Integrand size = 20, antiderivative size = 39 \[ \int \sqrt {x} \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=\frac {2}{3} a A x^{3/2}+\frac {2}{7} (A b+a B) x^{7/2}+\frac {2}{11} b B x^{11/2} \]
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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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {459} \[ \int \sqrt {x} \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=\frac {2}{7} x^{7/2} (a B+A b)+\frac {2}{3} a A x^{3/2}+\frac {2}{11} b B x^{11/2} \]
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Rule 459
Rubi steps \begin{align*} \text {integral}& = \int \left (a A \sqrt {x}+(A b+a B) x^{5/2}+b B x^{9/2}\right ) \, dx \\ & = \frac {2}{3} a A x^{3/2}+\frac {2}{7} (A b+a B) x^{7/2}+\frac {2}{11} b B x^{11/2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.90 \[ \int \sqrt {x} \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=\frac {2}{231} x^{3/2} \left (77 a A+33 A b x^2+33 a B x^2+21 b B x^4\right ) \]
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Time = 0.13 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.72
method | result | size |
derivativedivides | \(\frac {2 a A \,x^{\frac {3}{2}}}{3}+\frac {2 \left (A b +B a \right ) x^{\frac {7}{2}}}{7}+\frac {2 b B \,x^{\frac {11}{2}}}{11}\) | \(28\) |
default | \(\frac {2 a A \,x^{\frac {3}{2}}}{3}+\frac {2 \left (A b +B a \right ) x^{\frac {7}{2}}}{7}+\frac {2 b B \,x^{\frac {11}{2}}}{11}\) | \(28\) |
gosper | \(\frac {2 x^{\frac {3}{2}} \left (21 b B \,x^{4}+33 A b \,x^{2}+33 B a \,x^{2}+77 A a \right )}{231}\) | \(32\) |
trager | \(\frac {2 x^{\frac {3}{2}} \left (21 b B \,x^{4}+33 A b \,x^{2}+33 B a \,x^{2}+77 A a \right )}{231}\) | \(32\) |
risch | \(\frac {2 x^{\frac {3}{2}} \left (21 b B \,x^{4}+33 A b \,x^{2}+33 B a \,x^{2}+77 A a \right )}{231}\) | \(32\) |
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Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.77 \[ \int \sqrt {x} \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=\frac {2}{231} \, {\left (21 \, B b x^{5} + 33 \, {\left (B a + A b\right )} x^{3} + 77 \, A a x\right )} \sqrt {x} \]
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Time = 0.44 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.95 \[ \int \sqrt {x} \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=\frac {2 A a x^{\frac {3}{2}}}{3} + \frac {2 B b x^{\frac {11}{2}}}{11} + \frac {2 x^{\frac {7}{2}} \left (A b + B a\right )}{7} \]
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Time = 0.20 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.69 \[ \int \sqrt {x} \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=\frac {2}{11} \, B b x^{\frac {11}{2}} + \frac {2}{7} \, {\left (B a + A b\right )} x^{\frac {7}{2}} + \frac {2}{3} \, A a x^{\frac {3}{2}} \]
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Time = 0.29 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.74 \[ \int \sqrt {x} \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=\frac {2}{11} \, B b x^{\frac {11}{2}} + \frac {2}{7} \, B a x^{\frac {7}{2}} + \frac {2}{7} \, A b x^{\frac {7}{2}} + \frac {2}{3} \, A a x^{\frac {3}{2}} \]
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Time = 4.89 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.79 \[ \int \sqrt {x} \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=\frac {2\,x^{3/2}\,\left (77\,A\,a+33\,A\,b\,x^2+33\,B\,a\,x^2+21\,B\,b\,x^4\right )}{231} \]
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