Optimal. Leaf size=197 \[ \frac {1}{6 x^5 \sqrt {2+x^6}}-\frac {2 \sqrt {2+x^6}}{15 x^5}-\frac {2^{2/3} x \left (\sqrt [3]{2}+x^2\right ) \sqrt {\frac {2^{2/3}-\sqrt [3]{2} x^2+x^4}{\left (\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2\right )^2}} F\left (\cos ^{-1}\left (\frac {\sqrt [3]{2}+\left (1-\sqrt {3}\right ) x^2}{\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{15 \sqrt [4]{3} \sqrt {\frac {x^2 \left (\sqrt [3]{2}+x^2\right )}{\left (\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {2+x^6}} \]
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Rubi [A]
time = 0.04, antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {296, 331, 231}
\begin {gather*} -\frac {2^{2/3} x \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}} F\left (\text {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 )}{15 \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 {2 \sqrt {x^6+2}}{15 x^5}+\frac {1}{6 x^5 \sqrt {x^6+2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 231
Rule 296
Rule 331
Rubi steps
\begin {align*} \int \frac {1}{x^6 \left (2+x^6\right )^{3/2}} \, dx &=\frac {1}{6 x^5 \sqrt {2+x^6}}+\frac {4}{3} \int \frac {1}{x^6 \sqrt {2+x^6}} \, dx\\ &=\frac {1}{6 x^5 \sqrt {2+x^6}}-\frac {2 \sqrt {2+x^6}}{15 x^5}-\frac {4}{15} \int \frac {1}{\sqrt {2+x^6}} \, dx\\ &=\frac {1}{6 x^5 \sqrt {2+x^6}}-\frac {2 \sqrt {2+x^6}}{15 x^5}-\frac {2^{2/3} 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 )}{15 \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] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.01, size = 29, normalized size = 0.15 \begin {gather*} -\frac {\, _2F_1\left (-\frac {5}{6},\frac {3}{2};\frac {1}{6};-\frac {x^6}{2}\right )}{10 \sqrt {2} x^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order
4.
time = 0.19, size = 20, normalized size = 0.10
method | result | size |
meijerg | \(-\frac {\sqrt {2}\, \hypergeom \left (\left [-\frac {5}{6}, \frac {3}{2}\right ], \left [\frac {1}{6}\right ], -\frac {x^{6}}{2}\right )}{20 x^{5}}\) | \(20\) |
risch | \(-\frac {4 x^{6}+3}{30 x^{5} \sqrt {x^{6}+2}}-\frac {2 \sqrt {2}\, x \hypergeom \left (\left [\frac {1}{6}, \frac {1}{2}\right ], \left [\frac {7}{6}\right ], -\frac {x^{6}}{2}\right )}{15}\) | \(38\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.07, size = 26, normalized size = 0.13 \begin {gather*} {\rm integral}\left (\frac {\sqrt {x^{6} + 2}}{x^{18} + 4 \, x^{12} + 4 \, x^{6}}, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.50, size = 39, normalized size = 0.20 \begin {gather*} \frac {\sqrt {2} \Gamma \left (- \frac {5}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{6}, \frac {3}{2} \\ \frac {1}{6} \end {matrix}\middle | {\frac {x^{6} e^{i \pi }}{2}} \right )}}{24 x^{5} \Gamma \left (\frac {1}{6}\right )} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x^6\,{\left (x^6+2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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