Integrand size = 9, antiderivative size = 25 \[ \int x^m (a+b x) \, dx=\frac {a x^{1+m}}{1+m}+\frac {b x^{2+m}}{2+m} \]
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Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {45} \[ \int x^m (a+b x) \, dx=\frac {a x^{m+1}}{m+1}+\frac {b x^{m+2}}{m+2} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (a x^m+b x^{1+m}\right ) \, dx \\ & = \frac {a x^{1+m}}{1+m}+\frac {b x^{2+m}}{2+m} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int x^m (a+b x) \, dx=x^{1+m} \left (\frac {a}{1+m}+\frac {b x}{2+m}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20
method | result | size |
norman | \(\frac {a x \,{\mathrm e}^{m \ln \left (x \right )}}{1+m}+\frac {b \,x^{2} {\mathrm e}^{m \ln \left (x \right )}}{2+m}\) | \(30\) |
risch | \(\frac {x \left (b x m +a m +b x +2 a \right ) x^{m}}{\left (2+m \right ) \left (1+m \right )}\) | \(30\) |
gosper | \(\frac {x^{1+m} \left (b x m +a m +b x +2 a \right )}{\left (1+m \right ) \left (2+m \right )}\) | \(31\) |
parallelrisch | \(\frac {x^{2} x^{m} b m +x^{2} x^{m} b +x \,x^{m} a m +2 x \,x^{m} a}{\left (2+m \right ) \left (1+m \right )}\) | \(44\) |
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none
Time = 0.22 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int x^m (a+b x) \, dx=\frac {{\left ({\left (b m + b\right )} x^{2} + {\left (a m + 2 \, a\right )} x\right )} x^{m}}{m^{2} + 3 \, m + 2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (19) = 38\).
Time = 0.36 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.48 \[ \int x^m (a+b x) \, dx=\begin {cases} - \frac {a}{x} + b \log {\left (x \right )} & \text {for}\: m = -2 \\a \log {\left (x \right )} + b x & \text {for}\: m = -1 \\\frac {a m x x^{m}}{m^{2} + 3 m + 2} + \frac {2 a x x^{m}}{m^{2} + 3 m + 2} + \frac {b m x^{2} x^{m}}{m^{2} + 3 m + 2} + \frac {b x^{2} x^{m}}{m^{2} + 3 m + 2} & \text {otherwise} \end {cases} \]
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none
Time = 0.21 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int x^m (a+b x) \, dx=\frac {b x^{m + 2}}{m + 2} + \frac {a x^{m + 1}}{m + 1} \]
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none
Time = 0.29 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.72 \[ \int x^m (a+b x) \, dx=\frac {b m x^{2} x^{m} + a m x x^{m} + b x^{2} x^{m} + 2 \, a x x^{m}}{m^{2} + 3 \, m + 2} \]
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Time = 0.36 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20 \[ \int x^m (a+b x) \, dx=\frac {x^{m+1}\,\left (2\,a+a\,m+b\,x+b\,m\,x\right )}{m^2+3\,m+2} \]
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