Integrand size = 7, antiderivative size = 14 \[ \int (a+b x)^2 \, dx=\frac {(a+b x)^3}{3 b} \]
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Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {32} \[ \int (a+b x)^2 \, dx=\frac {(a+b x)^3}{3 b} \]
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Rule 32
Rubi steps \begin{align*} \text {integral}& = \frac {(a+b x)^3}{3 b} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int (a+b x)^2 \, dx=\frac {(a+b x)^3}{3 b} \]
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Time = 0.14 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93
method | result | size |
default | \(\frac {\left (b x +a \right )^{3}}{3 b}\) | \(13\) |
gosper | \(\frac {1}{3} b^{2} x^{3}+a b \,x^{2}+a^{2} x\) | \(21\) |
norman | \(\frac {1}{3} b^{2} x^{3}+a b \,x^{2}+a^{2} x\) | \(21\) |
parallelrisch | \(\frac {1}{3} b^{2} x^{3}+a b \,x^{2}+a^{2} x\) | \(21\) |
risch | \(\frac {b^{2} x^{3}}{3}+a b \,x^{2}+a^{2} x +\frac {a^{3}}{3 b}\) | \(29\) |
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none
Time = 0.21 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.43 \[ \int (a+b x)^2 \, dx=\frac {1}{3} \, b^{2} x^{3} + a b x^{2} + a^{2} x \]
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Leaf count of result is larger than twice the leaf count of optimal. 19 vs. \(2 (8) = 16\).
Time = 0.04 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.36 \[ \int (a+b x)^2 \, dx=a^{2} x + a b x^{2} + \frac {b^{2} x^{3}}{3} \]
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none
Time = 0.22 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.43 \[ \int (a+b x)^2 \, dx=\frac {1}{3} \, b^{2} x^{3} + a b x^{2} + a^{2} x \]
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none
Time = 0.29 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int (a+b x)^2 \, dx=\frac {{\left (b x + a\right )}^{3}}{3 \, b} \]
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Time = 0.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.43 \[ \int (a+b x)^2 \, dx=a^2\,x+a\,b\,x^2+\frac {b^2\,x^3}{3} \]
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