Integrand size = 13, antiderivative size = 51 \[ \int \sqrt {x} (a+b x)^3 \, dx=\frac {2}{3} a^3 x^{3/2}+\frac {6}{5} a^2 b x^{5/2}+\frac {6}{7} a b^2 x^{7/2}+\frac {2}{9} b^3 x^{9/2} \]
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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} \[ \int \sqrt {x} (a+b x)^3 \, dx=\frac {2}{3} a^3 x^{3/2}+\frac {6}{5} a^2 b x^{5/2}+\frac {6}{7} a b^2 x^{7/2}+\frac {2}{9} b^3 x^{9/2} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (a^3 \sqrt {x}+3 a^2 b x^{3/2}+3 a b^2 x^{5/2}+b^3 x^{7/2}\right ) \, dx \\ & = \frac {2}{3} a^3 x^{3/2}+\frac {6}{5} a^2 b x^{5/2}+\frac {6}{7} a b^2 x^{7/2}+\frac {2}{9} b^3 x^{9/2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.76 \[ \int \sqrt {x} (a+b x)^3 \, dx=\frac {2}{315} x^{3/2} \left (105 a^3+189 a^2 b x+135 a b^2 x^2+35 b^3 x^3\right ) \]
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Time = 0.07 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.71
method | result | size |
gosper | \(\frac {2 x^{\frac {3}{2}} \left (35 b^{3} x^{3}+135 a \,b^{2} x^{2}+189 a^{2} b x +105 a^{3}\right )}{315}\) | \(36\) |
derivativedivides | \(\frac {2 a^{3} x^{\frac {3}{2}}}{3}+\frac {6 a^{2} b \,x^{\frac {5}{2}}}{5}+\frac {6 a \,b^{2} x^{\frac {7}{2}}}{7}+\frac {2 b^{3} x^{\frac {9}{2}}}{9}\) | \(36\) |
default | \(\frac {2 a^{3} x^{\frac {3}{2}}}{3}+\frac {6 a^{2} b \,x^{\frac {5}{2}}}{5}+\frac {6 a \,b^{2} x^{\frac {7}{2}}}{7}+\frac {2 b^{3} x^{\frac {9}{2}}}{9}\) | \(36\) |
trager | \(\frac {2 x^{\frac {3}{2}} \left (35 b^{3} x^{3}+135 a \,b^{2} x^{2}+189 a^{2} b x +105 a^{3}\right )}{315}\) | \(36\) |
risch | \(\frac {2 x^{\frac {3}{2}} \left (35 b^{3} x^{3}+135 a \,b^{2} x^{2}+189 a^{2} b x +105 a^{3}\right )}{315}\) | \(36\) |
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none
Time = 0.22 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.75 \[ \int \sqrt {x} (a+b x)^3 \, dx=\frac {2}{315} \, {\left (35 \, b^{3} x^{4} + 135 \, a b^{2} x^{3} + 189 \, a^{2} b x^{2} + 105 \, a^{3} x\right )} \sqrt {x} \]
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Timed out. \[ \int \sqrt {x} (a+b x)^3 \, dx=\text {Timed out} \]
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none
Time = 0.21 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.69 \[ \int \sqrt {x} (a+b x)^3 \, dx=\frac {2}{9} \, b^{3} x^{\frac {9}{2}} + \frac {6}{7} \, a b^{2} x^{\frac {7}{2}} + \frac {6}{5} \, a^{2} b x^{\frac {5}{2}} + \frac {2}{3} \, a^{3} x^{\frac {3}{2}} \]
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none
Time = 0.28 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.69 \[ \int \sqrt {x} (a+b x)^3 \, dx=\frac {2}{9} \, b^{3} x^{\frac {9}{2}} + \frac {6}{7} \, a b^{2} x^{\frac {7}{2}} + \frac {6}{5} \, a^{2} b x^{\frac {5}{2}} + \frac {2}{3} \, a^{3} x^{\frac {3}{2}} \]
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Time = 0.05 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.69 \[ \int \sqrt {x} (a+b x)^3 \, dx=\frac {2\,a^3\,x^{3/2}}{3}+\frac {2\,b^3\,x^{9/2}}{9}+\frac {6\,a^2\,b\,x^{5/2}}{5}+\frac {6\,a\,b^2\,x^{7/2}}{7} \]
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