative size = 21 \[ \int (-a-b x)^{-n} (a+b x)^n \, dx=x (-a-b x)^{-n} (a+b x)^n \]
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Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {23, 8} \[ \int (-a-b x)^{-n} (a+b x)^n \, dx=x (-a-b x)^{-n} (a+b x)^n \]
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Rule 8
Rule 23
Rubi steps \begin{align*} \text {integral}& = \left ((-a-b x)^{-n} (a+b x)^n\right ) \int 1 \, dx \\ & = x (-a-b x)^{-n} (a+b x)^n \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int (-a-b x)^{-n} (a+b x)^n \, dx=\frac {(-a-b x)^{-n} (a+b x)^{1+n}}{b} \]
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Time = 1.46 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24
method | result | size |
norman | \(x \,{\mathrm e}^{n \ln \left (b x +a \right )} {\mathrm e}^{-n \ln \left (-b x -a \right )}\) | \(26\) |
risch | \(x \,{\mathrm e}^{-i n \pi \left (\operatorname {csgn}\left (i \left (b x +a \right )\right )^{3}-\operatorname {csgn}\left (i \left (b x +a \right )\right )^{2}+1\right )}\) | \(35\) |
parallelrisch | \(-\frac {\left (-x \left (b x +a \right )^{n} b n +\left (b x +a \right )^{n} a n \right ) \left (-b x -a \right )^{-n}}{n b}\) | \(44\) |
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Result contains complex when optimal does not.
Time = 0.22 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.33 \[ \int (-a-b x)^{-n} (a+b x)^n \, dx=x e^{\left (i \, \pi n\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (15) = 30\).
Time = 1.51 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.10 \[ \int (-a-b x)^{-n} (a+b x)^n \, dx=\begin {cases} - \frac {a \left (- a - b x\right )^{- n} \left (a + b x\right )^{n}}{b} + x \left (- a - b x\right )^{- n} \left (a + b x\right )^{n} & \text {for}\: b \neq 0 \\a^{n} x \left (- a\right )^{- n} & \text {otherwise} \end {cases} \]
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none
Time = 0.20 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.24 \[ \int (-a-b x)^{-n} (a+b x)^n \, dx=\left (-1\right )^{n} x \]
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none
Time = 0.28 (sec) , antiderivative size = 1, normalized size of antiderivative = 0.05 \[ \int (-a-b x)^{-n} (a+b x)^n \, dx=x \]
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Time = 1.33 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int (-a-b x)^{-n} (a+b x)^n \, dx=\frac {x\,{\left (a+b\,x\right )}^n}{{\left (-a-b\,x\right )}^n} \]
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