Integrand size = 13, antiderivative size = 16 \[ \int \frac {x^3}{\left (b x^2\right )^{3/2}} \, dx=\frac {x^2}{b \sqrt {b x^2}} \]
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Time = 0.00 (sec) , antiderivative size = 16, 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, 8} \[ \int \frac {x^3}{\left (b x^2\right )^{3/2}} \, dx=\frac {x^2}{b \sqrt {b x^2}} \]
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Rule 8
Rule 15
Rubi steps \begin{align*} \text {integral}& = \frac {x \int 1 \, dx}{b \sqrt {b x^2}} \\ & = \frac {x^2}{b \sqrt {b x^2}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81 \[ \int \frac {x^3}{\left (b x^2\right )^{3/2}} \, dx=\frac {\sqrt {b x^2}}{b^2} \]
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Time = 0.05 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75
method | result | size |
default | \(\frac {x^{4}}{\left (b \,x^{2}\right )^{\frac {3}{2}}}\) | \(12\) |
risch | \(\frac {x^{2}}{b \sqrt {b \,x^{2}}}\) | \(15\) |
pseudoelliptic | \(\frac {x^{2}}{b \sqrt {b \,x^{2}}}\) | \(15\) |
trager | \(\frac {\left (-1+x \right ) \sqrt {b \,x^{2}}}{b^{2} x}\) | \(18\) |
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none
Time = 0.26 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.69 \[ \int \frac {x^3}{\left (b x^2\right )^{3/2}} \, dx=\frac {\sqrt {b x^{2}}}{b^{2}} \]
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Time = 0.20 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.62 \[ \int \frac {x^3}{\left (b x^2\right )^{3/2}} \, dx=\frac {x^{4}}{\left (b x^{2}\right )^{\frac {3}{2}}} \]
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none
Time = 0.22 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {x^3}{\left (b x^2\right )^{3/2}} \, dx=\frac {x^{2}}{\sqrt {b x^{2}} b} \]
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none
Time = 0.30 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.56 \[ \int \frac {x^3}{\left (b x^2\right )^{3/2}} \, dx=\frac {x}{b^{\frac {3}{2}} \mathrm {sgn}\left (x\right )} \]
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Time = 6.13 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.38 \[ \int \frac {x^3}{\left (b x^2\right )^{3/2}} \, dx=\frac {\left |x\right |}{b^{3/2}} \]
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