Optimal. Leaf size=121 \[ -\frac {8\ 2^{3/4} \sqrt {-\frac {x^2}{\left (\sqrt {-3 x^2-2}+\sqrt {2}\right )^2}} \left (\sqrt {-3 x^2-2}+\sqrt {2}\right ) \operatorname {EllipticF}\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-3 x^2-2}}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{63 \sqrt {3} x}+\frac {8}{63} \sqrt [4]{-3 x^2-2} x-\frac {2}{21} \sqrt [4]{-3 x^2-2} x^3 \]
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Rubi [A] time = 0.05, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {321, 234, 220} \[ -\frac {2}{21} \sqrt [4]{-3 x^2-2} x^3+\frac {8}{63} \sqrt [4]{-3 x^2-2} x-\frac {8\ 2^{3/4} \sqrt {-\frac {x^2}{\left (\sqrt {-3 x^2-2}+\sqrt {2}\right )^2}} \left (\sqrt {-3 x^2-2}+\sqrt {2}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-3 x^2-2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{63 \sqrt {3} x} \]
Antiderivative was successfully verified.
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Rule 220
Rule 234
Rule 321
Rubi steps
\begin {align*} \int \frac {x^4}{\left (-2-3 x^2\right )^{3/4}} \, dx &=-\frac {2}{21} x^3 \sqrt [4]{-2-3 x^2}-\frac {4}{7} \int \frac {x^2}{\left (-2-3 x^2\right )^{3/4}} \, dx\\ &=\frac {8}{63} x \sqrt [4]{-2-3 x^2}-\frac {2}{21} x^3 \sqrt [4]{-2-3 x^2}+\frac {16}{63} \int \frac {1}{\left (-2-3 x^2\right )^{3/4}} \, dx\\ &=\frac {8}{63} x \sqrt [4]{-2-3 x^2}-\frac {2}{21} x^3 \sqrt [4]{-2-3 x^2}-\frac {\left (16 \sqrt {\frac {2}{3}} \sqrt {-x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{2}}} \, dx,x,\sqrt [4]{-2-3 x^2}\right )}{63 x}\\ &=\frac {8}{63} x \sqrt [4]{-2-3 x^2}-\frac {2}{21} x^3 \sqrt [4]{-2-3 x^2}-\frac {8\ 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 )}{63 \sqrt {3} x}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 63, normalized size = 0.52 \[ \frac {2 x \left (4 \sqrt [4]{2} \left (3 x^2+2\right )^{3/4} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {3}{2};-\frac {3 x^2}{2}\right )+9 x^4-6 x^2-8\right )}{63 \left (-3 x^2-2\right )^{3/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ -\frac {2}{63} \, {\left (3 \, x^{3} - 4 \, x\right )} {\left (-3 \, x^{2} - 2\right )}^{\frac {1}{4}} + {\rm integral}\left (-\frac {16 \, {\left (-3 \, x^{2} - 2\right )}^{\frac {1}{4}}}{63 \, {\left (3 \, x^{2} + 2\right )}}, x\right ) \]
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 \[ \int \frac {x^{4}}{{\left (-3 \, x^{2} - 2\right )}^{\frac {3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{\left (-3 x^{2}-2\right )^{\frac {3}{4}}}\, dx \]
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 \[ \int \frac {x^{4}}{{\left (-3 \, x^{2} - 2\right )}^{\frac {3}{4}}}\,{d x} \]
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 \[ \int \frac {x^4}{{\left (-3\,x^2-2\right )}^{3/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.76, size = 36, normalized size = 0.30 \[ \frac {\sqrt [4]{2} x^{5} e^{- \frac {3 i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {3 x^{2} e^{i \pi }}{2}} \right )}}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
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