Integrand size = 24, antiderivative size = 14 \[ \int \frac {\left (2+3 x^2\right ) \sqrt [3]{x+x^3}}{1+x^2} \, dx=\frac {3}{2} x \sqrt [3]{x+x^3} \]
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Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2081, 460} \[ \int \frac {\left (2+3 x^2\right ) \sqrt [3]{x+x^3}}{1+x^2} \, dx=\frac {3}{2} x \sqrt [3]{x^3+x} \]
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Rule 460
Rule 2081
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [3]{x+x^3} \int \frac {\sqrt [3]{x} \left (2+3 x^2\right )}{\left (1+x^2\right )^{2/3}} \, dx}{\sqrt [3]{x} \sqrt [3]{1+x^2}} \\ & = \frac {3}{2} x \sqrt [3]{x+x^3} \\ \end{align*}
Time = 1.47 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {\left (2+3 x^2\right ) \sqrt [3]{x+x^3}}{1+x^2} \, dx=\frac {3}{2} x \sqrt [3]{x+x^3} \]
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Time = 1.03 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.79
method | result | size |
gosper | \(\frac {3 x \left (x^{3}+x \right )^{\frac {1}{3}}}{2}\) | \(11\) |
trager | \(\frac {3 x \left (x^{3}+x \right )^{\frac {1}{3}}}{2}\) | \(11\) |
risch | \(\frac {3 {\left (\left (x^{2}+1\right ) x \right )}^{\frac {1}{3}} x}{2}\) | \(13\) |
pseudoelliptic | \(\frac {3 {\left (\left (x^{2}+1\right ) x \right )}^{\frac {1}{3}} x}{2}\) | \(13\) |
meijerg | \(\frac {3 x^{\frac {4}{3}} \operatorname {hypergeom}\left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], -x^{2}\right )}{2}+\frac {9 x^{\frac {10}{3}} \operatorname {hypergeom}\left (\left [\frac {2}{3}, \frac {5}{3}\right ], \left [\frac {8}{3}\right ], -x^{2}\right )}{10}\) | \(34\) |
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Time = 0.24 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \frac {\left (2+3 x^2\right ) \sqrt [3]{x+x^3}}{1+x^2} \, dx=\frac {3}{2} \, {\left (x^{3} + x\right )}^{\frac {1}{3}} x \]
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\[ \int \frac {\left (2+3 x^2\right ) \sqrt [3]{x+x^3}}{1+x^2} \, dx=\int \frac {\sqrt [3]{x \left (x^{2} + 1\right )} \left (3 x^{2} + 2\right )}{x^{2} + 1}\, dx \]
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\[ \int \frac {\left (2+3 x^2\right ) \sqrt [3]{x+x^3}}{1+x^2} \, dx=\int { \frac {{\left (x^{3} + x\right )}^{\frac {1}{3}} {\left (3 \, x^{2} + 2\right )}}{x^{2} + 1} \,d x } \]
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Time = 0.30 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {\left (2+3 x^2\right ) \sqrt [3]{x+x^3}}{1+x^2} \, dx=\frac {3}{2} \, x^{2} {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} \]
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Time = 5.11 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \frac {\left (2+3 x^2\right ) \sqrt [3]{x+x^3}}{1+x^2} \, dx=\frac {3\,x\,{\left (x^3+x\right )}^{1/3}}{2} \]
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