Integrand size = 17, antiderivative size = 63 \[ \int x^{-1+3 n} \left (a+b x^n\right )^3 \, dx=\frac {a^3 x^{3 n}}{3 n}+\frac {3 a^2 b x^{4 n}}{4 n}+\frac {3 a b^2 x^{5 n}}{5 n}+\frac {b^3 x^{6 n}}{6 n} \]
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Time = 0.02 (sec) , antiderivative size = 63, 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} \[ \int x^{-1+3 n} \left (a+b x^n\right )^3 \, dx=\frac {a^3 x^{3 n}}{3 n}+\frac {3 a^2 b x^{4 n}}{4 n}+\frac {3 a b^2 x^{5 n}}{5 n}+\frac {b^3 x^{6 n}}{6 n} \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int x^2 (a+b x)^3 \, dx,x,x^n\right )}{n} \\ & = \frac {\text {Subst}\left (\int \left (a^3 x^2+3 a^2 b x^3+3 a b^2 x^4+b^3 x^5\right ) \, dx,x,x^n\right )}{n} \\ & = \frac {a^3 x^{3 n}}{3 n}+\frac {3 a^2 b x^{4 n}}{4 n}+\frac {3 a b^2 x^{5 n}}{5 n}+\frac {b^3 x^{6 n}}{6 n} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.76 \[ \int x^{-1+3 n} \left (a+b x^n\right )^3 \, dx=\frac {x^{3 n} \left (20 a^3+45 a^2 b x^n+36 a b^2 x^{2 n}+10 b^3 x^{3 n}\right )}{60 n} \]
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Time = 3.68 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.89
method | result | size |
risch | \(\frac {a^{3} x^{3 n}}{3 n}+\frac {3 a^{2} b \,x^{4 n}}{4 n}+\frac {3 a \,b^{2} x^{5 n}}{5 n}+\frac {b^{3} x^{6 n}}{6 n}\) | \(56\) |
parallelrisch | \(\frac {10 x \,x^{3 n} x^{-1+3 n} b^{3}+36 x \,x^{2 n} x^{-1+3 n} a \,b^{2}+45 x \,x^{n} x^{-1+3 n} a^{2} b +20 x \,x^{-1+3 n} a^{3}}{60 n}\) | \(74\) |
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Time = 0.28 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.76 \[ \int x^{-1+3 n} \left (a+b x^n\right )^3 \, dx=\frac {10 \, b^{3} x^{6 \, n} + 36 \, a b^{2} x^{5 \, n} + 45 \, a^{2} b x^{4 \, n} + 20 \, a^{3} x^{3 \, n}}{60 \, n} \]
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Time = 0.37 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.40 \[ \int x^{-1+3 n} \left (a+b x^n\right )^3 \, dx=\begin {cases} \frac {a^{3} x x^{3 n - 1}}{3 n} + \frac {3 a^{2} b x x^{n} x^{3 n - 1}}{4 n} + \frac {3 a b^{2} x x^{2 n} x^{3 n - 1}}{5 n} + \frac {b^{3} x x^{3 n} x^{3 n - 1}}{6 n} & \text {for}\: n \neq 0 \\\left (a + b\right )^{3} \log {\left (x \right )} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.87 \[ \int x^{-1+3 n} \left (a+b x^n\right )^3 \, dx=\frac {b^{3} x^{6 \, n}}{6 \, n} + \frac {3 \, a b^{2} x^{5 \, n}}{5 \, n} + \frac {3 \, a^{2} b x^{4 \, n}}{4 \, n} + \frac {a^{3} x^{3 \, n}}{3 \, n} \]
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\[ \int x^{-1+3 n} \left (a+b x^n\right )^3 \, dx=\int { {\left (b x^{n} + a\right )}^{3} x^{3 \, n - 1} \,d x } \]
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Time = 5.79 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.87 \[ \int x^{-1+3 n} \left (a+b x^n\right )^3 \, dx=\frac {a^3\,x^{3\,n}}{3\,n}+\frac {b^3\,x^{6\,n}}{6\,n}+\frac {3\,a^2\,b\,x^{4\,n}}{4\,n}+\frac {3\,a\,b^2\,x^{5\,n}}{5\,n} \]
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