Integrand size = 13, antiderivative size = 17 \[ \int \frac {\left (b x^2\right )^{3/2}}{x} \, dx=\frac {1}{3} b x^2 \sqrt {b x^2} \]
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Time = 0.00 (sec) , antiderivative size = 17, 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} \[ \int \frac {\left (b x^2\right )^{3/2}}{x} \, dx=\frac {1}{3} b x^2 \sqrt {b x^2} \]
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Rule 15
Rule 30
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b \sqrt {b x^2}\right ) \int x^2 \, dx}{x} \\ & = \frac {1}{3} b x^2 \sqrt {b x^2} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {\left (b x^2\right )^{3/2}}{x} \, dx=\frac {1}{3} b x^2 \sqrt {b x^2} \]
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Time = 0.06 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.59
method | result | size |
gosper | \(\frac {\left (b \,x^{2}\right )^{\frac {3}{2}}}{3}\) | \(10\) |
derivativedivides | \(\frac {\left (b \,x^{2}\right )^{\frac {3}{2}}}{3}\) | \(10\) |
default | \(\frac {\left (b \,x^{2}\right )^{\frac {3}{2}}}{3}\) | \(10\) |
risch | \(\frac {b \,x^{2} \sqrt {b \,x^{2}}}{3}\) | \(14\) |
pseudoelliptic | \(\frac {b \,x^{2} \sqrt {b \,x^{2}}}{3}\) | \(14\) |
trager | \(\frac {b \left (x^{2}+x +1\right ) \left (-1+x \right ) \sqrt {b \,x^{2}}}{3 x}\) | \(23\) |
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none
Time = 0.25 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {\left (b x^2\right )^{3/2}}{x} \, dx=\frac {1}{3} \, \sqrt {b x^{2}} b x^{2} \]
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Time = 0.10 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.47 \[ \int \frac {\left (b x^2\right )^{3/2}}{x} \, dx=\frac {\left (b x^{2}\right )^{\frac {3}{2}}}{3} \]
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none
Time = 0.22 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.53 \[ \int \frac {\left (b x^2\right )^{3/2}}{x} \, dx=\frac {1}{3} \, \left (b x^{2}\right )^{\frac {3}{2}} \]
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none
Time = 0.28 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.59 \[ \int \frac {\left (b x^2\right )^{3/2}}{x} \, dx=\frac {1}{3} \, b^{\frac {3}{2}} x^{3} \mathrm {sgn}\left (x\right ) \]
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Time = 5.68 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.59 \[ \int \frac {\left (b x^2\right )^{3/2}}{x} \, dx=\frac {b^{3/2}\,\sqrt {x^6}}{3} \]
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