Integrand size = 13, antiderivative size = 394 \[ \int \frac {x^{15}}{\left (2+x^6\right )^{3/2}} \, dx=-\frac {x^{10}}{3 \sqrt {2+x^6}}+\frac {10}{21} x^4 \sqrt {2+x^6}-\frac {80 \sqrt {2+x^6}}{21 \left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )}+\frac {40 \sqrt [6]{2} \sqrt {2-\sqrt {3}} \left (\sqrt [3]{2}+x^2\right ) \sqrt {\frac {2^{2/3}-\sqrt [3]{2} x^2+x^4}{\left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{2} \left (1-\sqrt {3}\right )+x^2}{\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2}\right )|-7-4 \sqrt {3}\right )}{7\ 3^{3/4} \sqrt {\frac {\sqrt [3]{2}+x^2}{\left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )^2}} \sqrt {2+x^6}}-\frac {80\ 2^{2/3} \left (\sqrt [3]{2}+x^2\right ) \sqrt {\frac {2^{2/3}-\sqrt [3]{2} x^2+x^4}{\left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{2} \left (1-\sqrt {3}\right )+x^2}{\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2}\right ),-7-4 \sqrt {3}\right )}{21 \sqrt [4]{3} \sqrt {\frac {\sqrt [3]{2}+x^2}{\left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )^2}} \sqrt {2+x^6}} \]
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Time = 0.20 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {281, 294, 327, 309, 224, 1891} \[ \int \frac {x^{15}}{\left (2+x^6\right )^{3/2}} \, dx=-\frac {80\ 2^{2/3} \left (x^2+\sqrt [3]{2}\right ) \sqrt {\frac {x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x^2+\sqrt [3]{2} \left (1-\sqrt {3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )}\right ),-7-4 \sqrt {3}\right )}{21 \sqrt [4]{3} \sqrt {\frac {x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} \sqrt {x^6+2}}+\frac {40 \sqrt [6]{2} \sqrt {2-\sqrt {3}} \left (x^2+\sqrt [3]{2}\right ) \sqrt {\frac {x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} E\left (\arcsin \left (\frac {x^2+\sqrt [3]{2} \left (1-\sqrt {3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )}\right )|-7-4 \sqrt {3}\right )}{7\ 3^{3/4} \sqrt {\frac {x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} \sqrt {x^6+2}}-\frac {x^{10}}{3 \sqrt {x^6+2}}+\frac {10}{21} \sqrt {x^6+2} x^4-\frac {80 \sqrt {x^6+2}}{21 \left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )} \]
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Rule 224
Rule 281
Rule 294
Rule 309
Rule 327
Rule 1891
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x^7}{\left (2+x^3\right )^{3/2}} \, dx,x,x^2\right ) \\ & = -\frac {x^{10}}{3 \sqrt {2+x^6}}+\frac {5}{3} \text {Subst}\left (\int \frac {x^4}{\sqrt {2+x^3}} \, dx,x,x^2\right ) \\ & = -\frac {x^{10}}{3 \sqrt {2+x^6}}+\frac {10}{21} x^4 \sqrt {2+x^6}-\frac {40}{21} \text {Subst}\left (\int \frac {x}{\sqrt {2+x^3}} \, dx,x,x^2\right ) \\ & = -\frac {x^{10}}{3 \sqrt {2+x^6}}+\frac {10}{21} x^4 \sqrt {2+x^6}-\frac {40}{21} \text {Subst}\left (\int \frac {\sqrt [3]{2} \left (1-\sqrt {3}\right )+x}{\sqrt {2+x^3}} \, dx,x,x^2\right )+\frac {1}{21} \left (40 \sqrt [3]{2} \left (1-\sqrt {3}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2+x^3}} \, dx,x,x^2\right ) \\ & = -\frac {x^{10}}{3 \sqrt {2+x^6}}+\frac {10}{21} x^4 \sqrt {2+x^6}-\frac {80 \sqrt {2+x^6}}{21 \left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )}+\frac {40 \sqrt [6]{2} \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 )}{7\ 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 {80\ 2^{2/3} \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}} 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 )}{21 \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 = 8.99 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.14 \[ \int \frac {x^{15}}{\left (2+x^6\right )^{3/2}} \, dx=\frac {x^4 \left (-20+x^6+10 \sqrt {2} \sqrt {2+x^6} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {3}{2},\frac {5}{3},-\frac {x^6}{2}\right )\right )}{7 \sqrt {2+x^6}} \]
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Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 7.51 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.05
method | result | size |
meijerg | \(\frac {\sqrt {2}\, x^{16} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {3}{2},\frac {8}{3};\frac {11}{3};-\frac {x^{6}}{2}\right )}{64}\) | \(20\) |
risch | \(\frac {x^{4} \left (3 x^{6}+20\right )}{21 \sqrt {x^{6}+2}}-\frac {10 \sqrt {2}\, x^{4} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};-\frac {x^{6}}{2}\right )}{21}\) | \(40\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.11 \[ \int \frac {x^{15}}{\left (2+x^6\right )^{3/2}} \, dx=\frac {80 \, {\left (x^{6} + 2\right )} {\rm weierstrassZeta}\left (0, -8, {\rm weierstrassPInverse}\left (0, -8, x^{2}\right )\right ) + {\left (3 \, x^{10} + 20 \, x^{4}\right )} \sqrt {x^{6} + 2}}{21 \, {\left (x^{6} + 2\right )}} \]
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Time = 0.76 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.09 \[ \int \frac {x^{15}}{\left (2+x^6\right )^{3/2}} \, dx=\frac {\sqrt {2} x^{16} \Gamma \left (\frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {8}{3} \\ \frac {11}{3} \end {matrix}\middle | {\frac {x^{6} e^{i \pi }}{2}} \right )}}{24 \Gamma \left (\frac {11}{3}\right )} \]
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\[ \int \frac {x^{15}}{\left (2+x^6\right )^{3/2}} \, dx=\int { \frac {x^{15}}{{\left (x^{6} + 2\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x^{15}}{\left (2+x^6\right )^{3/2}} \, dx=\int { \frac {x^{15}}{{\left (x^{6} + 2\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^{15}}{\left (2+x^6\right )^{3/2}} \, dx=\int \frac {x^{15}}{{\left (x^6+2\right )}^{3/2}} \,d x \]
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