Integrand size = 13, antiderivative size = 18 \[ \int \frac {x^3}{\left (-1+x^2\right )^{3/4}} \, dx=\frac {2}{5} \sqrt [4]{-1+x^2} \left (4+x^2\right ) \]
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Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.39, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45} \[ \int \frac {x^3}{\left (-1+x^2\right )^{3/4}} \, dx=\frac {2}{5} \left (x^2-1\right )^{5/4}+2 \sqrt [4]{x^2-1} \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x}{(-1+x)^{3/4}} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{(-1+x)^{3/4}}+\sqrt [4]{-1+x}\right ) \, dx,x,x^2\right ) \\ & = 2 \sqrt [4]{-1+x^2}+\frac {2}{5} \left (-1+x^2\right )^{5/4} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {x^3}{\left (-1+x^2\right )^{3/4}} \, dx=\frac {2}{5} \sqrt [4]{-1+x^2} \left (4+x^2\right ) \]
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Time = 0.80 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83
method | result | size |
risch | \(\frac {2 \left (x^{2}-1\right )^{\frac {1}{4}} \left (x^{2}+4\right )}{5}\) | \(15\) |
pseudoelliptic | \(\frac {2 \left (x^{2}-1\right )^{\frac {1}{4}} \left (x^{2}+4\right )}{5}\) | \(15\) |
trager | \(\left (\frac {2 x^{2}}{5}+\frac {8}{5}\right ) \left (x^{2}-1\right )^{\frac {1}{4}}\) | \(16\) |
gosper | \(\frac {2 \left (x -1\right ) \left (1+x \right ) \left (x^{2}+4\right )}{5 \left (x^{2}-1\right )^{\frac {3}{4}}}\) | \(21\) |
meijerg | \(\frac {{\left (-\operatorname {signum}\left (x^{2}-1\right )\right )}^{\frac {3}{4}} x^{4} \operatorname {hypergeom}\left (\left [\frac {3}{4}, 2\right ], \left [3\right ], x^{2}\right )}{4 \operatorname {signum}\left (x^{2}-1\right )^{\frac {3}{4}}}\) | \(33\) |
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none
Time = 0.24 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {x^3}{\left (-1+x^2\right )^{3/4}} \, dx=\frac {2}{5} \, {\left (x^{2} + 4\right )} {\left (x^{2} - 1\right )}^{\frac {1}{4}} \]
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Result contains complex when optimal does not.
Time = 0.51 (sec) , antiderivative size = 75, normalized size of antiderivative = 4.17 \[ \int \frac {x^3}{\left (-1+x^2\right )^{3/4}} \, dx=\begin {cases} \frac {2 x^{2} \sqrt [4]{x^{2} - 1}}{5} + \frac {8 \sqrt [4]{x^{2} - 1}}{5} & \text {for}\: \left |{x^{2}}\right | > 1 \\- \frac {2 x^{2} \sqrt [4]{1 - x^{2}} e^{- \frac {3 i \pi }{4}}}{5} - \frac {8 \sqrt [4]{1 - x^{2}} e^{- \frac {3 i \pi }{4}}}{5} & \text {otherwise} \end {cases} \]
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none
Time = 0.19 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {x^3}{\left (-1+x^2\right )^{3/4}} \, dx=\frac {2}{5} \, {\left (x^{2} - 1\right )}^{\frac {5}{4}} + 2 \, {\left (x^{2} - 1\right )}^{\frac {1}{4}} \]
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Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {x^3}{\left (-1+x^2\right )^{3/4}} \, dx=\frac {2}{5} \, {\left (x^{2} - 1\right )}^{\frac {5}{4}} + 2 \, {\left (x^{2} - 1\right )}^{\frac {1}{4}} \]
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Time = 5.04 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {x^3}{\left (-1+x^2\right )^{3/4}} \, dx=\frac {2\,{\left (x^2-1\right )}^{1/4}\,\left (x^2+4\right )}{5} \]
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