Integrand size = 9, antiderivative size = 14 \[ \int \sqrt {b x^2} \, dx=\frac {1}{2} x \sqrt {b x^2} \]
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Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {15, 30} \[ \int \sqrt {b x^2} \, dx=\frac {1}{2} x \sqrt {b x^2} \]
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Rule 15
Rule 30
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {b x^2} \int x \, dx}{x} \\ & = \frac {1}{2} x \sqrt {b x^2} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \sqrt {b x^2} \, dx=\frac {1}{2} x \sqrt {b x^2} \]
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Time = 0.04 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.79
method | result | size |
gosper | \(\frac {x \sqrt {b \,x^{2}}}{2}\) | \(11\) |
default | \(\frac {x \sqrt {b \,x^{2}}}{2}\) | \(11\) |
risch | \(\frac {x \sqrt {b \,x^{2}}}{2}\) | \(11\) |
trager | \(\frac {\left (-1+x \right ) \left (1+x \right ) \sqrt {b \,x^{2}}}{2 x}\) | \(19\) |
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none
Time = 0.26 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \sqrt {b x^2} \, dx=\frac {1}{2} \, \sqrt {b x^{2}} x \]
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Time = 0.06 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \sqrt {b x^2} \, dx=\frac {x \sqrt {b x^{2}}}{2} \]
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none
Time = 0.20 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \sqrt {b x^2} \, dx=\frac {1}{2} \, \sqrt {b x^{2}} x \]
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none
Time = 0.27 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \sqrt {b x^2} \, dx=\frac {1}{2} \, \sqrt {b} x^{2} \mathrm {sgn}\left (x\right ) \]
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Time = 5.77 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.57 \[ \int \sqrt {b x^2} \, dx=\frac {\sqrt {b}\,x\,\left |x\right |}{2} \]
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