Optimal. Leaf size=164 \[ \frac {2}{27} x \sqrt [4]{2-3 x^2}+\frac {2\ 2^{3/4} \tan ^{-1}\left (\frac {2^{3/4}-\sqrt [4]{2} \sqrt {2-3 x^2}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{9 \sqrt {3}}-\frac {2\ 2^{3/4} \tanh ^{-1}\left (\frac {2^{3/4}+\sqrt [4]{2} \sqrt {2-3 x^2}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{9 \sqrt {3}}-\frac {4\ 2^{3/4} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{27 \sqrt {3}} \]
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Rubi [A]
time = 0.07, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {454, 238, 327,
409, 452} \begin {gather*} -\frac {4\ 2^{3/4} F\left (\left .\frac {1}{2} \text {ArcSin}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{27 \sqrt {3}}+\frac {2\ 2^{3/4} \text {ArcTan}\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 {2}{27} \sqrt [4]{2-3 x^2} x-\frac {2\ 2^{3/4} \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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 238
Rule 327
Rule 409
Rule 452
Rule 454
Rubi steps
\begin {align*} \int \frac {x^4}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx &=\int \left (-\frac {4}{9 \left (2-3 x^2\right )^{3/4}}-\frac {x^2}{3 \left (2-3 x^2\right )^{3/4}}+\frac {16}{9 \left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )}\right ) \, dx\\ &=-\left (\frac {1}{3} \int \frac {x^2}{\left (2-3 x^2\right )^{3/4}} \, dx\right )-\frac {4}{9} \int \frac {1}{\left (2-3 x^2\right )^{3/4}} \, dx+\frac {16}{9} \int \frac {1}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx\\ &=\frac {2}{27} x \sqrt [4]{2-3 x^2}-\frac {4\ 2^{3/4} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{9 \sqrt {3}}-\frac {4}{27} \int \frac {1}{\left (2-3 x^2\right )^{3/4}} \, dx+\frac {4}{9} \int \frac {1}{\left (2-3 x^2\right )^{3/4}} \, dx+\frac {4}{3} \int \frac {x^2}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx\\ &=\frac {2}{27} x \sqrt [4]{2-3 x^2}+\frac {2\ 2^{3/4} \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 {2\ 2^{3/4} \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 {4\ 2^{3/4} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{27 \sqrt {3}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in
optimal.
time = 6.83, size = 184, normalized size = 1.12 \begin {gather*} \frac {2}{27} x \left (\sqrt [4]{2} x^2 F_1\left (\frac {3}{2};\frac {3}{4},1;\frac {5}{2};\frac {3 x^2}{2},\frac {3 x^2}{4}\right )+\frac {2-3 x^2+\frac {32 F_1\left (\frac {1}{2};\frac {3}{4},1;\frac {3}{2};\frac {3 x^2}{2},\frac {3 x^2}{4}\right )}{\left (-4+3 x^2\right ) \left (4 F_1\left (\frac {1}{2};\frac {3}{4},1;\frac {3}{2};\frac {3 x^2}{2},\frac {3 x^2}{4}\right )+x^2 \left (2 F_1\left (\frac {3}{2};\frac {3}{4},2;\frac {5}{2};\frac {3 x^2}{2},\frac {3 x^2}{4}\right )+3 F_1\left (\frac {3}{2};\frac {7}{4},1;\frac {5}{2};\frac {3 x^2}{2},\frac {3 x^2}{4}\right )\right )\right )}}{\left (2-3 x^2\right )^{3/4}}\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {x^{4}}{\left (-3 x^{2}+2\right )^{\frac {3}{4}} \left (-3 x^{2}+4\right )}\, dx\]
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 \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^{4}}{3 x^{2} \left (2 - 3 x^{2}\right )^{\frac {3}{4}} - 4 \left (2 - 3 x^{2}\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^4}{{\left (2-3\,x^2\right )}^{3/4}\,\left (3\,x^2-4\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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