Optimal. Leaf size=93 \[ -\frac {45 \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{4 x^4+3}}\right )}{128 \sqrt {2}}+\frac {45 \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{4 x^4+3}}\right )}{128 \sqrt {2}}+\frac {1}{8} \left (4 x^4+3\right )^{5/4} x^3+\frac {15}{32} \sqrt [4]{4 x^4+3} x^3 \]
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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 = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {279, 331, 298, 203, 206} \[ \frac {1}{8} \left (4 x^4+3\right )^{5/4} x^3+\frac {15}{32} \sqrt [4]{4 x^4+3} x^3-\frac {45 \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{4 x^4+3}}\right )}{128 \sqrt {2}}+\frac {45 \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{4 x^4+3}}\right )}{128 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 279
Rule 298
Rule 331
Rubi steps
\begin {align*} \int x^2 \left (3+4 x^4\right )^{5/4} \, dx &=\frac {1}{8} x^3 \left (3+4 x^4\right )^{5/4}+\frac {15}{8} \int x^2 \sqrt [4]{3+4 x^4} \, dx\\ &=\frac {15}{32} x^3 \sqrt [4]{3+4 x^4}+\frac {1}{8} x^3 \left (3+4 x^4\right )^{5/4}+\frac {45}{32} \int \frac {x^2}{\left (3+4 x^4\right )^{3/4}} \, dx\\ &=\frac {15}{32} x^3 \sqrt [4]{3+4 x^4}+\frac {1}{8} x^3 \left (3+4 x^4\right )^{5/4}+\frac {45}{32} \operatorname {Subst}\left (\int \frac {x^2}{1-4 x^4} \, dx,x,\frac {x}{\sqrt [4]{3+4 x^4}}\right )\\ &=\frac {15}{32} x^3 \sqrt [4]{3+4 x^4}+\frac {1}{8} x^3 \left (3+4 x^4\right )^{5/4}+\frac {45}{128} \operatorname {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {x}{\sqrt [4]{3+4 x^4}}\right )-\frac {45}{128} \operatorname {Subst}\left (\int \frac {1}{1+2 x^2} \, dx,x,\frac {x}{\sqrt [4]{3+4 x^4}}\right )\\ &=\frac {15}{32} x^3 \sqrt [4]{3+4 x^4}+\frac {1}{8} x^3 \left (3+4 x^4\right )^{5/4}-\frac {45 \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{3+4 x^4}}\right )}{128 \sqrt {2}}+\frac {45 \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{3+4 x^4}}\right )}{128 \sqrt {2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 26, normalized size = 0.28 \[ \sqrt [4]{3} x^3 \, _2F_1\left (-\frac {5}{4},\frac {3}{4};\frac {7}{4};-\frac {4 x^4}{3}\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.18, size = 83, normalized size = 0.89 \[ -\frac {45 \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{4 x^4+3}}\right )}{128 \sqrt {2}}+\frac {45 \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{4 x^4+3}}\right )}{128 \sqrt {2}}+\frac {1}{32} \sqrt [4]{4 x^4+3} \left (16 x^7+27 x^3\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 105, normalized size = 1.13 \[ \frac {45}{256} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{2 \, x}\right ) + \frac {45}{512} \, \sqrt {2} \log \left (8 \, x^{4} + 4 \, \sqrt {2} {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {4 \, x^{4} + 3} x^{2} + 2 \, \sqrt {2} {\left (4 \, x^{4} + 3\right )}^{\frac {3}{4}} x + 3\right ) + \frac {1}{32} \, {\left (16 \, x^{7} + 27 \, x^{3}\right )} {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.65, size = 110, normalized size = 1.18 \[ \frac {1}{32} \, x^{8} {\left (\frac {9 \, {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}} {\left (\frac {3}{x^{4}} + 4\right )}}{x} - \frac {20 \, {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x}\right )} + \frac {45}{256} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{2 \, x}\right ) - \frac {45}{512} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x}}{\sqrt {2} + \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 3.41, size = 19, normalized size = 0.20
method | result | size |
meijerg | \(3^{\frac {1}{4}} x^{3} \hypergeom \left (\left [-\frac {5}{4}, \frac {3}{4}\right ], \left [\frac {7}{4}\right ], -\frac {4 x^{4}}{3}\right )\) | \(19\) |
risch | \(\frac {x^{3} \left (16 x^{4}+27\right ) \left (4 x^{4}+3\right )^{\frac {1}{4}}}{32}+\frac {5 \,3^{\frac {1}{4}} x^{3} \hypergeom \left (\left [\frac {3}{4}, \frac {3}{4}\right ], \left [\frac {7}{4}\right ], -\frac {4 x^{4}}{3}\right )}{32}\) | \(42\) |
trager | \(\frac {x^{3} \left (16 x^{4}+27\right ) \left (4 x^{4}+3\right )^{\frac {1}{4}}}{32}+\frac {45 \RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (4 \RootOf \left (\textit {\_Z}^{2}+2\right ) \sqrt {4 x^{4}+3}\, x^{2}-8 \RootOf \left (\textit {\_Z}^{2}+2\right ) x^{4}+4 \left (4 x^{4}+3\right )^{\frac {3}{4}} x -8 x^{3} \left (4 x^{4}+3\right )^{\frac {1}{4}}-3 \RootOf \left (\textit {\_Z}^{2}+2\right )\right )}{512}+\frac {45 \RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (-2 \RootOf \left (\textit {\_Z}^{2}-2\right ) \left (4 x^{4}+3\right )^{\frac {3}{4}} x -4 \left (4 x^{4}+3\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{3}-4 \sqrt {4 x^{4}+3}\, x^{2}-8 x^{4}-3\right )}{512}\) | \(166\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.24, size = 130, normalized size = 1.40 \[ \frac {45}{256} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{2 \, x}\right ) - \frac {45}{512} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x}}{\sqrt {2} + \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x}}\right ) + \frac {9 \, {\left (\frac {20 \, {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x} - \frac {9 \, {\left (4 \, x^{4} + 3\right )}^{\frac {5}{4}}}{x^{5}}\right )}}{32 \, {\left (\frac {8 \, {\left (4 \, x^{4} + 3\right )}}{x^{4}} - \frac {{\left (4 \, x^{4} + 3\right )}^{2}}{x^{8}} - 16\right )}} \]
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 \[ \int x^2\,{\left (4\,x^4+3\right )}^{5/4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.57, size = 41, normalized size = 0.44 \[ \frac {3 \sqrt [4]{3} x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {4 x^{4} e^{i \pi }}{3}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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