Integrand size = 16, antiderivative size = 39 \[ \int \sqrt {x} (a+b x) (A+B x) \, dx=\frac {2}{3} a A x^{3/2}+\frac {2}{5} (A b+a B) x^{5/2}+\frac {2}{7} b B x^{7/2} \]
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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} \[ \int \sqrt {x} (a+b x) (A+B x) \, dx=\frac {2}{5} x^{5/2} (a B+A b)+\frac {2}{3} a A x^{3/2}+\frac {2}{7} b B x^{7/2} \]
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Rule 77
Rubi steps \begin{align*} \text {integral}& = \int \left (a A \sqrt {x}+(A b+a B) x^{3/2}+b B x^{5/2}\right ) \, dx \\ & = \frac {2}{3} a A x^{3/2}+\frac {2}{5} (A b+a B) x^{5/2}+\frac {2}{7} b B x^{7/2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.85 \[ \int \sqrt {x} (a+b x) (A+B x) \, dx=\frac {2}{105} x^{3/2} (7 a (5 A+3 B x)+3 b x (7 A+5 B x)) \]
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Time = 0.10 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.72
method | result | size |
gosper | \(\frac {2 x^{\frac {3}{2}} \left (15 b B \,x^{2}+21 A b x +21 B a x +35 A a \right )}{105}\) | \(28\) |
derivativedivides | \(\frac {2 a A \,x^{\frac {3}{2}}}{3}+\frac {2 \left (A b +B a \right ) x^{\frac {5}{2}}}{5}+\frac {2 b B \,x^{\frac {7}{2}}}{7}\) | \(28\) |
default | \(\frac {2 a A \,x^{\frac {3}{2}}}{3}+\frac {2 \left (A b +B a \right ) x^{\frac {5}{2}}}{5}+\frac {2 b B \,x^{\frac {7}{2}}}{7}\) | \(28\) |
trager | \(\frac {2 x^{\frac {3}{2}} \left (15 b B \,x^{2}+21 A b x +21 B a x +35 A a \right )}{105}\) | \(28\) |
risch | \(\frac {2 x^{\frac {3}{2}} \left (15 b B \,x^{2}+21 A b x +21 B a x +35 A a \right )}{105}\) | \(28\) |
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none
Time = 0.23 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.77 \[ \int \sqrt {x} (a+b x) (A+B x) \, dx=\frac {2}{105} \, {\left (15 \, B b x^{3} + 35 \, A a x + 21 \, {\left (B a + A b\right )} x^{2}\right )} \sqrt {x} \]
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Time = 0.57 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.95 \[ \int \sqrt {x} (a+b x) (A+B x) \, dx=\frac {2 A a x^{\frac {3}{2}}}{3} + \frac {2 B b x^{\frac {7}{2}}}{7} + \frac {2 x^{\frac {5}{2}} \left (A b + B a\right )}{5} \]
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none
Time = 0.21 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.69 \[ \int \sqrt {x} (a+b x) (A+B x) \, dx=\frac {2}{7} \, B b x^{\frac {7}{2}} + \frac {2}{3} \, A a x^{\frac {3}{2}} + \frac {2}{5} \, {\left (B a + A b\right )} x^{\frac {5}{2}} \]
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none
Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.74 \[ \int \sqrt {x} (a+b x) (A+B x) \, dx=\frac {2}{7} \, B b x^{\frac {7}{2}} + \frac {2}{5} \, B a x^{\frac {5}{2}} + \frac {2}{5} \, A b x^{\frac {5}{2}} + \frac {2}{3} \, A a x^{\frac {3}{2}} \]
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Time = 0.36 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.69 \[ \int \sqrt {x} (a+b x) (A+B x) \, dx=\frac {2\,x^{3/2}\,\left (35\,A\,a+21\,A\,b\,x+21\,B\,a\,x+15\,B\,b\,x^2\right )}{105} \]
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