Optimal. Leaf size=164 \[ \frac {2}{45} \left (2-3 x^2\right )^{3/4} x+\frac {4 \sqrt [4]{2} \tan ^{-1}\left (\frac {2^{3/4}-\sqrt [4]{2} \sqrt {2-3 x^2}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{9 \sqrt {3}}+\frac {4 \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {2-3 x^2}+2^{3/4}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{9 \sqrt {3}}-\frac {16 \sqrt [4]{2} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{15 \sqrt {3}} \]
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Rubi [A] time = 0.07, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {440, 228, 321, 397} \[ \frac {2}{45} \left (2-3 x^2\right )^{3/4} x+\frac {4 \sqrt [4]{2} \tan ^{-1}\left (\frac {2^{3/4}-\sqrt [4]{2} \sqrt {2-3 x^2}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{9 \sqrt {3}}+\frac {4 \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {2-3 x^2}+2^{3/4}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{9 \sqrt {3}}-\frac {16 \sqrt [4]{2} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{15 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 228
Rule 321
Rule 397
Rule 440
Rubi steps
\begin {align*} \int \frac {x^4}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx &=\int \left (-\frac {4}{9 \sqrt [4]{2-3 x^2}}-\frac {x^2}{3 \sqrt [4]{2-3 x^2}}+\frac {16}{9 \sqrt [4]{2-3 x^2} \left (4-3 x^2\right )}\right ) \, dx\\ &=-\left (\frac {1}{3} \int \frac {x^2}{\sqrt [4]{2-3 x^2}} \, dx\right )-\frac {4}{9} \int \frac {1}{\sqrt [4]{2-3 x^2}} \, dx+\frac {16}{9} \int \frac {1}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx\\ &=\frac {2}{45} x \left (2-3 x^2\right )^{3/4}+\frac {4 \sqrt [4]{2} \tan ^{-1}\left (\frac {2^{3/4}-\sqrt [4]{2} \sqrt {2-3 x^2}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{9 \sqrt {3}}+\frac {4 \sqrt [4]{2} \tanh ^{-1}\left (\frac {2^{3/4}+\sqrt [4]{2} \sqrt {2-3 x^2}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{9 \sqrt {3}}-\frac {8 \sqrt [4]{2} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{9 \sqrt {3}}-\frac {4}{45} \int \frac {1}{\sqrt [4]{2-3 x^2}} \, dx\\ &=\frac {2}{45} x \left (2-3 x^2\right )^{3/4}+\frac {4 \sqrt [4]{2} \tan ^{-1}\left (\frac {2^{3/4}-\sqrt [4]{2} \sqrt {2-3 x^2}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{9 \sqrt {3}}+\frac {4 \sqrt [4]{2} \tanh ^{-1}\left (\frac {2^{3/4}+\sqrt [4]{2} \sqrt {2-3 x^2}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{9 \sqrt {3}}-\frac {16 \sqrt [4]{2} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{15 \sqrt {3}}\\ \end {align*}
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Mathematica [C] time = 0.26, size = 184, normalized size = 1.12 \[ \frac {1}{45} x \left (3\ 2^{3/4} x^2 F_1\left (\frac {3}{2};\frac {1}{4},1;\frac {5}{2};\frac {3 x^2}{2},\frac {3 x^2}{4}\right )+\frac {2 \left (\frac {32 F_1\left (\frac {1}{2};\frac {1}{4},1;\frac {3}{2};\frac {3 x^2}{2},\frac {3 x^2}{4}\right )}{\left (3 x^2-4\right ) \left (x^2 \left (2 F_1\left (\frac {3}{2};\frac {1}{4},2;\frac {5}{2};\frac {3 x^2}{2},\frac {3 x^2}{4}\right )+F_1\left (\frac {3}{2};\frac {5}{4},1;\frac {5}{2};\frac {3 x^2}{2},\frac {3 x^2}{4}\right )\right )+4 F_1\left (\frac {1}{2};\frac {1}{4},1;\frac {3}{2};\frac {3 x^2}{2},\frac {3 x^2}{4}\right )\right )}-3 x^2+2\right )}{\sqrt [4]{2-3 x^2}}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 8.22, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}} x^{4}}{9 \, x^{4} - 18 \, x^{2} + 8}, x\right ) \]
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 \[ \int -\frac {x^{4}}{{\left (3 \, x^{2} - 4\right )} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{\left (-3 x^{2}+2\right )^{\frac {1}{4}} \left (-3 x^{2}+4\right )}\, dx \]
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 \[ -\int \frac {x^{4}}{{\left (3 \, x^{2} - 4\right )} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}}\,{d x} \]
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 \[ -\int \frac {x^4}{{\left (2-3\,x^2\right )}^{1/4}\,\left (3\,x^2-4\right )} \,d x \]
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 \[ - \int \frac {x^{4}}{3 x^{2} \sqrt [4]{2 - 3 x^{2}} - 4 \sqrt [4]{2 - 3 x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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