Integrand size = 16, antiderivative size = 39 \[ \int x^{7/2} (a+b x) (A+B x) \, dx=\frac {2}{9} a A x^{9/2}+\frac {2}{11} (A b+a B) x^{11/2}+\frac {2}{13} b B x^{13/2} \]
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Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {77} \[ \int x^{7/2} (a+b x) (A+B x) \, dx=\frac {2}{11} x^{11/2} (a B+A b)+\frac {2}{9} a A x^{9/2}+\frac {2}{13} b B x^{13/2} \]
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Rule 77
Rubi steps \begin{align*} \text {integral}& = \int \left (a A x^{7/2}+(A b+a B) x^{9/2}+b B x^{11/2}\right ) \, dx \\ & = \frac {2}{9} a A x^{9/2}+\frac {2}{11} (A b+a B) x^{11/2}+\frac {2}{13} b B x^{13/2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.79 \[ \int x^{7/2} (a+b x) (A+B x) \, dx=\frac {2 x^{9/2} \left (143 a A+117 A b x+117 a B x+99 b B x^2\right )}{1287} \]
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Time = 0.12 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.72
method | result | size |
gosper | \(\frac {2 x^{\frac {9}{2}} \left (99 b B \,x^{2}+117 A b x +117 B a x +143 A a \right )}{1287}\) | \(28\) |
derivativedivides | \(\frac {2 a A \,x^{\frac {9}{2}}}{9}+\frac {2 \left (A b +B a \right ) x^{\frac {11}{2}}}{11}+\frac {2 b B \,x^{\frac {13}{2}}}{13}\) | \(28\) |
default | \(\frac {2 a A \,x^{\frac {9}{2}}}{9}+\frac {2 \left (A b +B a \right ) x^{\frac {11}{2}}}{11}+\frac {2 b B \,x^{\frac {13}{2}}}{13}\) | \(28\) |
trager | \(\frac {2 x^{\frac {9}{2}} \left (99 b B \,x^{2}+117 A b x +117 B a x +143 A a \right )}{1287}\) | \(28\) |
risch | \(\frac {2 x^{\frac {9}{2}} \left (99 b B \,x^{2}+117 A b x +117 B a x +143 A a \right )}{1287}\) | \(28\) |
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none
Time = 0.22 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.82 \[ \int x^{7/2} (a+b x) (A+B x) \, dx=\frac {2}{1287} \, {\left (99 \, B b x^{6} + 143 \, A a x^{4} + 117 \, {\left (B a + A b\right )} x^{5}\right )} \sqrt {x} \]
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Time = 0.38 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.18 \[ \int x^{7/2} (a+b x) (A+B x) \, dx=\frac {2 A a x^{\frac {9}{2}}}{9} + \frac {2 A b x^{\frac {11}{2}}}{11} + \frac {2 B a x^{\frac {11}{2}}}{11} + \frac {2 B b x^{\frac {13}{2}}}{13} \]
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none
Time = 0.20 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.69 \[ \int x^{7/2} (a+b x) (A+B x) \, dx=\frac {2}{13} \, B b x^{\frac {13}{2}} + \frac {2}{9} \, A a x^{\frac {9}{2}} + \frac {2}{11} \, {\left (B a + A b\right )} x^{\frac {11}{2}} \]
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none
Time = 0.29 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.74 \[ \int x^{7/2} (a+b x) (A+B x) \, dx=\frac {2}{13} \, B b x^{\frac {13}{2}} + \frac {2}{11} \, B a x^{\frac {11}{2}} + \frac {2}{11} \, A b x^{\frac {11}{2}} + \frac {2}{9} \, A a x^{\frac {9}{2}} \]
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Time = 0.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.69 \[ \int x^{7/2} (a+b x) (A+B x) \, dx=\frac {2\,x^{9/2}\,\left (143\,A\,a+117\,A\,b\,x+117\,B\,a\,x+99\,B\,b\,x^2\right )}{1287} \]
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