Integrand size = 17, antiderivative size = 40 \[ \int x^{-1+2 n} \left (a+b x^n\right )^3 \, dx=-\frac {a \left (a+b x^n\right )^4}{4 b^2 n}+\frac {\left (a+b x^n\right )^5}{5 b^2 n} \]
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Time = 0.01 (sec) , antiderivative size = 40, 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} \[ \int x^{-1+2 n} \left (a+b x^n\right )^3 \, dx=\frac {\left (a+b x^n\right )^5}{5 b^2 n}-\frac {a \left (a+b x^n\right )^4}{4 b^2 n} \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int x (a+b x)^3 \, dx,x,x^n\right )}{n} \\ & = \frac {\text {Subst}\left (\int \left (-\frac {a (a+b x)^3}{b}+\frac {(a+b x)^4}{b}\right ) \, dx,x,x^n\right )}{n} \\ & = -\frac {a \left (a+b x^n\right )^4}{4 b^2 n}+\frac {\left (a+b x^n\right )^5}{5 b^2 n} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.20 \[ \int x^{-1+2 n} \left (a+b x^n\right )^3 \, dx=\frac {x^{2 n} \left (10 a^3+20 a^2 b x^n+15 a b^2 x^{2 n}+4 b^3 x^{3 n}\right )}{20 n} \]
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Time = 3.82 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.38
method | result | size |
risch | \(\frac {a^{2} b \,x^{3 n}}{n}+\frac {a^{3} x^{2 n}}{2 n}+\frac {b^{3} x^{5 n}}{5 n}+\frac {3 a \,b^{2} x^{4 n}}{4 n}\) | \(55\) |
norman | \(\frac {a^{2} b \,{\mathrm e}^{3 n \ln \left (x \right )}}{n}+\frac {a^{3} {\mathrm e}^{2 n \ln \left (x \right )}}{2 n}+\frac {b^{3} {\mathrm e}^{5 n \ln \left (x \right )}}{5 n}+\frac {3 a \,b^{2} {\mathrm e}^{4 n \ln \left (x \right )}}{4 n}\) | \(63\) |
parallelrisch | \(\frac {4 x \,x^{3 n} x^{-1+2 n} b^{3}+15 x \,x^{2 n} x^{-1+2 n} a \,b^{2}+20 x \,x^{n} x^{-1+2 n} a^{2} b +10 x \,x^{-1+2 n} a^{3}}{20 n}\) | \(74\) |
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Time = 0.29 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.20 \[ \int x^{-1+2 n} \left (a+b x^n\right )^3 \, dx=\frac {4 \, b^{3} x^{5 \, n} + 15 \, a b^{2} x^{4 \, n} + 20 \, a^{2} b x^{3 \, n} + 10 \, a^{3} x^{2 \, n}}{20 \, n} \]
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Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (31) = 62\).
Time = 0.38 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.12 \[ \int x^{-1+2 n} \left (a+b x^n\right )^3 \, dx=\begin {cases} \frac {a^{3} x x^{2 n - 1}}{2 n} + \frac {a^{2} b x x^{n} x^{2 n - 1}}{n} + \frac {3 a b^{2} x x^{2 n} x^{2 n - 1}}{4 n} + \frac {b^{3} x x^{3 n} x^{2 n - 1}}{5 n} & \text {for}\: n \neq 0 \\\left (a + b\right )^{3} \log {\left (x \right )} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.35 \[ \int x^{-1+2 n} \left (a+b x^n\right )^3 \, dx=\frac {b^{3} x^{5 \, n}}{5 \, n} + \frac {3 \, a b^{2} x^{4 \, n}}{4 \, n} + \frac {a^{2} b x^{3 \, n}}{n} + \frac {a^{3} x^{2 \, n}}{2 \, n} \]
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\[ \int x^{-1+2 n} \left (a+b x^n\right )^3 \, dx=\int { {\left (b x^{n} + a\right )}^{3} x^{2 \, n - 1} \,d x } \]
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Time = 5.81 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.35 \[ \int x^{-1+2 n} \left (a+b x^n\right )^3 \, dx=\frac {a^3\,x^{2\,n}}{2\,n}+\frac {b^3\,x^{5\,n}}{5\,n}+\frac {a^2\,b\,x^{3\,n}}{n}+\frac {3\,a\,b^2\,x^{4\,n}}{4\,n} \]
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