Integrand size = 11, antiderivative size = 21 \[ \int x^{5/2} (a+b x) \, dx=\frac {2}{7} a x^{7/2}+\frac {2}{9} b x^{9/2} \]
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Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \[ \int x^{5/2} (a+b x) \, dx=\frac {2}{7} a x^{7/2}+\frac {2}{9} b x^{9/2} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (a x^{5/2}+b x^{7/2}\right ) \, dx \\ & = \frac {2}{7} a x^{7/2}+\frac {2}{9} b x^{9/2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int x^{5/2} (a+b x) \, dx=\frac {2}{63} x^{7/2} (9 a+7 b x) \]
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Time = 0.06 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67
method | result | size |
gosper | \(\frac {2 x^{\frac {7}{2}} \left (7 b x +9 a \right )}{63}\) | \(14\) |
derivativedivides | \(\frac {2 a \,x^{\frac {7}{2}}}{7}+\frac {2 b \,x^{\frac {9}{2}}}{9}\) | \(14\) |
default | \(\frac {2 a \,x^{\frac {7}{2}}}{7}+\frac {2 b \,x^{\frac {9}{2}}}{9}\) | \(14\) |
trager | \(\frac {2 x^{\frac {7}{2}} \left (7 b x +9 a \right )}{63}\) | \(14\) |
risch | \(\frac {2 x^{\frac {7}{2}} \left (7 b x +9 a \right )}{63}\) | \(14\) |
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Time = 0.22 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86 \[ \int x^{5/2} (a+b x) \, dx=\frac {2}{63} \, {\left (7 \, b x^{4} + 9 \, a x^{3}\right )} \sqrt {x} \]
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Time = 0.21 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int x^{5/2} (a+b x) \, dx=\frac {2 a x^{\frac {7}{2}}}{7} + \frac {2 b x^{\frac {9}{2}}}{9} \]
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Time = 0.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.62 \[ \int x^{5/2} (a+b x) \, dx=\frac {2}{9} \, b x^{\frac {9}{2}} + \frac {2}{7} \, a x^{\frac {7}{2}} \]
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Time = 0.30 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.62 \[ \int x^{5/2} (a+b x) \, dx=\frac {2}{9} \, b x^{\frac {9}{2}} + \frac {2}{7} \, a x^{\frac {7}{2}} \]
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Time = 0.15 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.62 \[ \int x^{5/2} (a+b x) \, dx=\frac {2\,x^{7/2}\,\left (9\,a+7\,b\,x\right )}{63} \]
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