Optimal. Leaf size=242 \[ -\frac {16 \sqrt [4]{2} \sqrt {-\frac {x^2}{\left (\sqrt {-3 x^2-2}+\sqrt {2}\right )^2}} \left (\sqrt {-3 x^2-2}+\sqrt {2}\right ) \operatorname {EllipticF}\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-3 x^2-2}}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{135 \sqrt {3} x}+\frac {8}{135} \left (-3 x^2-2\right )^{3/4} x+\frac {32 \sqrt [4]{-3 x^2-2} x}{135 \left (\sqrt {-3 x^2-2}+\sqrt {2}\right )}+\frac {32 \sqrt [4]{2} \sqrt {-\frac {x^2}{\left (\sqrt {-3 x^2-2}+\sqrt {2}\right )^2}} \left (\sqrt {-3 x^2-2}+\sqrt {2}\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-3 x^2-2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{135 \sqrt {3} x}-\frac {2}{27} \left (-3 x^2-2\right )^{3/4} x^3 \]
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Rubi [A] time = 0.11, antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {321, 230, 305, 220, 1196} \[ -\frac {2}{27} \left (-3 x^2-2\right )^{3/4} x^3+\frac {8}{135} \left (-3 x^2-2\right )^{3/4} x+\frac {32 \sqrt [4]{-3 x^2-2} x}{135 \left (\sqrt {-3 x^2-2}+\sqrt {2}\right )}-\frac {16 \sqrt [4]{2} \sqrt {-\frac {x^2}{\left (\sqrt {-3 x^2-2}+\sqrt {2}\right )^2}} \left (\sqrt {-3 x^2-2}+\sqrt {2}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-3 x^2-2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{135 \sqrt {3} x}+\frac {32 \sqrt [4]{2} \sqrt {-\frac {x^2}{\left (\sqrt {-3 x^2-2}+\sqrt {2}\right )^2}} \left (\sqrt {-3 x^2-2}+\sqrt {2}\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-3 x^2-2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{135 \sqrt {3} x} \]
Antiderivative was successfully verified.
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Rule 220
Rule 230
Rule 305
Rule 321
Rule 1196
Rubi steps
\begin {align*} \int \frac {x^4}{\sqrt [4]{-2-3 x^2}} \, dx &=-\frac {2}{27} x^3 \left (-2-3 x^2\right )^{3/4}-\frac {4}{9} \int \frac {x^2}{\sqrt [4]{-2-3 x^2}} \, dx\\ &=\frac {8}{135} x \left (-2-3 x^2\right )^{3/4}-\frac {2}{27} x^3 \left (-2-3 x^2\right )^{3/4}+\frac {16}{135} \int \frac {1}{\sqrt [4]{-2-3 x^2}} \, dx\\ &=\frac {8}{135} x \left (-2-3 x^2\right )^{3/4}-\frac {2}{27} x^3 \left (-2-3 x^2\right )^{3/4}-\frac {\left (16 \sqrt {\frac {2}{3}} \sqrt {-x^2}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+\frac {x^4}{2}}} \, dx,x,\sqrt [4]{-2-3 x^2}\right )}{135 x}\\ &=\frac {8}{135} x \left (-2-3 x^2\right )^{3/4}-\frac {2}{27} x^3 \left (-2-3 x^2\right )^{3/4}-\frac {\left (32 \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 )}{135 \sqrt {3} x}+\frac {\left (32 \sqrt {-x^2}\right ) \operatorname {Subst}\left (\int \frac {1-\frac {x^2}{\sqrt {2}}}{\sqrt {1+\frac {x^4}{2}}} \, dx,x,\sqrt [4]{-2-3 x^2}\right )}{135 \sqrt {3} x}\\ &=\frac {8}{135} x \left (-2-3 x^2\right )^{3/4}-\frac {2}{27} x^3 \left (-2-3 x^2\right )^{3/4}+\frac {32 x \sqrt [4]{-2-3 x^2}}{135 \left (\sqrt {2}+\sqrt {-2-3 x^2}\right )}+\frac {32 \sqrt [4]{2} \sqrt {-\frac {x^2}{\left (\sqrt {2}+\sqrt {-2-3 x^2}\right )^2}} \left (\sqrt {2}+\sqrt {-2-3 x^2}\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-2-3 x^2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{135 \sqrt {3} x}-\frac {16 \sqrt [4]{2} \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 )}{135 \sqrt {3} x}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 63, normalized size = 0.26 \[ \frac {2 x \left (4\ 2^{3/4} \sqrt [4]{3 x^2+2} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {3}{2};-\frac {3 x^2}{2}\right )+15 x^4-2 x^2-8\right )}{135 \sqrt [4]{-3 x^2-2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ \frac {405 \, x {\rm integral}\left (-\frac {64 \, {\left (-3 \, x^{2} - 2\right )}^{\frac {3}{4}}}{405 \, {\left (3 \, x^{4} + 2 \, x^{2}\right )}}, x\right ) - 2 \, {\left (15 \, x^{4} - 12 \, x^{2} + 16\right )} {\left (-3 \, x^{2} - 2\right )}^{\frac {3}{4}}}{405 \, x} \]
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 \[ \int \frac {x^{4}}{{\left (-3 \, x^{2} - 2\right )}^{\frac {1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.27, size = 48, normalized size = 0.20 \[ -\frac {8 \left (-1\right )^{\frac {3}{4}} 2^{\frac {3}{4}} x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], -\frac {3 x^{2}}{2}\right )}{135}+\frac {2 \left (5 x^{2}-4\right ) \left (3 x^{2}+2\right ) x}{135 \left (-3 x^{2}-2\right )^{\frac {1}{4}}} \]
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 \[ \int \frac {x^{4}}{{\left (-3 \, x^{2} - 2\right )}^{\frac {1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^4}{{\left (-3\,x^2-2\right )}^{1/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.77, size = 34, normalized size = 0.14 \[ \frac {2^{\frac {3}{4}} x^{5} e^{- \frac {i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {3 x^{2} e^{i \pi }}{2}} \right )}}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
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