Integrand size = 11, antiderivative size = 199 \[ \int \frac {1}{\sqrt [4]{-2+3 x^2}} \, dx=\frac {2 x \sqrt [4]{-2+3 x^2}}{\sqrt {2}+\sqrt {-2+3 x^2}}-\frac {2 \sqrt [4]{2} \sqrt {\frac {x^2}{\left (\sqrt {2}+\sqrt {-2+3 x^2}\right )^2}} \left (\sqrt {2}+\sqrt {-2+3 x^2}\right ) E\left (2 \arctan \left (\frac {\sqrt [4]{-2+3 x^2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{\sqrt {3} x}+\frac {\sqrt [4]{2} \sqrt {\frac {x^2}{\left (\sqrt {2}+\sqrt {-2+3 x^2}\right )^2}} \left (\sqrt {2}+\sqrt {-2+3 x^2}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{-2+3 x^2}}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{\sqrt {3} x} \]
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Time = 0.06 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {236, 311, 226, 1210} \[ \int \frac {1}{\sqrt [4]{-2+3 x^2}} \, dx=\frac {\sqrt [4]{2} \sqrt {\frac {x^2}{\left (\sqrt {3 x^2-2}+\sqrt {2}\right )^2}} \left (\sqrt {3 x^2-2}+\sqrt {2}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{\sqrt {3} x}-\frac {2 \sqrt [4]{2} \sqrt {\frac {x^2}{\left (\sqrt {3 x^2-2}+\sqrt {2}\right )^2}} \left (\sqrt {3 x^2-2}+\sqrt {2}\right ) E\left (2 \arctan \left (\frac {\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{\sqrt {3} x}+\frac {2 \sqrt [4]{3 x^2-2} x}{\sqrt {3 x^2-2}+\sqrt {2}} \]
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Rule 226
Rule 236
Rule 311
Rule 1210
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {\frac {2}{3}} \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 )}{x} \\ & = \frac {\left (2 \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 )}{\sqrt {3} x}-\frac {\left (2 \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 )}{\sqrt {3} x} \\ & = \frac {2 x \sqrt [4]{-2+3 x^2}}{\sqrt {2}+\sqrt {-2+3 x^2}}-\frac {2 \sqrt [4]{2} \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 )}{\sqrt {3} x}+\frac {\sqrt [4]{2} \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 )}{\sqrt {3} x} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 5.03 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.22 \[ \int \frac {1}{\sqrt [4]{-2+3 x^2}} \, dx=\frac {x \sqrt [4]{1-\frac {3 x^2}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {3}{2},\frac {3 x^2}{2}\right )}{\sqrt [4]{-2+3 x^2}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 2.12 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.20
method | result | size |
meijerg | \(\frac {2^{\frac {3}{4}} {\left (-\operatorname {signum}\left (-1+\frac {3 x^{2}}{2}\right )\right )}^{\frac {1}{4}} x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{4},\frac {1}{2};\frac {3}{2};\frac {3 x^{2}}{2}\right )}{2 \operatorname {signum}\left (-1+\frac {3 x^{2}}{2}\right )^{\frac {1}{4}}}\) | \(40\) |
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\[ \int \frac {1}{\sqrt [4]{-2+3 x^2}} \, dx=\int { \frac {1}{{\left (3 \, x^{2} - 2\right )}^{\frac {1}{4}}} \,d x } \]
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Result contains complex when optimal does not.
Time = 0.42 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.14 \[ \int \frac {1}{\sqrt [4]{-2+3 x^2}} \, dx=\frac {2^{\frac {3}{4}} x e^{- \frac {i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {3 x^{2}}{2}} \right )}}{2} \]
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\[ \int \frac {1}{\sqrt [4]{-2+3 x^2}} \, dx=\int { \frac {1}{{\left (3 \, x^{2} - 2\right )}^{\frac {1}{4}}} \,d x } \]
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\[ \int \frac {1}{\sqrt [4]{-2+3 x^2}} \, dx=\int { \frac {1}{{\left (3 \, x^{2} - 2\right )}^{\frac {1}{4}}} \,d x } \]
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Time = 4.41 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.17 \[ \int \frac {1}{\sqrt [4]{-2+3 x^2}} \, dx=\frac {2^{3/4}\,x\,{\left (2-3\,x^2\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{2};\ \frac {3}{2};\ \frac {3\,x^2}{2}\right )}{2\,{\left (3\,x^2-2\right )}^{1/4}} \]
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