Integrand size = 18, antiderivative size = 85 \[ \int x^{3/2} (a+b x)^3 (A+B x) \, dx=\frac {2}{5} a^3 A x^{5/2}+\frac {2}{7} a^2 (3 A b+a B) x^{7/2}+\frac {2}{3} a b (A b+a B) x^{9/2}+\frac {2}{11} b^2 (A b+3 a B) x^{11/2}+\frac {2}{13} b^3 B x^{13/2} \]
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Time = 0.03 (sec) , antiderivative size = 85, 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} \[ \int x^{3/2} (a+b x)^3 (A+B x) \, dx=\frac {2}{5} a^3 A x^{5/2}+\frac {2}{7} a^2 x^{7/2} (a B+3 A b)+\frac {2}{11} b^2 x^{11/2} (3 a B+A b)+\frac {2}{3} a b x^{9/2} (a B+A b)+\frac {2}{13} b^3 B x^{13/2} \]
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Rule 77
Rubi steps \begin{align*} \text {integral}& = \int \left (a^3 A x^{3/2}+a^2 (3 A b+a B) x^{5/2}+3 a b (A b+a B) x^{7/2}+b^2 (A b+3 a B) x^{9/2}+b^3 B x^{11/2}\right ) \, dx \\ & = \frac {2}{5} a^3 A x^{5/2}+\frac {2}{7} a^2 (3 A b+a B) x^{7/2}+\frac {2}{3} a b (A b+a B) x^{9/2}+\frac {2}{11} b^2 (A b+3 a B) x^{11/2}+\frac {2}{13} b^3 B x^{13/2} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.84 \[ \int x^{3/2} (a+b x)^3 (A+B x) \, dx=\frac {2 x^{5/2} \left (429 a^3 (7 A+5 B x)+715 a^2 b x (9 A+7 B x)+455 a b^2 x^2 (11 A+9 B x)+105 b^3 x^3 (13 A+11 B x)\right )}{15015} \]
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Time = 0.44 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.89
method | result | size |
gosper | \(\frac {2 x^{\frac {5}{2}} \left (1155 b^{3} B \,x^{4}+1365 A \,b^{3} x^{3}+4095 B a \,b^{2} x^{3}+5005 a A \,b^{2} x^{2}+5005 B \,a^{2} b \,x^{2}+6435 a^{2} A b x +2145 a^{3} B x +3003 a^{3} A \right )}{15015}\) | \(76\) |
derivativedivides | \(\frac {2 b^{3} B \,x^{\frac {13}{2}}}{13}+\frac {2 \left (b^{3} A +3 a \,b^{2} B \right ) x^{\frac {11}{2}}}{11}+\frac {2 \left (3 a \,b^{2} A +3 a^{2} b B \right ) x^{\frac {9}{2}}}{9}+\frac {2 \left (3 a^{2} b A +a^{3} B \right ) x^{\frac {7}{2}}}{7}+\frac {2 a^{3} A \,x^{\frac {5}{2}}}{5}\) | \(76\) |
default | \(\frac {2 b^{3} B \,x^{\frac {13}{2}}}{13}+\frac {2 \left (b^{3} A +3 a \,b^{2} B \right ) x^{\frac {11}{2}}}{11}+\frac {2 \left (3 a \,b^{2} A +3 a^{2} b B \right ) x^{\frac {9}{2}}}{9}+\frac {2 \left (3 a^{2} b A +a^{3} B \right ) x^{\frac {7}{2}}}{7}+\frac {2 a^{3} A \,x^{\frac {5}{2}}}{5}\) | \(76\) |
trager | \(\frac {2 x^{\frac {5}{2}} \left (1155 b^{3} B \,x^{4}+1365 A \,b^{3} x^{3}+4095 B a \,b^{2} x^{3}+5005 a A \,b^{2} x^{2}+5005 B \,a^{2} b \,x^{2}+6435 a^{2} A b x +2145 a^{3} B x +3003 a^{3} A \right )}{15015}\) | \(76\) |
risch | \(\frac {2 x^{\frac {5}{2}} \left (1155 b^{3} B \,x^{4}+1365 A \,b^{3} x^{3}+4095 B a \,b^{2} x^{3}+5005 a A \,b^{2} x^{2}+5005 B \,a^{2} b \,x^{2}+6435 a^{2} A b x +2145 a^{3} B x +3003 a^{3} A \right )}{15015}\) | \(76\) |
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Time = 0.22 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.92 \[ \int x^{3/2} (a+b x)^3 (A+B x) \, dx=\frac {2}{15015} \, {\left (1155 \, B b^{3} x^{6} + 3003 \, A a^{3} x^{2} + 1365 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{5} + 5005 \, {\left (B a^{2} b + A a b^{2}\right )} x^{4} + 2145 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{3}\right )} \sqrt {x} \]
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Time = 0.25 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.34 \[ \int x^{3/2} (a+b x)^3 (A+B x) \, dx=\frac {2 A a^{3} x^{\frac {5}{2}}}{5} + \frac {6 A a^{2} b x^{\frac {7}{2}}}{7} + \frac {2 A a b^{2} x^{\frac {9}{2}}}{3} + \frac {2 A b^{3} x^{\frac {11}{2}}}{11} + \frac {2 B a^{3} x^{\frac {7}{2}}}{7} + \frac {2 B a^{2} b x^{\frac {9}{2}}}{3} + \frac {6 B a b^{2} x^{\frac {11}{2}}}{11} + \frac {2 B b^{3} x^{\frac {13}{2}}}{13} \]
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Time = 0.20 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.86 \[ \int x^{3/2} (a+b x)^3 (A+B x) \, dx=\frac {2}{13} \, B b^{3} x^{\frac {13}{2}} + \frac {2}{5} \, A a^{3} x^{\frac {5}{2}} + \frac {2}{11} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac {11}{2}} + \frac {2}{3} \, {\left (B a^{2} b + A a b^{2}\right )} x^{\frac {9}{2}} + \frac {2}{7} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac {7}{2}} \]
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Time = 0.28 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.91 \[ \int x^{3/2} (a+b x)^3 (A+B x) \, dx=\frac {2}{13} \, B b^{3} x^{\frac {13}{2}} + \frac {6}{11} \, B a b^{2} x^{\frac {11}{2}} + \frac {2}{11} \, A b^{3} x^{\frac {11}{2}} + \frac {2}{3} \, B a^{2} b x^{\frac {9}{2}} + \frac {2}{3} \, A a b^{2} x^{\frac {9}{2}} + \frac {2}{7} \, B a^{3} x^{\frac {7}{2}} + \frac {6}{7} \, A a^{2} b x^{\frac {7}{2}} + \frac {2}{5} \, A a^{3} x^{\frac {5}{2}} \]
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Time = 0.03 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.81 \[ \int x^{3/2} (a+b x)^3 (A+B x) \, dx=x^{7/2}\,\left (\frac {2\,B\,a^3}{7}+\frac {6\,A\,b\,a^2}{7}\right )+x^{11/2}\,\left (\frac {2\,A\,b^3}{11}+\frac {6\,B\,a\,b^2}{11}\right )+\frac {2\,A\,a^3\,x^{5/2}}{5}+\frac {2\,B\,b^3\,x^{13/2}}{13}+\frac {2\,a\,b\,x^{9/2}\,\left (A\,b+B\,a\right )}{3} \]
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