Integrand size = 20, antiderivative size = 39 \[ \int x^{5/2} \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=\frac {2}{7} a A x^{7/2}+\frac {2}{11} (A b+a B) x^{11/2}+\frac {2}{15} b B x^{15/2} \]
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Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {459} \[ \int x^{5/2} \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=\frac {2}{11} x^{11/2} (a B+A b)+\frac {2}{7} a A x^{7/2}+\frac {2}{15} b B x^{15/2} \]
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Rule 459
Rubi steps \begin{align*} \text {integral}& = \int \left (a A x^{5/2}+(A b+a B) x^{9/2}+b B x^{13/2}\right ) \, dx \\ & = \frac {2}{7} a A x^{7/2}+\frac {2}{11} (A b+a B) x^{11/2}+\frac {2}{15} b B x^{15/2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.90 \[ \int x^{5/2} \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=\frac {2 x^{7/2} \left (165 a A+105 A b x^2+105 a B x^2+77 b B x^4\right )}{1155} \]
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Time = 0.16 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.72
method | result | size |
derivativedivides | \(\frac {2 a A \,x^{\frac {7}{2}}}{7}+\frac {2 \left (A b +B a \right ) x^{\frac {11}{2}}}{11}+\frac {2 b B \,x^{\frac {15}{2}}}{15}\) | \(28\) |
default | \(\frac {2 a A \,x^{\frac {7}{2}}}{7}+\frac {2 \left (A b +B a \right ) x^{\frac {11}{2}}}{11}+\frac {2 b B \,x^{\frac {15}{2}}}{15}\) | \(28\) |
gosper | \(\frac {2 x^{\frac {7}{2}} \left (77 b B \,x^{4}+105 A b \,x^{2}+105 B a \,x^{2}+165 A a \right )}{1155}\) | \(32\) |
trager | \(\frac {2 x^{\frac {7}{2}} \left (77 b B \,x^{4}+105 A b \,x^{2}+105 B a \,x^{2}+165 A a \right )}{1155}\) | \(32\) |
risch | \(\frac {2 x^{\frac {7}{2}} \left (77 b B \,x^{4}+105 A b \,x^{2}+105 B a \,x^{2}+165 A a \right )}{1155}\) | \(32\) |
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Time = 0.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.82 \[ \int x^{5/2} \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=\frac {2}{1155} \, {\left (77 \, B b x^{7} + 105 \, {\left (B a + A b\right )} x^{5} + 165 \, A a x^{3}\right )} \sqrt {x} \]
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Time = 0.41 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.18 \[ \int x^{5/2} \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=\frac {2 A a x^{\frac {7}{2}}}{7} + \frac {2 A b x^{\frac {11}{2}}}{11} + \frac {2 B a x^{\frac {11}{2}}}{11} + \frac {2 B b x^{\frac {15}{2}}}{15} \]
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Time = 0.20 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.69 \[ \int x^{5/2} \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=\frac {2}{15} \, B b x^{\frac {15}{2}} + \frac {2}{11} \, {\left (B a + A b\right )} x^{\frac {11}{2}} + \frac {2}{7} \, A a x^{\frac {7}{2}} \]
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Time = 0.29 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.74 \[ \int x^{5/2} \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=\frac {2}{15} \, B b x^{\frac {15}{2}} + \frac {2}{11} \, B a x^{\frac {11}{2}} + \frac {2}{11} \, A b x^{\frac {11}{2}} + \frac {2}{7} \, A a x^{\frac {7}{2}} \]
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Time = 5.20 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.79 \[ \int x^{5/2} \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=\frac {2\,x^{7/2}\,\left (165\,A\,a+105\,A\,b\,x^2+105\,B\,a\,x^2+77\,B\,b\,x^4\right )}{1155} \]
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