Integrand size = 21, antiderivative size = 26 \[ \int x (-a-b x)^{-n} (a+b x)^n \, dx=\frac {1}{2} x^2 (-a-b x)^{-n} (a+b x)^n \]
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Time = 0.00 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {23, 30} \[ \int x (-a-b x)^{-n} (a+b x)^n \, dx=\frac {1}{2} x^2 (-a-b x)^{-n} (a+b x)^n \]
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Rule 23
Rule 30
Rubi steps \begin{align*} \text {integral}& = \left ((-a-b x)^{-n} (a+b x)^n\right ) \int x \, dx \\ & = \frac {1}{2} x^2 (-a-b x)^{-n} (a+b x)^n \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int x (-a-b x)^{-n} (a+b x)^n \, dx=\frac {1}{2} x^2 (-a-b x)^{-n} (a+b x)^n \]
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Time = 1.78 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96
method | result | size |
gosper | \(\frac {x^{2} \left (b x +a \right )^{n} \left (-b x -a \right )^{-n}}{2}\) | \(25\) |
parallelrisch | \(\frac {x^{2} \left (b x +a \right )^{n} \left (-b x -a \right )^{-n}}{2}\) | \(25\) |
risch | \(\frac {x^{2} {\mathrm e}^{-i n \pi \left (\operatorname {csgn}\left (i \left (b x +a \right )\right )^{3}-\operatorname {csgn}\left (i \left (b x +a \right )\right )^{2}+1\right )}}{2}\) | \(38\) |
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Result contains complex when optimal does not.
Time = 0.22 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.38 \[ \int x (-a-b x)^{-n} (a+b x)^n \, dx=\frac {1}{2} \, x^{2} e^{\left (i \, \pi n\right )} \]
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Time = 1.74 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.73 \[ \int x (-a-b x)^{-n} (a+b x)^n \, dx=\frac {x^{2} \left (- a - b x\right )^{- n} \left (a + b x\right )^{n}}{2} \]
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none
Time = 0.19 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.31 \[ \int x (-a-b x)^{-n} (a+b x)^n \, dx=\frac {1}{2} \, \left (-1\right )^{n} x^{2} \]
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none
Time = 0.28 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.19 \[ \int x (-a-b x)^{-n} (a+b x)^n \, dx=\frac {1}{2} \, x^{2} \]
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Time = 1.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int x (-a-b x)^{-n} (a+b x)^n \, dx=\frac {x^2\,{\left (a+b\,x\right )}^n}{2\,{\left (-a-b\,x\right )}^n} \]
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