Integrand size = 15, antiderivative size = 27 \[ \int x^{-1+3 n} \left (a+b x^n\right ) \, dx=\frac {a x^{3 n}}{3 n}+\frac {b x^{4 n}}{4 n} \]
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Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {14} \[ \int x^{-1+3 n} \left (a+b x^n\right ) \, dx=\frac {a x^{3 n}}{3 n}+\frac {b x^{4 n}}{4 n} \]
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Rule 14
Rubi steps \begin{align*} \text {integral}& = \int \left (a x^{-1+3 n}+b x^{-1+4 n}\right ) \, dx \\ & = \frac {a x^{3 n}}{3 n}+\frac {b x^{4 n}}{4 n} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int x^{-1+3 n} \left (a+b x^n\right ) \, dx=\frac {x^{3 n} \left (4 a+3 b x^n\right )}{12 n} \]
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Time = 0.04 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89
method | result | size |
risch | \(\frac {a \,x^{3 n}}{3 n}+\frac {b \,x^{4 n}}{4 n}\) | \(24\) |
norman | \(\frac {a \,{\mathrm e}^{3 n \ln \left (x \right )}}{3 n}+\frac {b \,{\mathrm e}^{4 n \ln \left (x \right )}}{4 n}\) | \(28\) |
parallelrisch | \(\frac {3 x \,x^{n} x^{-1+3 n} b +4 x \,x^{-1+3 n} a}{12 n}\) | \(32\) |
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Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int x^{-1+3 n} \left (a+b x^n\right ) \, dx=\frac {3 \, b x^{4 \, n} + 4 \, a x^{3 \, n}}{12 \, n} \]
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Time = 0.17 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int x^{-1+3 n} \left (a+b x^n\right ) \, dx=\begin {cases} \frac {a x x^{3 n - 1}}{3 n} + \frac {b x x^{n} x^{3 n - 1}}{4 n} & \text {for}\: n \neq 0 \\\left (a + b\right ) \log {\left (x \right )} & \text {otherwise} \end {cases} \]
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Time = 0.23 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int x^{-1+3 n} \left (a+b x^n\right ) \, dx=\frac {b x^{4 \, n}}{4 \, n} + \frac {a x^{3 \, n}}{3 \, n} \]
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\[ \int x^{-1+3 n} \left (a+b x^n\right ) \, dx=\int { {\left (b x^{n} + a\right )} x^{3 \, n - 1} \,d x } \]
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Time = 5.96 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int x^{-1+3 n} \left (a+b x^n\right ) \, dx=\frac {x^{3\,n}\,\left (4\,a+3\,b\,x^n\right )}{12\,n} \]
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