Optimal. Leaf size=392 \[ -\frac {x^5}{3 \sqrt {x^6+2}}+\frac {5 \left (1+\sqrt {3}\right ) \sqrt {x^6+2} x}{6 \left (\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}\right )}-\frac {5 \left (1-\sqrt {3}\right ) \left (x^2+\sqrt [3]{2}\right ) \sqrt {\frac {x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}\right )^2}} x F\left (\cos ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{6\ 2^{2/3} \sqrt [4]{3} \sqrt {\frac {x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt {x^6+2}}-\frac {5 \left (x^2+\sqrt [3]{2}\right ) \sqrt {\frac {x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}\right )^2}} x E\left (\cos ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2^{2/3} 3^{3/4} \sqrt {\frac {x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt {x^6+2}} \]
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Rubi [A] time = 0.11, antiderivative size = 392, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {288, 308, 225, 1881} \[ -\frac {x^5}{3 \sqrt {x^6+2}}+\frac {5 \left (1+\sqrt {3}\right ) \sqrt {x^6+2} x}{6 \left (\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}\right )}-\frac {5 \left (1-\sqrt {3}\right ) \left (x^2+\sqrt [3]{2}\right ) \sqrt {\frac {x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}\right )^2}} x F\left (\cos ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{6\ 2^{2/3} \sqrt [4]{3} \sqrt {\frac {x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt {x^6+2}}-\frac {5 \left (x^2+\sqrt [3]{2}\right ) \sqrt {\frac {x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}\right )^2}} x E\left (\cos ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2^{2/3} 3^{3/4} \sqrt {\frac {x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt {x^6+2}} \]
Antiderivative was successfully verified.
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Rule 225
Rule 288
Rule 308
Rule 1881
Rubi steps
\begin {align*} \int \frac {x^{10}}{\left (2+x^6\right )^{3/2}} \, dx &=-\frac {x^5}{3 \sqrt {2+x^6}}+\frac {5}{3} \int \frac {x^4}{\sqrt {2+x^6}} \, dx\\ &=-\frac {x^5}{3 \sqrt {2+x^6}}-\frac {5}{6} \int \frac {2^{2/3} \left (-1+\sqrt {3}\right )-2 x^4}{\sqrt {2+x^6}} \, dx-\frac {\left (5 \left (1-\sqrt {3}\right )\right ) \int \frac {1}{\sqrt {2+x^6}} \, dx}{3 \sqrt [3]{2}}\\ &=-\frac {x^5}{3 \sqrt {2+x^6}}+\frac {5 \left (1+\sqrt {3}\right ) x \sqrt {2+x^6}}{6 \left (\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2\right )}-\frac {5 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 {5 \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*}
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Mathematica [C] time = 0.02, size = 43, normalized size = 0.11 \[ \frac {1}{4} x^5 \left (\frac {2}{\sqrt {x^6+2}}-\sqrt {2} \, _2F_1\left (\frac {5}{6},\frac {3}{2};\frac {11}{6};-\frac {x^6}{2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x^{6} + 2} x^{10}}{x^{12} + 4 \, x^{6} + 4}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{10}}{{\left (x^{6} + 2\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.13, size = 33, normalized size = 0.08 \[ \frac {\sqrt {2}\, x^{5} \hypergeom \left (\left [\frac {1}{2}, \frac {5}{6}\right ], \left [\frac {11}{6}\right ], -\frac {x^{6}}{2}\right )}{6}-\frac {x^{5}}{3 \sqrt {x^{6}+2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{10}}{{\left (x^{6} + 2\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^{10}}{{\left (x^6+2\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.98, size = 36, normalized size = 0.09 \[ \frac {\sqrt {2} x^{11} \Gamma \left (\frac {11}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {11}{6} \\ \frac {17}{6} \end {matrix}\middle | {\frac {x^{6} e^{i \pi }}{2}} \right )}}{24 \Gamma \left (\frac {17}{6}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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