Integrand size = 13, antiderivative size = 18 \[ \int \frac {x^3}{\left (1+x^2\right )^{2/3}} \, dx=\frac {3}{8} \left (-3+x^2\right ) \sqrt [3]{1+x^2} \]
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Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.50, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45} \[ \int \frac {x^3}{\left (1+x^2\right )^{2/3}} \, dx=\frac {3}{8} \left (x^2+1\right )^{4/3}-\frac {3}{2} \sqrt [3]{x^2+1} \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x}{(1+x)^{2/3}} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (-\frac {1}{(1+x)^{2/3}}+\sqrt [3]{1+x}\right ) \, dx,x,x^2\right ) \\ & = -\frac {3}{2} \sqrt [3]{1+x^2}+\frac {3}{8} \left (1+x^2\right )^{4/3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {x^3}{\left (1+x^2\right )^{2/3}} \, dx=\frac {3}{8} \left (-3+x^2\right ) \sqrt [3]{1+x^2} \]
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Time = 0.76 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83
method | result | size |
gosper | \(\frac {3 \left (x^{2}-3\right ) \left (x^{2}+1\right )^{\frac {1}{3}}}{8}\) | \(15\) |
risch | \(\frac {3 \left (x^{2}-3\right ) \left (x^{2}+1\right )^{\frac {1}{3}}}{8}\) | \(15\) |
pseudoelliptic | \(\frac {3 \left (x^{2}-3\right ) \left (x^{2}+1\right )^{\frac {1}{3}}}{8}\) | \(15\) |
trager | \(\left (\frac {3 x^{2}}{8}-\frac {9}{8}\right ) \left (x^{2}+1\right )^{\frac {1}{3}}\) | \(16\) |
meijerg | \(\frac {x^{4} \operatorname {hypergeom}\left (\left [\frac {2}{3}, 2\right ], \left [3\right ], -x^{2}\right )}{4}\) | \(17\) |
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Time = 0.23 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {x^3}{\left (1+x^2\right )^{2/3}} \, dx=\frac {3}{8} \, {\left (x^{2} + 1\right )}^{\frac {1}{3}} {\left (x^{2} - 3\right )} \]
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Time = 0.47 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.44 \[ \int \frac {x^3}{\left (1+x^2\right )^{2/3}} \, dx=\frac {3 x^{2} \sqrt [3]{x^{2} + 1}}{8} - \frac {9 \sqrt [3]{x^{2} + 1}}{8} \]
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Time = 0.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {x^3}{\left (1+x^2\right )^{2/3}} \, dx=\frac {3}{8} \, {\left (x^{2} + 1\right )}^{\frac {4}{3}} - \frac {3}{2} \, {\left (x^{2} + 1\right )}^{\frac {1}{3}} \]
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Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {x^3}{\left (1+x^2\right )^{2/3}} \, dx=\frac {3}{8} \, {\left (x^{2} + 1\right )}^{\frac {4}{3}} - \frac {3}{2} \, {\left (x^{2} + 1\right )}^{\frac {1}{3}} \]
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Time = 4.98 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {x^3}{\left (1+x^2\right )^{2/3}} \, dx=\frac {3\,{\left (x^2+1\right )}^{1/3}\,\left (x^2-3\right )}{8} \]
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