Integrand size = 9, antiderivative size = 19 \[ \int \frac {1}{\left (b x^2\right )^{3/2}} \, dx=-\frac {1}{2 b x \sqrt {b x^2}} \]
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Time = 0.00 (sec) , antiderivative size = 19, 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} \[ \int \frac {1}{\left (b x^2\right )^{3/2}} \, dx=-\frac {1}{2 b x \sqrt {b x^2}} \]
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Rule 15
Rule 30
Rubi steps \begin{align*} \text {integral}& = \frac {x \int \frac {1}{x^3} \, dx}{b \sqrt {b x^2}} \\ & = -\frac {1}{2 b x \sqrt {b x^2}} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74 \[ \int \frac {1}{\left (b x^2\right )^{3/2}} \, dx=-\frac {x}{2 \left (b x^2\right )^{3/2}} \]
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Time = 0.04 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.58
method | result | size |
gosper | \(-\frac {x}{2 \left (b \,x^{2}\right )^{\frac {3}{2}}}\) | \(11\) |
default | \(-\frac {x}{2 \left (b \,x^{2}\right )^{\frac {3}{2}}}\) | \(11\) |
risch | \(-\frac {1}{2 b x \sqrt {b \,x^{2}}}\) | \(16\) |
trager | \(\frac {\left (-1+x \right ) \left (1+x \right ) \sqrt {b \,x^{2}}}{2 b^{2} x^{3}}\) | \(22\) |
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none
Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {1}{\left (b x^2\right )^{3/2}} \, dx=-\frac {\sqrt {b x^{2}}}{2 \, b^{2} x^{3}} \]
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Time = 0.21 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.63 \[ \int \frac {1}{\left (b x^2\right )^{3/2}} \, dx=- \frac {x}{2 \left (b x^{2}\right )^{\frac {3}{2}}} \]
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none
Time = 0.19 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.42 \[ \int \frac {1}{\left (b x^2\right )^{3/2}} \, dx=-\frac {1}{2 \, b^{\frac {3}{2}} x^{2}} \]
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none
Time = 0.28 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.63 \[ \int \frac {1}{\left (b x^2\right )^{3/2}} \, dx=-\frac {1}{2 \, b^{\frac {3}{2}} x^{2} \mathrm {sgn}\left (x\right )} \]
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Time = 5.71 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int \frac {1}{\left (b x^2\right )^{3/2}} \, dx=-\frac {1}{2\,b^{3/2}\,x\,\sqrt {x^2}} \]
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