Integrand size = 18, antiderivative size = 16 \[ \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^5} \, dx=\frac {3 \left (1+x^4\right )^{4/3}}{4 x^4} \]
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Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {457, 75} \[ \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^5} \, dx=\frac {3 \left (x^4+1\right )^{4/3}}{4 x^4} \]
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Rule 75
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \text {Subst}\left (\int \frac {(-3+x) \sqrt [3]{1+x}}{x^2} \, dx,x,x^4\right ) \\ & = \frac {3 \left (1+x^4\right )^{4/3}}{4 x^4} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^5} \, dx=\frac {3 \left (1+x^4\right )^{4/3}}{4 x^4} \]
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Time = 0.99 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81
method | result | size |
gosper | \(\frac {3 \left (x^{4}+1\right )^{\frac {4}{3}}}{4 x^{4}}\) | \(13\) |
trager | \(\frac {3 \left (x^{4}+1\right )^{\frac {4}{3}}}{4 x^{4}}\) | \(13\) |
pseudoelliptic | \(\frac {3 \left (x^{4}+1\right )^{\frac {4}{3}}}{4 x^{4}}\) | \(13\) |
risch | \(\frac {\frac {3}{2} x^{4}+\frac {3}{4}+\frac {3}{4} x^{8}}{\left (x^{4}+1\right )^{\frac {2}{3}} x^{4}}\) | \(23\) |
meijerg | \(\frac {\frac {\Gamma \left (\frac {2}{3}\right ) x^{4} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{3}\right ], \left [2, 3\right ], -x^{4}\right )}{3}-\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}-1+4 \ln \left (x \right )\right ) \Gamma \left (\frac {2}{3}\right )+\frac {3 \Gamma \left (\frac {2}{3}\right )}{x^{4}}}{4 \Gamma \left (\frac {2}{3}\right )}-\frac {-\Gamma \left (\frac {2}{3}\right ) x^{4} \operatorname {hypergeom}\left (\left [\frac {2}{3}, 1, 1\right ], \left [2, 2\right ], -x^{4}\right )-3 \left (3+\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+4 \ln \left (x \right )\right ) \Gamma \left (\frac {2}{3}\right )}{12 \Gamma \left (\frac {2}{3}\right )}\) | \(103\) |
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Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^5} \, dx=\frac {3 \, {\left (x^{4} + 1\right )}^{\frac {4}{3}}}{4 \, x^{4}} \]
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Result contains complex when optimal does not.
Time = 65.65 (sec) , antiderivative size = 71, normalized size of antiderivative = 4.44 \[ \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^5} \, dx=- \frac {x^{\frac {4}{3}} \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, - \frac {1}{3} \\ \frac {2}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{4}}} \right )}}{4 \Gamma \left (\frac {2}{3}\right )} + \frac {3 \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{4}}} \right )}}{4 x^{\frac {8}{3}} \Gamma \left (\frac {5}{3}\right )} \]
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Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.38 \[ \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^5} \, dx=\frac {3}{4} \, {\left (x^{4} + 1\right )}^{\frac {1}{3}} + \frac {3 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{4 \, x^{4}} \]
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Time = 0.29 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.38 \[ \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^5} \, dx=\frac {3}{4} \, {\left (x^{4} + 1\right )}^{\frac {1}{3}} + \frac {3 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{4 \, x^{4}} \]
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Time = 0.12 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^5} \, dx=\frac {3\,{\left (x^4+1\right )}^{4/3}}{4\,x^4} \]
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