Integrand size = 18, antiderivative size = 14 \[ \int \frac {4+x^3}{x^2 \left (1+x^3\right )^{3/4}} \, dx=-\frac {4 \sqrt [4]{1+x^3}}{x} \]
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Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {460} \[ \int \frac {4+x^3}{x^2 \left (1+x^3\right )^{3/4}} \, dx=-\frac {4 \sqrt [4]{x^3+1}}{x} \]
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Rule 460
Rubi steps \begin{align*} \text {integral}& = -\frac {4 \sqrt [4]{1+x^3}}{x} \\ \end{align*}
Time = 1.90 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {4+x^3}{x^2 \left (1+x^3\right )^{3/4}} \, dx=-\frac {4 \sqrt [4]{1+x^3}}{x} \]
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Time = 0.87 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93
method | result | size |
trager | \(-\frac {4 \left (x^{3}+1\right )^{\frac {1}{4}}}{x}\) | \(13\) |
risch | \(-\frac {4 \left (x^{3}+1\right )^{\frac {1}{4}}}{x}\) | \(13\) |
gosper | \(-\frac {4 \left (1+x \right ) \left (x^{2}-x +1\right )}{x \left (x^{3}+1\right )^{\frac {3}{4}}}\) | \(24\) |
meijerg | \(\frac {x^{2} \operatorname {hypergeom}\left (\left [\frac {2}{3}, \frac {3}{4}\right ], \left [\frac {5}{3}\right ], -x^{3}\right )}{2}-\frac {4 \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {3}{4}\right ], \left [\frac {2}{3}\right ], -x^{3}\right )}{x}\) | \(34\) |
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none
Time = 0.23 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {4+x^3}{x^2 \left (1+x^3\right )^{3/4}} \, dx=-\frac {4 \, {\left (x^{3} + 1\right )}^{\frac {1}{4}}}{x} \]
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Result contains complex when optimal does not.
Time = 0.80 (sec) , antiderivative size = 63, normalized size of antiderivative = 4.50 \[ \int \frac {4+x^3}{x^2 \left (1+x^3\right )^{3/4}} \, dx=\frac {x^{2} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {3}{4} \\ \frac {5}{3} \end {matrix}\middle | {x^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac {5}{3}\right )} + \frac {4 \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {3}{4} \\ \frac {2}{3} \end {matrix}\middle | {x^{3} e^{i \pi }} \right )}}{3 x \Gamma \left (\frac {2}{3}\right )} \]
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none
Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.43 \[ \int \frac {4+x^3}{x^2 \left (1+x^3\right )^{3/4}} \, dx=-\frac {4 \, {\left (x^{2} - x + 1\right )}^{\frac {1}{4}} {\left (x + 1\right )}^{\frac {1}{4}}}{x} \]
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\[ \int \frac {4+x^3}{x^2 \left (1+x^3\right )^{3/4}} \, dx=\int { \frac {x^{3} + 4}{{\left (x^{3} + 1\right )}^{\frac {3}{4}} x^{2}} \,d x } \]
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Time = 0.09 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {4+x^3}{x^2 \left (1+x^3\right )^{3/4}} \, dx=-\frac {4\,{\left (x^3+1\right )}^{1/4}}{x} \]
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