Integrand size = 18, antiderivative size = 63 \[ \int x^{5/2} (a+b x)^2 (A+B x) \, dx=\frac {2}{7} a^2 A x^{7/2}+\frac {2}{9} a (2 A b+a B) x^{9/2}+\frac {2}{11} b (A b+2 a B) x^{11/2}+\frac {2}{13} b^2 B x^{13/2} \]
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Time = 0.02 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {77} \[ \int x^{5/2} (a+b x)^2 (A+B x) \, dx=\frac {2}{7} a^2 A x^{7/2}+\frac {2}{11} b x^{11/2} (2 a B+A b)+\frac {2}{9} a x^{9/2} (a B+2 A b)+\frac {2}{13} b^2 B x^{13/2} \]
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Rule 77
Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 A x^{5/2}+a (2 A b+a B) x^{7/2}+b (A b+2 a B) x^{9/2}+b^2 B x^{11/2}\right ) \, dx \\ & = \frac {2}{7} a^2 A x^{7/2}+\frac {2}{9} a (2 A b+a B) x^{9/2}+\frac {2}{11} b (A b+2 a B) x^{11/2}+\frac {2}{13} b^2 B x^{13/2} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.83 \[ \int x^{5/2} (a+b x)^2 (A+B x) \, dx=\frac {2 x^{7/2} \left (143 a^2 (9 A+7 B x)+182 a b x (11 A+9 B x)+63 b^2 x^2 (13 A+11 B x)\right )}{9009} \]
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Time = 1.05 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.83
method | result | size |
gosper | \(\frac {2 x^{\frac {7}{2}} \left (693 b^{2} B \,x^{3}+819 A \,b^{2} x^{2}+1638 B a b \,x^{2}+2002 a A b x +1001 a^{2} B x +1287 a^{2} A \right )}{9009}\) | \(52\) |
derivativedivides | \(\frac {2 b^{2} B \,x^{\frac {13}{2}}}{13}+\frac {2 \left (b^{2} A +2 a b B \right ) x^{\frac {11}{2}}}{11}+\frac {2 \left (2 a b A +a^{2} B \right ) x^{\frac {9}{2}}}{9}+\frac {2 a^{2} A \,x^{\frac {7}{2}}}{7}\) | \(52\) |
default | \(\frac {2 b^{2} B \,x^{\frac {13}{2}}}{13}+\frac {2 \left (b^{2} A +2 a b B \right ) x^{\frac {11}{2}}}{11}+\frac {2 \left (2 a b A +a^{2} B \right ) x^{\frac {9}{2}}}{9}+\frac {2 a^{2} A \,x^{\frac {7}{2}}}{7}\) | \(52\) |
trager | \(\frac {2 x^{\frac {7}{2}} \left (693 b^{2} B \,x^{3}+819 A \,b^{2} x^{2}+1638 B a b \,x^{2}+2002 a A b x +1001 a^{2} B x +1287 a^{2} A \right )}{9009}\) | \(52\) |
risch | \(\frac {2 x^{\frac {7}{2}} \left (693 b^{2} B \,x^{3}+819 A \,b^{2} x^{2}+1638 B a b \,x^{2}+2002 a A b x +1001 a^{2} B x +1287 a^{2} A \right )}{9009}\) | \(52\) |
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none
Time = 0.22 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.89 \[ \int x^{5/2} (a+b x)^2 (A+B x) \, dx=\frac {2}{9009} \, {\left (693 \, B b^{2} x^{6} + 1287 \, A a^{2} x^{3} + 819 \, {\left (2 \, B a b + A b^{2}\right )} x^{5} + 1001 \, {\left (B a^{2} + 2 \, A a b\right )} x^{4}\right )} \sqrt {x} \]
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Time = 0.30 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.27 \[ \int x^{5/2} (a+b x)^2 (A+B x) \, dx=\frac {2 A a^{2} x^{\frac {7}{2}}}{7} + \frac {4 A a b x^{\frac {9}{2}}}{9} + \frac {2 A b^{2} x^{\frac {11}{2}}}{11} + \frac {2 B a^{2} x^{\frac {9}{2}}}{9} + \frac {4 B a b x^{\frac {11}{2}}}{11} + \frac {2 B b^{2} x^{\frac {13}{2}}}{13} \]
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none
Time = 0.20 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.81 \[ \int x^{5/2} (a+b x)^2 (A+B x) \, dx=\frac {2}{13} \, B b^{2} x^{\frac {13}{2}} + \frac {2}{7} \, A a^{2} x^{\frac {7}{2}} + \frac {2}{11} \, {\left (2 \, B a b + A b^{2}\right )} x^{\frac {11}{2}} + \frac {2}{9} \, {\left (B a^{2} + 2 \, A a b\right )} x^{\frac {9}{2}} \]
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Time = 0.28 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.84 \[ \int x^{5/2} (a+b x)^2 (A+B x) \, dx=\frac {2}{13} \, B b^{2} x^{\frac {13}{2}} + \frac {4}{11} \, B a b x^{\frac {11}{2}} + \frac {2}{11} \, A b^{2} x^{\frac {11}{2}} + \frac {2}{9} \, B a^{2} x^{\frac {9}{2}} + \frac {4}{9} \, A a b x^{\frac {9}{2}} + \frac {2}{7} \, A a^{2} x^{\frac {7}{2}} \]
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Time = 0.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.81 \[ \int x^{5/2} (a+b x)^2 (A+B x) \, dx=x^{9/2}\,\left (\frac {2\,B\,a^2}{9}+\frac {4\,A\,b\,a}{9}\right )+x^{11/2}\,\left (\frac {2\,A\,b^2}{11}+\frac {4\,B\,a\,b}{11}\right )+\frac {2\,A\,a^2\,x^{7/2}}{7}+\frac {2\,B\,b^2\,x^{13/2}}{13} \]
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