Integrand size = 17, antiderivative size = 13 \[ \int \left (1+3 x^2\right ) \sqrt [3]{x+x^3} \, dx=\frac {3}{4} \left (x+x^3\right )^{4/3} \]
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Time = 0.00 (sec) , antiderivative size = 13, 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 = {1602} \[ \int \left (1+3 x^2\right ) \sqrt [3]{x+x^3} \, dx=\frac {3}{4} \left (x^3+x\right )^{4/3} \]
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Rule 1602
Rubi steps \begin{align*} \text {integral}& = \frac {3}{4} \left (x+x^3\right )^{4/3} \\ \end{align*}
Time = 0.86 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.46 \[ \int \left (1+3 x^2\right ) \sqrt [3]{x+x^3} \, dx=\frac {3}{4} x \left (1+x^2\right ) \sqrt [3]{x+x^3} \]
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Time = 0.81 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77
method | result | size |
derivativedivides | \(\frac {3 \left (x^{3}+x \right )^{\frac {4}{3}}}{4}\) | \(10\) |
default | \(\frac {3 \left (x^{3}+x \right )^{\frac {4}{3}}}{4}\) | \(10\) |
gosper | \(\frac {3 \left (x^{2}+1\right ) x \left (x^{3}+x \right )^{\frac {1}{3}}}{4}\) | \(16\) |
trager | \(\frac {3 \left (x^{2}+1\right ) x \left (x^{3}+x \right )^{\frac {1}{3}}}{4}\) | \(16\) |
risch | \(\frac {3 {\left (\left (x^{2}+1\right ) x \right )}^{\frac {1}{3}} \left (x^{2}+1\right ) x}{4}\) | \(18\) |
pseudoelliptic | \(\frac {3 {\left (\left (x^{2}+1\right ) x \right )}^{\frac {1}{3}} \left (x^{2}+1\right ) x}{4}\) | \(18\) |
meijerg | \(\frac {3 x^{\frac {4}{3}} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], -x^{2}\right )}{4}+\frac {9 x^{\frac {10}{3}} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {5}{3}\right ], \left [\frac {8}{3}\right ], -x^{2}\right )}{10}\) | \(34\) |
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Time = 0.23 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \left (1+3 x^2\right ) \sqrt [3]{x+x^3} \, dx=\frac {3}{4} \, {\left (x^{3} + x\right )}^{\frac {4}{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (10) = 20\).
Time = 0.07 (sec) , antiderivative size = 27, normalized size of antiderivative = 2.08 \[ \int \left (1+3 x^2\right ) \sqrt [3]{x+x^3} \, dx=\frac {3 x^{3} \sqrt [3]{x^{3} + x}}{4} + \frac {3 x \sqrt [3]{x^{3} + x}}{4} \]
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Time = 0.18 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \left (1+3 x^2\right ) \sqrt [3]{x+x^3} \, dx=\frac {3}{4} \, {\left (x^{3} + x\right )}^{\frac {4}{3}} \]
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Time = 0.27 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \left (1+3 x^2\right ) \sqrt [3]{x+x^3} \, dx=\frac {3}{4} \, {\left (x^{3} + x\right )}^{\frac {4}{3}} \]
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Time = 4.88 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \left (1+3 x^2\right ) \sqrt [3]{x+x^3} \, dx=\frac {3\,{\left (x^3+x\right )}^{4/3}}{4} \]
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