Integrand size = 13, antiderivative size = 19 \[ \int \frac {1}{x^3 \left (b x^2\right )^{3/2}} \, dx=-\frac {1}{5 b x^4 \sqrt {b x^2}} \]
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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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {15, 30} \[ \int \frac {1}{x^3 \left (b x^2\right )^{3/2}} \, dx=-\frac {1}{5 b x^4 \sqrt {b x^2}} \]
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Rule 15
Rule 30
Rubi steps \begin{align*} \text {integral}& = \frac {x \int \frac {1}{x^6} \, dx}{b \sqrt {b x^2}} \\ & = -\frac {1}{5 b x^4 \sqrt {b x^2}} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x^3 \left (b x^2\right )^{3/2}} \, dx=-\frac {1}{5 x^2 \left (b x^2\right )^{3/2}} \]
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Time = 0.06 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68
method | result | size |
gosper | \(-\frac {1}{5 x^{2} \left (b \,x^{2}\right )^{\frac {3}{2}}}\) | \(13\) |
default | \(-\frac {1}{5 x^{2} \left (b \,x^{2}\right )^{\frac {3}{2}}}\) | \(13\) |
risch | \(-\frac {1}{5 b \,x^{4} \sqrt {b \,x^{2}}}\) | \(16\) |
pseudoelliptic | \(-\frac {1}{5 b \,x^{4} \sqrt {b \,x^{2}}}\) | \(16\) |
trager | \(\frac {\left (-1+x \right ) \left (x^{4}+x^{3}+x^{2}+x +1\right ) \sqrt {b \,x^{2}}}{5 b^{2} x^{6}}\) | \(31\) |
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none
Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {1}{x^3 \left (b x^2\right )^{3/2}} \, dx=-\frac {\sqrt {b x^{2}}}{5 \, b^{2} x^{6}} \]
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Time = 0.24 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {1}{x^3 \left (b x^2\right )^{3/2}} \, dx=- \frac {1}{5 x^{2} \left (b x^{2}\right )^{\frac {3}{2}}} \]
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none
Time = 0.21 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.42 \[ \int \frac {1}{x^3 \left (b x^2\right )^{3/2}} \, dx=-\frac {1}{5 \, b^{\frac {3}{2}} x^{5}} \]
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none
Time = 0.28 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.63 \[ \int \frac {1}{x^3 \left (b x^2\right )^{3/2}} \, dx=-\frac {1}{5 \, b^{\frac {3}{2}} x^{5} \mathrm {sgn}\left (x\right )} \]
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Time = 5.74 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.53 \[ \int \frac {1}{x^3 \left (b x^2\right )^{3/2}} \, dx=-\frac {1}{5\,b^{3/2}\,{\left (x^2\right )}^{5/2}} \]
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