Integrand size = 15, antiderivative size = 93 \[ \int x^2 \left (3+4 x^4\right )^{5/4} \, dx=\frac {15}{32} x^3 \sqrt [4]{3+4 x^4}+\frac {1}{8} x^3 \left (3+4 x^4\right )^{5/4}-\frac {45 \arctan \left (\frac {\sqrt {2} x}{\sqrt [4]{3+4 x^4}}\right )}{128 \sqrt {2}}+\frac {45 \text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt [4]{3+4 x^4}}\right )}{128 \sqrt {2}} \]
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Time = 0.02 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {285, 338, 304, 209, 212} \[ \int x^2 \left (3+4 x^4\right )^{5/4} \, dx=-\frac {45 \arctan \left (\frac {\sqrt {2} x}{\sqrt [4]{4 x^4+3}}\right )}{128 \sqrt {2}}+\frac {45 \text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt [4]{4 x^4+3}}\right )}{128 \sqrt {2}}+\frac {1}{8} \left (4 x^4+3\right )^{5/4} x^3+\frac {15}{32} \sqrt [4]{4 x^4+3} x^3 \]
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Rule 209
Rule 212
Rule 285
Rule 304
Rule 338
Rubi steps \begin{align*} \text {integral}& = \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} \text {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} \text {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {x}{\sqrt [4]{3+4 x^4}}\right )-\frac {45}{128} \text {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 \arctan \left (\frac {\sqrt {2} x}{\sqrt [4]{3+4 x^4}}\right )}{128 \sqrt {2}}+\frac {45 \text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt [4]{3+4 x^4}}\right )}{128 \sqrt {2}} \\ \end{align*}
Time = 0.39 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.88 \[ \int x^2 \left (3+4 x^4\right )^{5/4} \, dx=\frac {1}{32} x^3 \sqrt [4]{3+4 x^4} \left (27+16 x^4\right )-\frac {45 \arctan \left (\frac {\sqrt {2} x}{\sqrt [4]{3+4 x^4}}\right )}{128 \sqrt {2}}+\frac {45 \text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt [4]{3+4 x^4}}\right )}{128 \sqrt {2}} \]
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Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 2.23 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.20
method | result | size |
meijerg | \(3^{\frac {1}{4}} x^{3} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (-\frac {5}{4},\frac {3}{4};\frac {7}{4};-\frac {4 x^{4}}{3}\right )\) | \(19\) |
risch | \(\frac {x^{3} \left (16 x^{4}+27\right ) \left (4 x^{4}+3\right )^{\frac {1}{4}}}{32}+\frac {5 \,3^{\frac {1}{4}} x^{3} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};-\frac {4 x^{4}}{3}\right )}{32}\) | \(42\) |
pseudoelliptic | \(\frac {\frac {9 x^{7} \left (4 x^{4}+3\right )^{\frac {1}{4}}}{2}+\frac {243 x^{3} \left (4 x^{4}+3\right )^{\frac {1}{4}}}{32}+\frac {405 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (4 x^{4}+3\right )^{\frac {1}{4}} \sqrt {2}}{2 x}\right )}{256}+\frac {405 \sqrt {2}\, \arctan \left (\frac {\left (4 x^{4}+3\right )^{\frac {1}{4}} \sqrt {2}}{2 x}\right )}{256}}{\left (-2 x^{2}+\sqrt {4 x^{4}+3}\right )^{2} \left (2 x^{2}+\sqrt {4 x^{4}+3}\right )^{2}}\) | \(112\) |
trager | \(\frac {x^{3} \left (16 x^{4}+27\right ) \left (4 x^{4}+3\right )^{\frac {1}{4}}}{32}-\frac {45 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-4 \sqrt {4 x^{4}+3}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{2}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{4}+4 \left (4 x^{4}+3\right )^{\frac {3}{4}} x +8 x^{3} \left (4 x^{4}+3\right )^{\frac {1}{4}}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )\right )}{512}-\frac {45 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \sqrt {4 x^{4}+3}\, x^{2}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{4}+4 \left (4 x^{4}+3\right )^{\frac {3}{4}} x -8 x^{3} \left (4 x^{4}+3\right )^{\frac {1}{4}}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right )\right )}{512}\) | \(173\) |
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Time = 0.26 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.13 \[ \int x^2 \left (3+4 x^4\right )^{5/4} \, dx=\frac {45}{256} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{2 \, x}\right ) + \frac {45}{512} \, \sqrt {2} \log \left (8 \, x^{4} + 4 \, \sqrt {2} {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {4 \, x^{4} + 3} x^{2} + 2 \, \sqrt {2} {\left (4 \, x^{4} + 3\right )}^{\frac {3}{4}} x + 3\right ) + \frac {1}{32} \, {\left (16 \, x^{7} + 27 \, x^{3}\right )} {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}} \]
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Result contains complex when optimal does not.
Time = 0.92 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.44 \[ \int x^2 \left (3+4 x^4\right )^{5/4} \, dx=\frac {3 \cdot \sqrt [4]{3} x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {4 x^{4} e^{i \pi }}{3}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} \]
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Time = 0.27 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.40 \[ \int x^2 \left (3+4 x^4\right )^{5/4} \, dx=\frac {45}{256} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{2 \, x}\right ) - \frac {45}{512} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x}}{\sqrt {2} + \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x}}\right ) + \frac {9 \, {\left (\frac {20 \, {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x} - \frac {9 \, {\left (4 \, x^{4} + 3\right )}^{\frac {5}{4}}}{x^{5}}\right )}}{32 \, {\left (\frac {8 \, {\left (4 \, x^{4} + 3\right )}}{x^{4}} - \frac {{\left (4 \, x^{4} + 3\right )}^{2}}{x^{8}} - 16\right )}} \]
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Time = 0.31 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.18 \[ \int x^2 \left (3+4 x^4\right )^{5/4} \, dx=\frac {1}{32} \, x^{8} {\left (\frac {9 \, {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}} {\left (\frac {3}{x^{4}} + 4\right )}}{x} - \frac {20 \, {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x}\right )} + \frac {45}{256} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{2 \, x}\right ) - \frac {45}{512} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x}}{\sqrt {2} + \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x}}\right ) \]
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Timed out. \[ \int x^2 \left (3+4 x^4\right )^{5/4} \, dx=\int x^2\,{\left (4\,x^4+3\right )}^{5/4} \,d x \]
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