Integrand size = 13, antiderivative size = 51 \[ \int x^{5/2} (a+b x)^3 \, dx=\frac {2}{7} a^3 x^{7/2}+\frac {2}{3} a^2 b x^{9/2}+\frac {6}{11} a b^2 x^{11/2}+\frac {2}{13} b^3 x^{13/2} \]
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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} \[ \int x^{5/2} (a+b x)^3 \, dx=\frac {2}{7} a^3 x^{7/2}+\frac {2}{3} a^2 b x^{9/2}+\frac {6}{11} a b^2 x^{11/2}+\frac {2}{13} b^3 x^{13/2} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (a^3 x^{5/2}+3 a^2 b x^{7/2}+3 a b^2 x^{9/2}+b^3 x^{11/2}\right ) \, dx \\ & = \frac {2}{7} a^3 x^{7/2}+\frac {2}{3} a^2 b x^{9/2}+\frac {6}{11} a b^2 x^{11/2}+\frac {2}{13} b^3 x^{13/2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.76 \[ \int x^{5/2} (a+b x)^3 \, dx=\frac {2 x^{7/2} \left (429 a^3+1001 a^2 b x+819 a b^2 x^2+231 b^3 x^3\right )}{3003} \]
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Time = 0.09 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.71
method | result | size |
gosper | \(\frac {2 x^{\frac {7}{2}} \left (231 b^{3} x^{3}+819 a \,b^{2} x^{2}+1001 a^{2} b x +429 a^{3}\right )}{3003}\) | \(36\) |
derivativedivides | \(\frac {2 a^{3} x^{\frac {7}{2}}}{7}+\frac {2 a^{2} b \,x^{\frac {9}{2}}}{3}+\frac {6 a \,b^{2} x^{\frac {11}{2}}}{11}+\frac {2 b^{3} x^{\frac {13}{2}}}{13}\) | \(36\) |
default | \(\frac {2 a^{3} x^{\frac {7}{2}}}{7}+\frac {2 a^{2} b \,x^{\frac {9}{2}}}{3}+\frac {6 a \,b^{2} x^{\frac {11}{2}}}{11}+\frac {2 b^{3} x^{\frac {13}{2}}}{13}\) | \(36\) |
trager | \(\frac {2 x^{\frac {7}{2}} \left (231 b^{3} x^{3}+819 a \,b^{2} x^{2}+1001 a^{2} b x +429 a^{3}\right )}{3003}\) | \(36\) |
risch | \(\frac {2 x^{\frac {7}{2}} \left (231 b^{3} x^{3}+819 a \,b^{2} x^{2}+1001 a^{2} b x +429 a^{3}\right )}{3003}\) | \(36\) |
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none
Time = 0.22 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.78 \[ \int x^{5/2} (a+b x)^3 \, dx=\frac {2}{3003} \, {\left (231 \, b^{3} x^{6} + 819 \, a b^{2} x^{5} + 1001 \, a^{2} b x^{4} + 429 \, a^{3} x^{3}\right )} \sqrt {x} \]
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Time = 0.37 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.96 \[ \int x^{5/2} (a+b x)^3 \, dx=\frac {2 a^{3} x^{\frac {7}{2}}}{7} + \frac {2 a^{2} b x^{\frac {9}{2}}}{3} + \frac {6 a b^{2} x^{\frac {11}{2}}}{11} + \frac {2 b^{3} x^{\frac {13}{2}}}{13} \]
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none
Time = 0.20 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.69 \[ \int x^{5/2} (a+b x)^3 \, dx=\frac {2}{13} \, b^{3} x^{\frac {13}{2}} + \frac {6}{11} \, a b^{2} x^{\frac {11}{2}} + \frac {2}{3} \, a^{2} b x^{\frac {9}{2}} + \frac {2}{7} \, a^{3} x^{\frac {7}{2}} \]
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none
Time = 0.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.69 \[ \int x^{5/2} (a+b x)^3 \, dx=\frac {2}{13} \, b^{3} x^{\frac {13}{2}} + \frac {6}{11} \, a b^{2} x^{\frac {11}{2}} + \frac {2}{3} \, a^{2} b x^{\frac {9}{2}} + \frac {2}{7} \, a^{3} x^{\frac {7}{2}} \]
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Time = 0.05 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.69 \[ \int x^{5/2} (a+b x)^3 \, dx=\frac {2\,a^3\,x^{7/2}}{7}+\frac {2\,b^3\,x^{13/2}}{13}+\frac {2\,a^2\,b\,x^{9/2}}{3}+\frac {6\,a\,b^2\,x^{11/2}}{11} \]
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