Optimal. Leaf size=165 \[ \frac {40}{567} x \sqrt [4]{-1+3 x^2}+\frac {2}{63} x^3 \sqrt [4]{-1+3 x^2}+\frac {2}{27} \sqrt {\frac {2}{3}} \tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )-\frac {2}{27} \sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )+\frac {40 \sqrt {\frac {x^2}{\left (1+\sqrt {-1+3 x^2}\right )^2}} \left (1+\sqrt {-1+3 x^2}\right ) F\left (2 \tan ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )|\frac {1}{2}\right )}{567 \sqrt {3} x} \]
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Rubi [A]
time = 0.13, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {454, 240, 226,
327, 409, 453} \begin {gather*} \frac {2}{27} \sqrt {\frac {2}{3}} \text {ArcTan}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )+\frac {40 \sqrt {\frac {x^2}{\left (\sqrt {3 x^2-1}+1\right )^2}} \left (\sqrt {3 x^2-1}+1\right ) F\left (2 \text {ArcTan}\left (\sqrt [4]{3 x^2-1}\right )|\frac {1}{2}\right )}{567 \sqrt {3} x}+\frac {40}{567} \sqrt [4]{3 x^2-1} x-\frac {2}{27} \sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )+\frac {2}{63} \sqrt [4]{3 x^2-1} x^3 \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 240
Rule 327
Rule 409
Rule 453
Rule 454
Rubi steps
\begin {align*} \int \frac {x^6}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx &=\int \left (\frac {4}{27 \left (-1+3 x^2\right )^{3/4}}+\frac {2 x^2}{9 \left (-1+3 x^2\right )^{3/4}}+\frac {x^4}{3 \left (-1+3 x^2\right )^{3/4}}+\frac {8}{27 \left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}}\right ) \, dx\\ &=\frac {4}{27} \int \frac {1}{\left (-1+3 x^2\right )^{3/4}} \, dx+\frac {2}{9} \int \frac {x^2}{\left (-1+3 x^2\right )^{3/4}} \, dx+\frac {8}{27} \int \frac {1}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx+\frac {1}{3} \int \frac {x^4}{\left (-1+3 x^2\right )^{3/4}} \, dx\\ &=\frac {4}{81} x \sqrt [4]{-1+3 x^2}+\frac {2}{63} x^3 \sqrt [4]{-1+3 x^2}+\frac {4}{81} \int \frac {1}{\left (-1+3 x^2\right )^{3/4}} \, dx+\frac {2}{21} \int \frac {x^2}{\left (-1+3 x^2\right )^{3/4}} \, dx-\frac {4}{27} \int \frac {1}{\left (-1+3 x^2\right )^{3/4}} \, dx+\frac {4}{9} \int \frac {x^2}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx+\frac {\left (8 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,\sqrt [4]{-1+3 x^2}\right )}{27 \sqrt {3} x}\\ &=\frac {40}{567} x \sqrt [4]{-1+3 x^2}+\frac {2}{63} x^3 \sqrt [4]{-1+3 x^2}+\frac {2}{27} \sqrt {\frac {2}{3}} \tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )-\frac {2}{27} \sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )+\frac {4 \sqrt {\frac {x^2}{\left (1+\sqrt {-1+3 x^2}\right )^2}} \left (1+\sqrt {-1+3 x^2}\right ) F\left (2 \tan ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )|\frac {1}{2}\right )}{27 \sqrt {3} x}+\frac {4}{189} \int \frac {1}{\left (-1+3 x^2\right )^{3/4}} \, dx+\frac {\left (8 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,\sqrt [4]{-1+3 x^2}\right )}{81 \sqrt {3} x}-\frac {\left (8 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,\sqrt [4]{-1+3 x^2}\right )}{27 \sqrt {3} x}\\ &=\frac {40}{567} x \sqrt [4]{-1+3 x^2}+\frac {2}{63} x^3 \sqrt [4]{-1+3 x^2}+\frac {2}{27} \sqrt {\frac {2}{3}} \tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )-\frac {2}{27} \sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )+\frac {4 \sqrt {\frac {x^2}{\left (1+\sqrt {-1+3 x^2}\right )^2}} \left (1+\sqrt {-1+3 x^2}\right ) F\left (2 \tan ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )|\frac {1}{2}\right )}{81 \sqrt {3} x}+\frac {\left (8 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,\sqrt [4]{-1+3 x^2}\right )}{189 \sqrt {3} x}\\ &=\frac {40}{567} x \sqrt [4]{-1+3 x^2}+\frac {2}{63} x^3 \sqrt [4]{-1+3 x^2}+\frac {2}{27} \sqrt {\frac {2}{3}} \tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )-\frac {2}{27} \sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )+\frac {40 \sqrt {\frac {x^2}{\left (1+\sqrt {-1+3 x^2}\right )^2}} \left (1+\sqrt {-1+3 x^2}\right ) F\left (2 \tan ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )|\frac {1}{2}\right )}{567 \sqrt {3} x}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in
optimal.
time = 6.81, size = 184, normalized size = 1.12 \begin {gather*} \frac {2 x \left (-20+51 x^2+27 x^4-31 x^2 \left (1-3 x^2\right )^{3/4} F_1\left (\frac {3}{2};\frac {3}{4},1;\frac {5}{2};3 x^2,\frac {3 x^2}{2}\right )-\frac {80 F_1\left (\frac {1}{2};\frac {3}{4},1;\frac {3}{2};3 x^2,\frac {3 x^2}{2}\right )}{\left (-2+3 x^2\right ) \left (2 F_1\left (\frac {1}{2};\frac {3}{4},1;\frac {3}{2};3 x^2,\frac {3 x^2}{2}\right )+x^2 \left (2 F_1\left (\frac {3}{2};\frac {3}{4},2;\frac {5}{2};3 x^2,\frac {3 x^2}{2}\right )+3 F_1\left (\frac {3}{2};\frac {7}{4},1;\frac {5}{2};3 x^2,\frac {3 x^2}{2}\right )\right )\right )}\right )}{567 \left (-1+3 x^2\right )^{3/4}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x^{6}}{\left (3 x^{2}-2\right ) \left (3 x^{2}-1\right )^{\frac {3}{4}}}\, dx\]
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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{6}}{\left (3 x^{2} - 2\right ) \left (3 x^{2} - 1\right )^{\frac {3}{4}}}\, dx \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^6}{{\left (3\,x^2-1\right )}^{3/4}\,\left (3\,x^2-2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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