Integrand size = 15, antiderivative size = 260 \[ \int \frac {1}{x^6 \sqrt [4]{-2+3 x^2}} \, dx=\frac {\left (-2+3 x^2\right )^{3/4}}{10 x^5}+\frac {7 \left (-2+3 x^2\right )^{3/4}}{40 x^3}+\frac {63 \left (-2+3 x^2\right )^{3/4}}{160 x}-\frac {189 x \sqrt [4]{-2+3 x^2}}{160 \left (\sqrt {2}+\sqrt {-2+3 x^2}\right )}+\frac {63 \sqrt {3} \sqrt {\frac {x^2}{\left (\sqrt {2}+\sqrt {-2+3 x^2}\right )^2}} \left (\sqrt {2}+\sqrt {-2+3 x^2}\right ) E\left (2 \arctan \left (\frac {\sqrt [4]{-2+3 x^2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{80\ 2^{3/4} x}-\frac {63 \sqrt {3} \sqrt {\frac {x^2}{\left (\sqrt {2}+\sqrt {-2+3 x^2}\right )^2}} \left (\sqrt {2}+\sqrt {-2+3 x^2}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{-2+3 x^2}}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{160\ 2^{3/4} x} \]
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Time = 0.10 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {331, 236, 311, 226, 1210} \[ \int \frac {1}{x^6 \sqrt [4]{-2+3 x^2}} \, dx=-\frac {63 \sqrt {3} \sqrt {\frac {x^2}{\left (\sqrt {3 x^2-2}+\sqrt {2}\right )^2}} \left (\sqrt {3 x^2-2}+\sqrt {2}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{160\ 2^{3/4} x}+\frac {63 \sqrt {3} \sqrt {\frac {x^2}{\left (\sqrt {3 x^2-2}+\sqrt {2}\right )^2}} \left (\sqrt {3 x^2-2}+\sqrt {2}\right ) E\left (2 \arctan \left (\frac {\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{80\ 2^{3/4} x}-\frac {189 \sqrt [4]{3 x^2-2} x}{160 \left (\sqrt {3 x^2-2}+\sqrt {2}\right )}+\frac {63 \left (3 x^2-2\right )^{3/4}}{160 x}+\frac {\left (3 x^2-2\right )^{3/4}}{10 x^5}+\frac {7 \left (3 x^2-2\right )^{3/4}}{40 x^3} \]
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Rule 226
Rule 236
Rule 311
Rule 331
Rule 1210
Rubi steps \begin{align*} \text {integral}& = \frac {\left (-2+3 x^2\right )^{3/4}}{10 x^5}+\frac {21}{20} \int \frac {1}{x^4 \sqrt [4]{-2+3 x^2}} \, dx \\ & = \frac {\left (-2+3 x^2\right )^{3/4}}{10 x^5}+\frac {7 \left (-2+3 x^2\right )^{3/4}}{40 x^3}+\frac {63}{80} \int \frac {1}{x^2 \sqrt [4]{-2+3 x^2}} \, dx \\ & = \frac {\left (-2+3 x^2\right )^{3/4}}{10 x^5}+\frac {7 \left (-2+3 x^2\right )^{3/4}}{40 x^3}+\frac {63 \left (-2+3 x^2\right )^{3/4}}{160 x}-\frac {189}{320} \int \frac {1}{\sqrt [4]{-2+3 x^2}} \, dx \\ & = \frac {\left (-2+3 x^2\right )^{3/4}}{10 x^5}+\frac {7 \left (-2+3 x^2\right )^{3/4}}{40 x^3}+\frac {63 \left (-2+3 x^2\right )^{3/4}}{160 x}-\frac {\left (63 \sqrt {\frac {3}{2}} \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1+\frac {x^4}{2}}} \, dx,x,\sqrt [4]{-2+3 x^2}\right )}{160 x} \\ & = \frac {\left (-2+3 x^2\right )^{3/4}}{10 x^5}+\frac {7 \left (-2+3 x^2\right )^{3/4}}{40 x^3}+\frac {63 \left (-2+3 x^2\right )^{3/4}}{160 x}-\frac {\left (63 \sqrt {3} \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{2}}} \, dx,x,\sqrt [4]{-2+3 x^2}\right )}{160 x}+\frac {\left (63 \sqrt {3} \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {1-\frac {x^2}{\sqrt {2}}}{\sqrt {1+\frac {x^4}{2}}} \, dx,x,\sqrt [4]{-2+3 x^2}\right )}{160 x} \\ & = \frac {\left (-2+3 x^2\right )^{3/4}}{10 x^5}+\frac {7 \left (-2+3 x^2\right )^{3/4}}{40 x^3}+\frac {63 \left (-2+3 x^2\right )^{3/4}}{160 x}-\frac {189 x \sqrt [4]{-2+3 x^2}}{160 \left (\sqrt {2}+\sqrt {-2+3 x^2}\right )}+\frac {63 \sqrt {3} \sqrt {\frac {x^2}{\left (\sqrt {2}+\sqrt {-2+3 x^2}\right )^2}} \left (\sqrt {2}+\sqrt {-2+3 x^2}\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-2+3 x^2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{80\ 2^{3/4} x}-\frac {63 \sqrt {3} \sqrt {\frac {x^2}{\left (\sqrt {2}+\sqrt {-2+3 x^2}\right )^2}} \left (\sqrt {2}+\sqrt {-2+3 x^2}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-2+3 x^2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{160\ 2^{3/4} x} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.01 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.18 \[ \int \frac {1}{x^6 \sqrt [4]{-2+3 x^2}} \, dx=-\frac {\sqrt [4]{1-\frac {3 x^2}{2}} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1}{4},-\frac {3}{2},\frac {3 x^2}{2}\right )}{5 x^5 \sqrt [4]{-2+3 x^2}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 2.21 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.16
method | result | size |
meijerg | \(-\frac {2^{\frac {3}{4}} {\left (-\operatorname {signum}\left (-1+\frac {3 x^{2}}{2}\right )\right )}^{\frac {1}{4}} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (-\frac {5}{2},\frac {1}{4};-\frac {3}{2};\frac {3 x^{2}}{2}\right )}{10 \operatorname {signum}\left (-1+\frac {3 x^{2}}{2}\right )^{\frac {1}{4}} x^{5}}\) | \(42\) |
risch | \(\frac {189 x^{6}-42 x^{4}-8 x^{2}-32}{160 x^{5} \left (3 x^{2}-2\right )^{\frac {1}{4}}}-\frac {189 \,2^{\frac {3}{4}} {\left (-\operatorname {signum}\left (-1+\frac {3 x^{2}}{2}\right )\right )}^{\frac {1}{4}} x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{4},\frac {1}{2};\frac {3}{2};\frac {3 x^{2}}{2}\right )}{640 \operatorname {signum}\left (-1+\frac {3 x^{2}}{2}\right )^{\frac {1}{4}}}\) | \(72\) |
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\[ \int \frac {1}{x^6 \sqrt [4]{-2+3 x^2}} \, dx=\int { \frac {1}{{\left (3 \, x^{2} - 2\right )}^{\frac {1}{4}} x^{6}} \,d x } \]
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Result contains complex when optimal does not.
Time = 0.60 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.13 \[ \int \frac {1}{x^6 \sqrt [4]{-2+3 x^2}} \, dx=\frac {2^{\frac {3}{4}} e^{\frac {3 i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{2}, \frac {1}{4} \\ - \frac {3}{2} \end {matrix}\middle | {\frac {3 x^{2}}{2}} \right )}}{10 x^{5}} \]
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\[ \int \frac {1}{x^6 \sqrt [4]{-2+3 x^2}} \, dx=\int { \frac {1}{{\left (3 \, x^{2} - 2\right )}^{\frac {1}{4}} x^{6}} \,d x } \]
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\[ \int \frac {1}{x^6 \sqrt [4]{-2+3 x^2}} \, dx=\int { \frac {1}{{\left (3 \, x^{2} - 2\right )}^{\frac {1}{4}} x^{6}} \,d x } \]
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Timed out. \[ \int \frac {1}{x^6 \sqrt [4]{-2+3 x^2}} \, dx=\int \frac {1}{x^6\,{\left (3\,x^2-2\right )}^{1/4}} \,d x \]
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