Integrand size = 13, antiderivative size = 378 \[ \int \frac {x^3}{\left (2+x^6\right )^{3/2}} \, dx=\frac {x^4}{6 \sqrt {2+x^6}}-\frac {\sqrt {2+x^6}}{6 \left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )}+\frac {\sqrt {2-\sqrt {3}} \left (\sqrt [3]{2}+x^2\right ) \sqrt {\frac {2^{2/3}-\sqrt [3]{2} x^2+x^4}{\left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{2} \left (1-\sqrt {3}\right )+x^2}{\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2}\right )|-7-4 \sqrt {3}\right )}{2\ 2^{5/6} 3^{3/4} \sqrt {\frac {\sqrt [3]{2}+x^2}{\left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )^2}} \sqrt {2+x^6}}-\frac {\left (\sqrt [3]{2}+x^2\right ) \sqrt {\frac {2^{2/3}-\sqrt [3]{2} x^2+x^4}{\left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{2} \left (1-\sqrt {3}\right )+x^2}{\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [3]{2} \sqrt [4]{3} \sqrt {\frac {\sqrt [3]{2}+x^2}{\left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )^2}} \sqrt {2+x^6}} \]
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Time = 0.18 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {281, 296, 309, 224, 1891} \[ \int \frac {x^3}{\left (2+x^6\right )^{3/2}} \, dx=-\frac {\left (x^2+\sqrt [3]{2}\right ) \sqrt {\frac {x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x^2+\sqrt [3]{2} \left (1-\sqrt {3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [3]{2} \sqrt [4]{3} \sqrt {\frac {x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} \sqrt {x^6+2}}+\frac {\sqrt {2-\sqrt {3}} \left (x^2+\sqrt [3]{2}\right ) \sqrt {\frac {x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} E\left (\arcsin \left (\frac {x^2+\sqrt [3]{2} \left (1-\sqrt {3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )}\right )|-7-4 \sqrt {3}\right )}{2\ 2^{5/6} 3^{3/4} \sqrt {\frac {x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} \sqrt {x^6+2}}+\frac {x^4}{6 \sqrt {x^6+2}}-\frac {\sqrt {x^6+2}}{6 \left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )} \]
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Rule 224
Rule 281
Rule 296
Rule 309
Rule 1891
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x}{\left (2+x^3\right )^{3/2}} \, dx,x,x^2\right ) \\ & = \frac {x^4}{6 \sqrt {2+x^6}}-\frac {1}{12} \text {Subst}\left (\int \frac {x}{\sqrt {2+x^3}} \, dx,x,x^2\right ) \\ & = \frac {x^4}{6 \sqrt {2+x^6}}-\frac {1}{12} \text {Subst}\left (\int \frac {\sqrt [3]{2} \left (1-\sqrt {3}\right )+x}{\sqrt {2+x^3}} \, dx,x,x^2\right )+\frac {\left (1-\sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2+x^3}} \, dx,x,x^2\right )}{6\ 2^{2/3}} \\ & = \frac {x^4}{6 \sqrt {2+x^6}}-\frac {\sqrt {2+x^6}}{6 \left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )}+\frac {\sqrt {2-\sqrt {3}} \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 (\sin ^{-1}\left (\frac {\sqrt [3]{2} \left (1-\sqrt {3}\right )+x^2}{\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2}\right )|-7-4 \sqrt {3}\right )}{2\ 2^{5/6} 3^{3/4} \sqrt {\frac {\sqrt [3]{2}+x^2}{\left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )^2}} \sqrt {2+x^6}}-\frac {\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 (\sin ^{-1}\left (\frac {\sqrt [3]{2} \left (1-\sqrt {3}\right )+x^2}{\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2}\right )|-7-4 \sqrt {3}\right )}{3 \sqrt [3]{2} \sqrt [4]{3} \sqrt {\frac {\sqrt [3]{2}+x^2}{\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 = 6.06 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.08 \[ \int \frac {x^3}{\left (2+x^6\right )^{3/2}} \, dx=\frac {x^4 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {3}{2},\frac {5}{3},-\frac {x^6}{2}\right )}{8 \sqrt {2}} \]
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Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 5.37 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.05
method | result | size |
meijerg | \(\frac {\sqrt {2}\, x^{4} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {2}{3},\frac {3}{2};\frac {5}{3};-\frac {x^{6}}{2}\right )}{16}\) | \(20\) |
risch | \(\frac {x^{4}}{6 \sqrt {x^{6}+2}}-\frac {\sqrt {2}\, x^{4} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};-\frac {x^{6}}{2}\right )}{48}\) | \(33\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.10 \[ \int \frac {x^3}{\left (2+x^6\right )^{3/2}} \, dx=\frac {\sqrt {x^{6} + 2} x^{4} + {\left (x^{6} + 2\right )} {\rm weierstrassZeta}\left (0, -8, {\rm weierstrassPInverse}\left (0, -8, x^{2}\right )\right )}{6 \, {\left (x^{6} + 2\right )}} \]
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Time = 0.38 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.10 \[ \int \frac {x^3}{\left (2+x^6\right )^{3/2}} \, dx=\frac {\sqrt {2} x^{4} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {3}{2} \\ \frac {5}{3} \end {matrix}\middle | {\frac {x^{6} e^{i \pi }}{2}} \right )}}{24 \Gamma \left (\frac {5}{3}\right )} \]
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\[ \int \frac {x^3}{\left (2+x^6\right )^{3/2}} \, dx=\int { \frac {x^{3}}{{\left (x^{6} + 2\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x^3}{\left (2+x^6\right )^{3/2}} \, dx=\int { \frac {x^{3}}{{\left (x^{6} + 2\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^3}{\left (2+x^6\right )^{3/2}} \, dx=\int \frac {x^3}{{\left (x^6+2\right )}^{3/2}} \,d x \]
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