Integrand size = 13, antiderivative size = 51 \[ \int x^{3/2} (a+b x)^3 \, dx=\frac {2}{5} a^3 x^{5/2}+\frac {6}{7} a^2 b x^{7/2}+\frac {2}{3} a b^2 x^{9/2}+\frac {2}{11} b^3 x^{11/2} \]
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Time = 0.01 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {45} \[ \int x^{3/2} (a+b x)^3 \, dx=\frac {2}{5} a^3 x^{5/2}+\frac {6}{7} a^2 b x^{7/2}+\frac {2}{3} a b^2 x^{9/2}+\frac {2}{11} b^3 x^{11/2} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (a^3 x^{3/2}+3 a^2 b x^{5/2}+3 a b^2 x^{7/2}+b^3 x^{9/2}\right ) \, dx \\ & = \frac {2}{5} a^3 x^{5/2}+\frac {6}{7} a^2 b x^{7/2}+\frac {2}{3} a b^2 x^{9/2}+\frac {2}{11} b^3 x^{11/2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.76 \[ \int x^{3/2} (a+b x)^3 \, dx=\frac {2 x^{5/2} \left (231 a^3+495 a^2 b x+385 a b^2 x^2+105 b^3 x^3\right )}{1155} \]
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Time = 0.07 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.71
method | result | size |
gosper | \(\frac {2 x^{\frac {5}{2}} \left (105 b^{3} x^{3}+385 a \,b^{2} x^{2}+495 a^{2} b x +231 a^{3}\right )}{1155}\) | \(36\) |
derivativedivides | \(\frac {2 a^{3} x^{\frac {5}{2}}}{5}+\frac {6 a^{2} b \,x^{\frac {7}{2}}}{7}+\frac {2 a \,b^{2} x^{\frac {9}{2}}}{3}+\frac {2 b^{3} x^{\frac {11}{2}}}{11}\) | \(36\) |
default | \(\frac {2 a^{3} x^{\frac {5}{2}}}{5}+\frac {6 a^{2} b \,x^{\frac {7}{2}}}{7}+\frac {2 a \,b^{2} x^{\frac {9}{2}}}{3}+\frac {2 b^{3} x^{\frac {11}{2}}}{11}\) | \(36\) |
trager | \(\frac {2 x^{\frac {5}{2}} \left (105 b^{3} x^{3}+385 a \,b^{2} x^{2}+495 a^{2} b x +231 a^{3}\right )}{1155}\) | \(36\) |
risch | \(\frac {2 x^{\frac {5}{2}} \left (105 b^{3} x^{3}+385 a \,b^{2} x^{2}+495 a^{2} b x +231 a^{3}\right )}{1155}\) | \(36\) |
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none
Time = 0.23 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.78 \[ \int x^{3/2} (a+b x)^3 \, dx=\frac {2}{1155} \, {\left (105 \, b^{3} x^{5} + 385 \, a b^{2} x^{4} + 495 \, a^{2} b x^{3} + 231 \, a^{3} x^{2}\right )} \sqrt {x} \]
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Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.96 \[ \int x^{3/2} (a+b x)^3 \, dx=\frac {2 a^{3} x^{\frac {5}{2}}}{5} + \frac {6 a^{2} b x^{\frac {7}{2}}}{7} + \frac {2 a b^{2} x^{\frac {9}{2}}}{3} + \frac {2 b^{3} x^{\frac {11}{2}}}{11} \]
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none
Time = 0.24 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.69 \[ \int x^{3/2} (a+b x)^3 \, dx=\frac {2}{11} \, b^{3} x^{\frac {11}{2}} + \frac {2}{3} \, a b^{2} x^{\frac {9}{2}} + \frac {6}{7} \, a^{2} b x^{\frac {7}{2}} + \frac {2}{5} \, a^{3} x^{\frac {5}{2}} \]
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none
Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.69 \[ \int x^{3/2} (a+b x)^3 \, dx=\frac {2}{11} \, b^{3} x^{\frac {11}{2}} + \frac {2}{3} \, a b^{2} x^{\frac {9}{2}} + \frac {6}{7} \, a^{2} b x^{\frac {7}{2}} + \frac {2}{5} \, a^{3} x^{\frac {5}{2}} \]
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Time = 0.05 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.69 \[ \int x^{3/2} (a+b x)^3 \, dx=\frac {2\,a^3\,x^{5/2}}{5}+\frac {2\,b^3\,x^{11/2}}{11}+\frac {6\,a^2\,b\,x^{7/2}}{7}+\frac {2\,a\,b^2\,x^{9/2}}{3} \]
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