Optimal. Leaf size=93 \[ \frac {27 \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{4 x^4+3}}\right )}{512 \sqrt {2}}-\frac {27 \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{4 x^4+3}}\right )}{512 \sqrt {2}}+\frac {1}{8} \sqrt [4]{4 x^4+3} x^7+\frac {3}{128} \sqrt [4]{4 x^4+3} x^3 \]
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Rubi [A] time = 0.03, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {279, 321, 331, 298, 203, 206} \[ \frac {1}{8} \sqrt [4]{4 x^4+3} x^7+\frac {3}{128} \sqrt [4]{4 x^4+3} x^3+\frac {27 \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{4 x^4+3}}\right )}{512 \sqrt {2}}-\frac {27 \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{4 x^4+3}}\right )}{512 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 279
Rule 298
Rule 321
Rule 331
Rubi steps
\begin {align*} \int x^6 \sqrt [4]{3+4 x^4} \, dx &=\frac {1}{8} x^7 \sqrt [4]{3+4 x^4}+\frac {3}{8} \int \frac {x^6}{\left (3+4 x^4\right )^{3/4}} \, dx\\ &=\frac {3}{128} x^3 \sqrt [4]{3+4 x^4}+\frac {1}{8} x^7 \sqrt [4]{3+4 x^4}-\frac {27}{128} \int \frac {x^2}{\left (3+4 x^4\right )^{3/4}} \, dx\\ &=\frac {3}{128} x^3 \sqrt [4]{3+4 x^4}+\frac {1}{8} x^7 \sqrt [4]{3+4 x^4}-\frac {27}{128} \operatorname {Subst}\left (\int \frac {x^2}{1-4 x^4} \, dx,x,\frac {x}{\sqrt [4]{3+4 x^4}}\right )\\ &=\frac {3}{128} x^3 \sqrt [4]{3+4 x^4}+\frac {1}{8} x^7 \sqrt [4]{3+4 x^4}-\frac {27}{512} \operatorname {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {x}{\sqrt [4]{3+4 x^4}}\right )+\frac {27}{512} \operatorname {Subst}\left (\int \frac {1}{1+2 x^2} \, dx,x,\frac {x}{\sqrt [4]{3+4 x^4}}\right )\\ &=\frac {3}{128} x^3 \sqrt [4]{3+4 x^4}+\frac {1}{8} x^7 \sqrt [4]{3+4 x^4}+\frac {27 \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{3+4 x^4}}\right )}{512 \sqrt {2}}-\frac {27 \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{3+4 x^4}}\right )}{512 \sqrt {2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 43, normalized size = 0.46 \[ \frac {1}{32} x^3 \left (\left (4 x^4+3\right )^{5/4}-3 \sqrt [4]{3} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};-\frac {4 x^4}{3}\right )\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.18, size = 83, normalized size = 0.89 \[ \frac {27 \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{4 x^4+3}}\right )}{512 \sqrt {2}}-\frac {27 \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{4 x^4+3}}\right )}{512 \sqrt {2}}+\frac {1}{128} \sqrt [4]{4 x^4+3} \left (16 x^7+3 x^3\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 105, normalized size = 1.13 \[ -\frac {27}{1024} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{2 \, x}\right ) + \frac {27}{2048} \, \sqrt {2} \log \left (8 \, x^{4} - 4 \, \sqrt {2} {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {4 \, x^{4} + 3} x^{2} - 2 \, \sqrt {2} {\left (4 \, x^{4} + 3\right )}^{\frac {3}{4}} x + 3\right ) + \frac {1}{128} \, {\left (16 \, x^{7} + 3 \, x^{3}\right )} {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.67, size = 109, normalized size = 1.17 \[ \frac {1}{128} \, x^{8} {\left (\frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}} {\left (\frac {3}{x^{4}} + 4\right )}}{x} + \frac {12 \, {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x}\right )} - \frac {27}{1024} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{2 \, x}\right ) + \frac {27}{2048} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x}}{\sqrt {2} + \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 3.32, size = 20, normalized size = 0.22
method | result | size |
meijerg | \(\frac {3^{\frac {1}{4}} x^{7} \hypergeom \left (\left [-\frac {1}{4}, \frac {7}{4}\right ], \left [\frac {11}{4}\right ], -\frac {4 x^{4}}{3}\right )}{7}\) | \(20\) |
risch | \(\frac {x^{3} \left (16 x^{4}+3\right ) \left (4 x^{4}+3\right )^{\frac {1}{4}}}{128}-\frac {3 \,3^{\frac {1}{4}} x^{3} \hypergeom \left (\left [\frac {3}{4}, \frac {3}{4}\right ], \left [\frac {7}{4}\right ], -\frac {4 x^{4}}{3}\right )}{128}\) | \(42\) |
trager | \(\frac {x^{3} \left (16 x^{4}+3\right ) \left (4 x^{4}+3\right )^{\frac {1}{4}}}{128}+\frac {27 \RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (-4 \RootOf \left (\textit {\_Z}^{2}-2\right ) \sqrt {4 x^{4}+3}\, x^{2}-8 \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{4}+4 \left (4 x^{4}+3\right )^{\frac {3}{4}} x +8 x^{3} \left (4 x^{4}+3\right )^{\frac {1}{4}}-3 \RootOf \left (\textit {\_Z}^{2}-2\right )\right )}{2048}+\frac {27 \RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (-4 \RootOf \left (\textit {\_Z}^{2}+2\right ) \sqrt {4 x^{4}+3}\, x^{2}+8 \RootOf \left (\textit {\_Z}^{2}+2\right ) x^{4}+4 \left (4 x^{4}+3\right )^{\frac {3}{4}} x -8 x^{3} \left (4 x^{4}+3\right )^{\frac {1}{4}}+3 \RootOf \left (\textit {\_Z}^{2}+2\right )\right )}{2048}\) | \(173\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.20, size = 129, normalized size = 1.39 \[ -\frac {27}{1024} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{2 \, x}\right ) + \frac {27}{2048} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x}}{\sqrt {2} + \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x}}\right ) - \frac {9 \, {\left (\frac {12 \, {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x} + \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {5}{4}}}{x^{5}}\right )}}{128 \, {\left (\frac {8 \, {\left (4 \, x^{4} + 3\right )}}{x^{4}} - \frac {{\left (4 \, x^{4} + 3\right )}^{2}}{x^{8}} - 16\right )}} \]
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 \[ \int x^6\,{\left (4\,x^4+3\right )}^{1/4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.32, size = 39, normalized size = 0.42 \[ \frac {\sqrt [4]{3} x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {4 x^{4} e^{i \pi }}{3}} \right )}}{4 \Gamma \left (\frac {11}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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