Integrand size = 9, antiderivative size = 30 \[ \int x (a+b x)^3 \, dx=-\frac {a (a+b x)^4}{4 b^2}+\frac {(a+b x)^5}{5 b^2} \]
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Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {45} \[ \int x (a+b x)^3 \, dx=\frac {(a+b x)^5}{5 b^2}-\frac {a (a+b x)^4}{4 b^2} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a (a+b x)^3}{b}+\frac {(a+b x)^4}{b}\right ) \, dx \\ & = -\frac {a (a+b x)^4}{4 b^2}+\frac {(a+b x)^5}{5 b^2} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.33 \[ \int x (a+b x)^3 \, dx=\frac {a^3 x^2}{2}+a^2 b x^3+\frac {3}{4} a b^2 x^4+\frac {b^3 x^5}{5} \]
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Time = 0.17 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17
method | result | size |
gosper | \(\frac {1}{5} b^{3} x^{5}+\frac {3}{4} a \,b^{2} x^{4}+a^{2} b \,x^{3}+\frac {1}{2} a^{3} x^{2}\) | \(35\) |
default | \(\frac {1}{5} b^{3} x^{5}+\frac {3}{4} a \,b^{2} x^{4}+a^{2} b \,x^{3}+\frac {1}{2} a^{3} x^{2}\) | \(35\) |
norman | \(\frac {1}{5} b^{3} x^{5}+\frac {3}{4} a \,b^{2} x^{4}+a^{2} b \,x^{3}+\frac {1}{2} a^{3} x^{2}\) | \(35\) |
risch | \(\frac {1}{5} b^{3} x^{5}+\frac {3}{4} a \,b^{2} x^{4}+a^{2} b \,x^{3}+\frac {1}{2} a^{3} x^{2}\) | \(35\) |
parallelrisch | \(\frac {1}{5} b^{3} x^{5}+\frac {3}{4} a \,b^{2} x^{4}+a^{2} b \,x^{3}+\frac {1}{2} a^{3} x^{2}\) | \(35\) |
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none
Time = 0.21 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \[ \int x (a+b x)^3 \, dx=\frac {1}{5} \, b^{3} x^{5} + \frac {3}{4} \, a b^{2} x^{4} + a^{2} b x^{3} + \frac {1}{2} \, a^{3} x^{2} \]
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Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20 \[ \int x (a+b x)^3 \, dx=\frac {a^{3} x^{2}}{2} + a^{2} b x^{3} + \frac {3 a b^{2} x^{4}}{4} + \frac {b^{3} x^{5}}{5} \]
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none
Time = 0.20 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \[ \int x (a+b x)^3 \, dx=\frac {1}{5} \, b^{3} x^{5} + \frac {3}{4} \, a b^{2} x^{4} + a^{2} b x^{3} + \frac {1}{2} \, a^{3} x^{2} \]
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none
Time = 0.29 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \[ \int x (a+b x)^3 \, dx=\frac {1}{5} \, b^{3} x^{5} + \frac {3}{4} \, a b^{2} x^{4} + a^{2} b x^{3} + \frac {1}{2} \, a^{3} x^{2} \]
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Time = 0.02 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \[ \int x (a+b x)^3 \, dx=\frac {a^3\,x^2}{2}+a^2\,b\,x^3+\frac {3\,a\,b^2\,x^4}{4}+\frac {b^3\,x^5}{5} \]
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