Integrand size = 11, antiderivative size = 21 \[ \int x^{3/2} (a+b x) \, dx=\frac {2}{5} a x^{5/2}+\frac {2}{7} b x^{7/2} \]
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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} \[ \int x^{3/2} (a+b x) \, dx=\frac {2}{5} a x^{5/2}+\frac {2}{7} b x^{7/2} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (a x^{3/2}+b x^{5/2}\right ) \, dx \\ & = \frac {2}{5} a x^{5/2}+\frac {2}{7} b x^{7/2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int x^{3/2} (a+b x) \, dx=\frac {2}{35} x^{5/2} (7 a+5 b x) \]
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Time = 0.06 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67
method | result | size |
gosper | \(\frac {2 x^{\frac {5}{2}} \left (5 b x +7 a \right )}{35}\) | \(14\) |
derivativedivides | \(\frac {2 a \,x^{\frac {5}{2}}}{5}+\frac {2 b \,x^{\frac {7}{2}}}{7}\) | \(14\) |
default | \(\frac {2 a \,x^{\frac {5}{2}}}{5}+\frac {2 b \,x^{\frac {7}{2}}}{7}\) | \(14\) |
trager | \(\frac {2 x^{\frac {5}{2}} \left (5 b x +7 a \right )}{35}\) | \(14\) |
risch | \(\frac {2 x^{\frac {5}{2}} \left (5 b x +7 a \right )}{35}\) | \(14\) |
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Time = 0.22 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86 \[ \int x^{3/2} (a+b x) \, dx=\frac {2}{35} \, {\left (5 \, b x^{3} + 7 \, a x^{2}\right )} \sqrt {x} \]
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Time = 0.18 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int x^{3/2} (a+b x) \, dx=\frac {2 a x^{\frac {5}{2}}}{5} + \frac {2 b x^{\frac {7}{2}}}{7} \]
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none
Time = 0.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.62 \[ \int x^{3/2} (a+b x) \, dx=\frac {2}{7} \, b x^{\frac {7}{2}} + \frac {2}{5} \, a x^{\frac {5}{2}} \]
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Time = 0.31 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.62 \[ \int x^{3/2} (a+b x) \, dx=\frac {2}{7} \, b x^{\frac {7}{2}} + \frac {2}{5} \, a x^{\frac {5}{2}} \]
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Time = 0.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.62 \[ \int x^{3/2} (a+b x) \, dx=\frac {2\,x^{5/2}\,\left (7\,a+5\,b\,x\right )}{35} \]
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