Integrand size = 18, antiderivative size = 16 \[ \int \frac {4+x^3}{x^4 \sqrt [4]{1+x^3}} \, dx=-\frac {4 \left (1+x^3\right )^{3/4}}{3 x^3} \]
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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} \[ \int \frac {4+x^3}{x^4 \sqrt [4]{1+x^3}} \, dx=-\frac {4 \left (x^3+1\right )^{3/4}}{3 x^3} \]
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Rule 75
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {4+x}{x^2 \sqrt [4]{1+x}} \, dx,x,x^3\right ) \\ & = -\frac {4 \left (1+x^3\right )^{3/4}}{3 x^3} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {4+x^3}{x^4 \sqrt [4]{1+x^3}} \, dx=-\frac {4 \left (1+x^3\right )^{3/4}}{3 x^3} \]
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Time = 0.97 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81
method | result | size |
trager | \(-\frac {4 \left (x^{3}+1\right )^{\frac {3}{4}}}{3 x^{3}}\) | \(13\) |
risch | \(-\frac {4 \left (x^{3}+1\right )^{\frac {3}{4}}}{3 x^{3}}\) | \(13\) |
pseudoelliptic | \(-\frac {4 \left (x^{3}+1\right )^{\frac {3}{4}}}{3 x^{3}}\) | \(13\) |
gosper | \(-\frac {4 \left (1+x \right ) \left (x^{2}-x +1\right )}{3 x^{3} \left (x^{3}+1\right )^{\frac {1}{4}}}\) | \(24\) |
meijerg | \(\frac {2 \sqrt {2}\, \Gamma \left (\frac {3}{4}\right ) \left (\frac {5 \pi \sqrt {2}\, x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {9}{4}\right ], \left [2, 3\right ], -x^{3}\right )}{32 \Gamma \left (\frac {3}{4}\right )}-\frac {\left (3-3 \ln \left (2\right )-\frac {\pi }{2}+3 \ln \left (x \right )\right ) \pi \sqrt {2}}{4 \Gamma \left (\frac {3}{4}\right )}-\frac {\pi \sqrt {2}}{\Gamma \left (\frac {3}{4}\right ) x^{3}}\right )}{3 \pi }+\frac {\sqrt {2}\, \Gamma \left (\frac {3}{4}\right ) \left (-\frac {\pi \sqrt {2}\, x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{4}\right ], \left [2, 2\right ], -x^{3}\right )}{4 \Gamma \left (\frac {3}{4}\right )}+\frac {\left (-3 \ln \left (2\right )-\frac {\pi }{2}+3 \ln \left (x \right )\right ) \pi \sqrt {2}}{\Gamma \left (\frac {3}{4}\right )}\right )}{6 \pi }\) | \(133\) |
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Time = 0.23 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {4+x^3}{x^4 \sqrt [4]{1+x^3}} \, dx=-\frac {4 \, {\left (x^{3} + 1\right )}^{\frac {3}{4}}}{3 \, x^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (15) = 30\).
Time = 8.53 (sec) , antiderivative size = 49, normalized size of antiderivative = 3.06 \[ \int \frac {4+x^3}{x^4 \sqrt [4]{1+x^3}} \, dx=- \frac {2 \sqrt [4]{x^{3} + 1}}{3 \left (\sqrt {x^{3} + 1} + 1\right )} - \frac {1}{3 \left (\sqrt [4]{x^{3} + 1} + 1\right )} - \frac {1}{3 \left (\sqrt [4]{x^{3} + 1} - 1\right )} \]
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Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {4+x^3}{x^4 \sqrt [4]{1+x^3}} \, dx=-\frac {4 \, {\left (x^{3} + 1\right )}^{\frac {3}{4}}}{3 \, x^{3}} \]
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Time = 0.29 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {4+x^3}{x^4 \sqrt [4]{1+x^3}} \, dx=-\frac {4 \, {\left (x^{3} + 1\right )}^{\frac {3}{4}}}{3 \, x^{3}} \]
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Time = 0.08 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {4+x^3}{x^4 \sqrt [4]{1+x^3}} \, dx=-\frac {4\,{\left (x^3+1\right )}^{3/4}}{3\,x^3} \]
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