Optimal. Leaf size=186 \[ \frac {1}{5} x^2 \sqrt {2+x^6}-\frac {2\ 2^{5/6} \sqrt {2+\sqrt {3}} \left (\sqrt [3]{2}+x^2\right ) \sqrt {\frac {2^{2/3}-\sqrt [3]{2} x^2+x^4}{\left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )^2}} F\left (\sin ^{-1}\left (\frac {\sqrt [3]{2} \left (1-\sqrt {3}\right )+x^2}{\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2}\right )|-7-4 \sqrt {3}\right )}{5 \sqrt [4]{3} \sqrt {\frac {\sqrt [3]{2}+x^2}{\left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )^2}} \sqrt {2+x^6}} \]
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Rubi [A]
time = 0.09, antiderivative size = 186, 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 = {281, 327, 224}
\begin {gather*} \frac {1}{5} x^2 \sqrt {x^6+2}-\frac {2\ 2^{5/6} \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}} F\left (\text {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 )}{5 \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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 224
Rule 281
Rule 327
Rubi steps
\begin {align*} \int \frac {x^7}{\sqrt {2+x^6}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^3}{\sqrt {2+x^3}} \, dx,x,x^2\right )\\ &=\frac {1}{5} x^2 \sqrt {2+x^6}-\frac {2}{5} \text {Subst}\left (\int \frac {1}{\sqrt {2+x^3}} \, dx,x,x^2\right )\\ &=\frac {1}{5} x^2 \sqrt {2+x^6}-\frac {2\ 2^{5/6} \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}} 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 )}{5 \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*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.03, size = 41, normalized size = 0.22 \begin {gather*} \frac {1}{5} x^2 \left (\sqrt {2+x^6}-\sqrt {2} \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {4}{3};-\frac {x^6}{2}\right )\right ) \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 = 0.18, size = 20, normalized size = 0.11
method | result | size |
meijerg | \(\frac {\sqrt {2}\, x^{8} \hypergeom \left (\left [\frac {1}{2}, \frac {4}{3}\right ], \left [\frac {7}{3}\right ], -\frac {x^{6}}{2}\right )}{16}\) | \(20\) |
risch | \(\frac {x^{2} \sqrt {x^{6}+2}}{5}-\frac {\sqrt {2}\, x^{2} \hypergeom \left (\left [\frac {1}{3}, \frac {1}{2}\right ], \left [\frac {4}{3}\right ], -\frac {x^{6}}{2}\right )}{5}\) | \(33\) |
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 [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.07, size = 21, normalized size = 0.11 \begin {gather*} \frac {1}{5} \, \sqrt {x^{6} + 2} x^{2} - \frac {4}{5} \, {\rm weierstrassPInverse}\left (0, -8, x^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.39, size = 36, normalized size = 0.19 \begin {gather*} \frac {\sqrt {2} x^{8} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {x^{6} e^{i \pi }}{2}} \right )}}{12 \Gamma \left (\frac {7}{3}\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 {x^7}{\sqrt {x^6+2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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