Integrand size = 17, antiderivative size = 20 \[ \int x^2 \sqrt {-b+a x^3} \, dx=\frac {2 \left (-b+a x^3\right )^{3/2}}{9 a} \]
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Time = 0.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {267} \[ \int x^2 \sqrt {-b+a x^3} \, dx=\frac {2 \left (a x^3-b\right )^{3/2}}{9 a} \]
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Rule 267
Rubi steps \begin{align*} \text {integral}& = \frac {2 \left (-b+a x^3\right )^{3/2}}{9 a} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int x^2 \sqrt {-b+a x^3} \, dx=\frac {2 \left (-b+a x^3\right )^{3/2}}{9 a} \]
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Time = 0.79 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85
method | result | size |
gosper | \(\frac {2 \left (a \,x^{3}-b \right )^{\frac {3}{2}}}{9 a}\) | \(17\) |
derivativedivides | \(\frac {2 \left (a \,x^{3}-b \right )^{\frac {3}{2}}}{9 a}\) | \(17\) |
default | \(\frac {2 \left (a \,x^{3}-b \right )^{\frac {3}{2}}}{9 a}\) | \(17\) |
trager | \(\frac {2 \left (a \,x^{3}-b \right )^{\frac {3}{2}}}{9 a}\) | \(17\) |
risch | \(\frac {2 \left (a \,x^{3}-b \right )^{\frac {3}{2}}}{9 a}\) | \(17\) |
pseudoelliptic | \(\frac {2 \left (a \,x^{3}-b \right )^{\frac {3}{2}}}{9 a}\) | \(17\) |
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none
Time = 0.25 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int x^2 \sqrt {-b+a x^3} \, dx=\frac {2 \, {\left (a x^{3} - b\right )}^{\frac {3}{2}}}{9 \, a} \]
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Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (14) = 28\).
Time = 0.08 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.20 \[ \int x^2 \sqrt {-b+a x^3} \, dx=\begin {cases} \frac {2 x^{3} \sqrt {a x^{3} - b}}{9} - \frac {2 b \sqrt {a x^{3} - b}}{9 a} & \text {for}\: a \neq 0 \\\frac {x^{3} \sqrt {- b}}{3} & \text {otherwise} \end {cases} \]
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none
Time = 0.20 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int x^2 \sqrt {-b+a x^3} \, dx=\frac {2 \, {\left (a x^{3} - b\right )}^{\frac {3}{2}}}{9 \, a} \]
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none
Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int x^2 \sqrt {-b+a x^3} \, dx=\frac {2 \, {\left (a x^{3} - b\right )}^{\frac {3}{2}}}{9 \, a} \]
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Time = 5.08 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int x^2 \sqrt {-b+a x^3} \, dx=\frac {2\,{\left (a\,x^3-b\right )}^{3/2}}{9\,a} \]
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