Integrand size = 13, antiderivative size = 38 \[ \int \frac {x^5}{\left (-4+x^2\right )^{13/6}} \, dx=-\frac {48}{7 \left (-4+x^2\right )^{7/6}}-\frac {24}{\sqrt [6]{-4+x^2}}+\frac {3}{5} \left (-4+x^2\right )^{5/6} \]
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Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45} \[ \int \frac {x^5}{\left (-4+x^2\right )^{13/6}} \, dx=\frac {3}{5} \left (x^2-4\right )^{5/6}-\frac {24}{\sqrt [6]{x^2-4}}-\frac {48}{7 \left (x^2-4\right )^{7/6}} \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x^2}{(-4+x)^{13/6}} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {16}{(-4+x)^{13/6}}+\frac {8}{(-4+x)^{7/6}}+\frac {1}{\sqrt [6]{-4+x}}\right ) \, dx,x,x^2\right ) \\ & = -\frac {48}{7 \left (-4+x^2\right )^{7/6}}-\frac {24}{\sqrt [6]{-4+x^2}}+\frac {3}{5} \left (-4+x^2\right )^{5/6} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.66 \[ \int \frac {x^5}{\left (-4+x^2\right )^{13/6}} \, dx=\frac {3 \left (1152-336 x^2+7 x^4\right )}{35 \left (-4+x^2\right )^{7/6}} \]
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Time = 0.30 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.53
method | result | size |
pseudoelliptic | \(\frac {\frac {3}{5} x^{4}-\frac {144}{5} x^{2}+\frac {3456}{35}}{\left (x^{2}-4\right )^{\frac {7}{6}}}\) | \(20\) |
trager | \(\frac {\frac {3}{5} x^{4}-\frac {144}{5} x^{2}+\frac {3456}{35}}{\left (x^{2}-4\right )^{\frac {7}{6}}}\) | \(22\) |
risch | \(\frac {\frac {3}{5} x^{4}-\frac {144}{5} x^{2}+\frac {3456}{35}}{\left (x^{2}-4\right )^{\frac {7}{6}}}\) | \(22\) |
gosper | \(\frac {3 \left (-2+x \right ) \left (2+x \right ) \left (7 x^{4}-336 x^{2}+1152\right )}{35 \left (x^{2}-4\right )^{\frac {13}{6}}}\) | \(28\) |
meijerg | \(\frac {2^{\frac {2}{3}} {\left (-\operatorname {signum}\left (-1+\frac {x^{2}}{4}\right )\right )}^{\frac {13}{6}} x^{6} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {13}{6},3;4;\frac {x^{2}}{4}\right )}{192 \operatorname {signum}\left (-1+\frac {x^{2}}{4}\right )^{\frac {13}{6}}}\) | \(42\) |
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Time = 0.24 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.87 \[ \int \frac {x^5}{\left (-4+x^2\right )^{13/6}} \, dx=\frac {3 \, {\left (7 \, x^{4} - 336 \, x^{2} + 1152\right )} {\left (x^{2} - 4\right )}^{\frac {5}{6}}}{35 \, {\left (x^{4} - 8 \, x^{2} + 16\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (32) = 64\).
Time = 0.79 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.16 \[ \int \frac {x^5}{\left (-4+x^2\right )^{13/6}} \, dx=\frac {21 x^{4}}{35 x^{2} \sqrt [6]{x^{2} - 4} - 140 \sqrt [6]{x^{2} - 4}} - \frac {1008 x^{2}}{35 x^{2} \sqrt [6]{x^{2} - 4} - 140 \sqrt [6]{x^{2} - 4}} + \frac {3456}{35 x^{2} \sqrt [6]{x^{2} - 4} - 140 \sqrt [6]{x^{2} - 4}} \]
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Time = 0.19 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.74 \[ \int \frac {x^5}{\left (-4+x^2\right )^{13/6}} \, dx=\frac {3}{5} \, {\left (x^{2} - 4\right )}^{\frac {5}{6}} - \frac {24}{{\left (x^{2} - 4\right )}^{\frac {1}{6}}} - \frac {48}{7 \, {\left (x^{2} - 4\right )}^{\frac {7}{6}}} \]
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Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.68 \[ \int \frac {x^5}{\left (-4+x^2\right )^{13/6}} \, dx=\frac {3}{5} \, {\left (x^{2} - 4\right )}^{\frac {5}{6}} - \frac {24 \, {\left (7 \, x^{2} - 26\right )}}{7 \, {\left (x^{2} - 4\right )}^{\frac {7}{6}}} \]
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Time = 0.46 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.55 \[ \int \frac {x^5}{\left (-4+x^2\right )^{13/6}} \, dx=\frac {3\,\left (7\,x^4-336\,x^2+1152\right )}{35\,{\left (x^2-4\right )}^{7/6}} \]
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