Integrand size = 17, antiderivative size = 45 \[ \int x^{-1-4 n} \left (a+b x^n\right )^2 \, dx=-\frac {a^2 x^{-4 n}}{4 n}-\frac {2 a b x^{-3 n}}{3 n}-\frac {b^2 x^{-2 n}}{2 n} \]
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Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {272, 45} \[ \int x^{-1-4 n} \left (a+b x^n\right )^2 \, dx=-\frac {a^2 x^{-4 n}}{4 n}-\frac {2 a b x^{-3 n}}{3 n}-\frac {b^2 x^{-2 n}}{2 n} \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(a+b x)^2}{x^5} \, dx,x,x^n\right )}{n} \\ & = \frac {\text {Subst}\left (\int \left (\frac {a^2}{x^5}+\frac {2 a b}{x^4}+\frac {b^2}{x^3}\right ) \, dx,x,x^n\right )}{n} \\ & = -\frac {a^2 x^{-4 n}}{4 n}-\frac {2 a b x^{-3 n}}{3 n}-\frac {b^2 x^{-2 n}}{2 n} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.78 \[ \int x^{-1-4 n} \left (a+b x^n\right )^2 \, dx=\frac {x^{-4 n} \left (-3 a^2-8 a b x^n-6 b^2 x^{2 n}\right )}{12 n} \]
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Time = 3.59 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.89
method | result | size |
risch | \(-\frac {b^{2} x^{-2 n}}{2 n}-\frac {2 a b \,x^{-3 n}}{3 n}-\frac {a^{2} x^{-4 n}}{4 n}\) | \(40\) |
norman | \(\left (-\frac {a^{2}}{4 n}-\frac {b^{2} {\mathrm e}^{2 n \ln \left (x \right )}}{2 n}-\frac {2 a b \,{\mathrm e}^{n \ln \left (x \right )}}{3 n}\right ) {\mathrm e}^{-4 n \ln \left (x \right )}\) | \(45\) |
parallelrisch | \(\frac {-6 x \,x^{2 n} x^{-1-4 n} b^{2}-8 x \,x^{n} x^{-1-4 n} a b -3 x \,x^{-1-4 n} a^{2}}{12 n}\) | \(53\) |
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Time = 0.28 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.78 \[ \int x^{-1-4 n} \left (a+b x^n\right )^2 \, dx=-\frac {6 \, b^{2} x^{2 \, n} + 8 \, a b x^{n} + 3 \, a^{2}}{12 \, n x^{4 \, n}} \]
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Time = 0.25 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.56 \[ \int x^{-1-4 n} \left (a+b x^n\right )^2 \, dx=\begin {cases} - \frac {a^{2} x x^{- 4 n - 1}}{4 n} - \frac {2 a b x x^{n} x^{- 4 n - 1}}{3 n} - \frac {b^{2} x x^{2 n} x^{- 4 n - 1}}{2 n} & \text {for}\: n \neq 0 \\\left (a + b\right )^{2} \log {\left (x \right )} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00 \[ \int x^{-1-4 n} \left (a+b x^n\right )^2 \, dx=-\frac {a^{2}}{4 \, n x^{4 \, n}} - \frac {2 \, a b}{3 \, n x^{3 \, n}} - \frac {b^{2}}{2 \, n x^{2 \, n}} \]
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Time = 0.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.78 \[ \int x^{-1-4 n} \left (a+b x^n\right )^2 \, dx=-\frac {6 \, b^{2} x^{2 \, n} + 8 \, a b x^{n} + 3 \, a^{2}}{12 \, n x^{4 \, n}} \]
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Time = 5.90 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.78 \[ \int x^{-1-4 n} \left (a+b x^n\right )^2 \, dx=-\frac {3\,a^2+6\,b^2\,x^{2\,n}+8\,a\,b\,x^n}{12\,n\,x^{4\,n}} \]
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