Integrand size = 13, antiderivative size = 391 \[ \int \frac {x^4}{\left (2+x^6\right )^{3/2}} \, dx=\frac {x^5}{6 \sqrt {2+x^6}}-\frac {\left (1+\sqrt {3}\right ) x \sqrt {2+x^6}}{6 \left (\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2\right )}+\frac {x \left (\sqrt [3]{2}+x^2\right ) \sqrt {\frac {2^{2/3}-\sqrt [3]{2} x^2+x^4}{\left (\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2\right )^2}} E\left (\arccos \left (\frac {\sqrt [3]{2}+\left (1-\sqrt {3}\right ) x^2}{\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2^{2/3} 3^{3/4} \sqrt {\frac {x^2 \left (\sqrt [3]{2}+x^2\right )}{\left (\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {2+x^6}}+\frac {\left (1-\sqrt {3}\right ) x \left (\sqrt [3]{2}+x^2\right ) \sqrt {\frac {2^{2/3}-\sqrt [3]{2} x^2+x^4}{\left (\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{2}+\left (1-\sqrt {3}\right ) x^2}{\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{6\ 2^{2/3} \sqrt [4]{3} \sqrt {\frac {x^2 \left (\sqrt [3]{2}+x^2\right )}{\left (\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {2+x^6}} \]
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Time = 0.10 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {296, 314, 231, 1895} \[ \int \frac {x^4}{\left (2+x^6\right )^{3/2}} \, dx=\frac {\left (1-\sqrt {3}\right ) \left (x^2+\sqrt [3]{2}\right ) \sqrt {\frac {x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}\right )^2}} x \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{6\ 2^{2/3} \sqrt [4]{3} \sqrt {\frac {x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt {x^6+2}}+\frac {\left (x^2+\sqrt [3]{2}\right ) \sqrt {\frac {x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}\right )^2}} x E\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2^{2/3} 3^{3/4} \sqrt {\frac {x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt {x^6+2}}+\frac {x^5}{6 \sqrt {x^6+2}}-\frac {\left (1+\sqrt {3}\right ) \sqrt {x^6+2} x}{6 \left (\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}\right )} \]
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Rule 231
Rule 296
Rule 314
Rule 1895
Rubi steps \begin{align*} \text {integral}& = \frac {x^5}{6 \sqrt {2+x^6}}-\frac {1}{3} \int \frac {x^4}{\sqrt {2+x^6}} \, dx \\ & = \frac {x^5}{6 \sqrt {2+x^6}}+\frac {1}{6} \int \frac {2^{2/3} \left (-1+\sqrt {3}\right )-2 x^4}{\sqrt {2+x^6}} \, dx+\frac {\left (1-\sqrt {3}\right ) \int \frac {1}{\sqrt {2+x^6}} \, dx}{3 \sqrt [3]{2}} \\ & = \frac {x^5}{6 \sqrt {2+x^6}}-\frac {\left (1+\sqrt {3}\right ) x \sqrt {2+x^6}}{6 \left (\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2\right )}+\frac {x \left (\sqrt [3]{2}+x^2\right ) \sqrt {\frac {2^{2/3}-\sqrt [3]{2} x^2+x^4}{\left (\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2\right )^2}} E\left (\cos ^{-1}\left (\frac {\sqrt [3]{2}+\left (1-\sqrt {3}\right ) x^2}{\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2^{2/3} 3^{3/4} \sqrt {\frac {x^2 \left (\sqrt [3]{2}+x^2\right )}{\left (\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {2+x^6}}+\frac {\left (1-\sqrt {3}\right ) x \left (\sqrt [3]{2}+x^2\right ) \sqrt {\frac {2^{2/3}-\sqrt [3]{2} x^2+x^4}{\left (\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2\right )^2}} F\left (\cos ^{-1}\left (\frac {\sqrt [3]{2}+\left (1-\sqrt {3}\right ) x^2}{\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{6\ 2^{2/3} \sqrt [4]{3} \sqrt {\frac {x^2 \left (\sqrt [3]{2}+x^2\right )}{\left (\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {2+x^6}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 6.43 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.07 \[ \int \frac {x^4}{\left (2+x^6\right )^{3/2}} \, dx=\frac {x^5 \operatorname {Hypergeometric2F1}\left (\frac {5}{6},\frac {3}{2},\frac {11}{6},-\frac {x^6}{2}\right )}{10 \sqrt {2}} \]
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Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 5.36 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.05
method | result | size |
meijerg | \(\frac {\sqrt {2}\, x^{5} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {5}{6},\frac {3}{2};\frac {11}{6};-\frac {x^{6}}{2}\right )}{20}\) | \(20\) |
risch | \(\frac {x^{5}}{6 \sqrt {x^{6}+2}}-\frac {\sqrt {2}\, x^{5} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{2},\frac {5}{6};\frac {11}{6};-\frac {x^{6}}{2}\right )}{30}\) | \(33\) |
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\[ \int \frac {x^4}{\left (2+x^6\right )^{3/2}} \, dx=\int { \frac {x^{4}}{{\left (x^{6} + 2\right )}^{\frac {3}{2}}} \,d x } \]
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Result contains complex when optimal does not.
Time = 0.40 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.09 \[ \int \frac {x^4}{\left (2+x^6\right )^{3/2}} \, dx=\frac {\sqrt {2} x^{5} \Gamma \left (\frac {5}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{6}, \frac {3}{2} \\ \frac {11}{6} \end {matrix}\middle | {\frac {x^{6} e^{i \pi }}{2}} \right )}}{24 \Gamma \left (\frac {11}{6}\right )} \]
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\[ \int \frac {x^4}{\left (2+x^6\right )^{3/2}} \, dx=\int { \frac {x^{4}}{{\left (x^{6} + 2\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x^4}{\left (2+x^6\right )^{3/2}} \, dx=\int { \frac {x^{4}}{{\left (x^{6} + 2\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^4}{\left (2+x^6\right )^{3/2}} \, dx=\int \frac {x^4}{{\left (x^6+2\right )}^{3/2}} \,d x \]
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