Optimal. Leaf size=102 \[ \frac {2}{9} x \sqrt [4]{-2+3 x^2}+\frac {2\ 2^{3/4} \sqrt {\frac {x^2}{\left (\sqrt {2}+\sqrt {-2+3 x^2}\right )^2}} \left (\sqrt {2}+\sqrt {-2+3 x^2}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-2+3 x^2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{9 \sqrt {3} x} \]
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Rubi [A]
time = 0.03, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {327, 240, 226}
\begin {gather*} \frac {2\ 2^{3/4} \sqrt {\frac {x^2}{\left (\sqrt {3 x^2-2}+\sqrt {2}\right )^2}} \left (\sqrt {3 x^2-2}+\sqrt {2}\right ) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{9 \sqrt {3} x}+\frac {2}{9} \sqrt [4]{3 x^2-2} x \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 240
Rule 327
Rubi steps
\begin {align*} \int \frac {x^2}{\left (-2+3 x^2\right )^{3/4}} \, dx &=\frac {2}{9} x \sqrt [4]{-2+3 x^2}+\frac {4}{9} \int \frac {1}{\left (-2+3 x^2\right )^{3/4}} \, dx\\ &=\frac {2}{9} x \sqrt [4]{-2+3 x^2}+\frac {\left (4 \sqrt {\frac {2}{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 )}{9 x}\\ &=\frac {2}{9} x \sqrt [4]{-2+3 x^2}+\frac {2\ 2^{3/4} \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 )}{9 \sqrt {3} x}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 5.58, size = 57, normalized size = 0.56 \begin {gather*} \frac {2 x \left (-2+3 x^2+\sqrt [4]{2} \left (2-3 x^2\right )^{3/4} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {3}{2};\frac {3 x^2}{2}\right )\right )}{9 \left (-2+3 x^2\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.09, size = 42, normalized size = 0.41
method | result | size |
meijerg | \(\frac {2^{\frac {1}{4}} \left (-\mathrm {signum}\left (-1+\frac {3 x^{2}}{2}\right )\right )^{\frac {3}{4}} x^{3} \hypergeom \left (\left [\frac {3}{4}, \frac {3}{2}\right ], \left [\frac {5}{2}\right ], \frac {3 x^{2}}{2}\right )}{6 \mathrm {signum}\left (-1+\frac {3 x^{2}}{2}\right )^{\frac {3}{4}}}\) | \(42\) |
risch | \(\frac {2 x \left (3 x^{2}-2\right )^{\frac {1}{4}}}{9}+\frac {2 \,2^{\frac {1}{4}} \left (-\mathrm {signum}\left (-1+\frac {3 x^{2}}{2}\right )\right )^{\frac {3}{4}} x \hypergeom \left (\left [\frac {1}{2}, \frac {3}{4}\right ], \left [\frac {3}{2}\right ], \frac {3 x^{2}}{2}\right )}{9 \mathrm {signum}\left (-1+\frac {3 x^{2}}{2}\right )^{\frac {3}{4}}}\) | \(53\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.38, size = 31, normalized size = 0.30 \begin {gather*} \frac {\sqrt [4]{2} x^{3} e^{- \frac {3 i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {3 x^{2}}{2}} \right )}}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x^2}{{\left (3\,x^2-2\right )}^{3/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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