Optimal. Leaf size=242 \[ -\frac{16 \sqrt [4]{2} \sqrt{-\frac{x^2}{\left (\sqrt{-3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{-3 x^2-2}+\sqrt{2}\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{-3 x^2-2}}{\sqrt [4]{2}}\right ),\frac{1}{2}\right )}{135 \sqrt{3} x}-\frac{2}{27} \left (-3 x^2-2\right )^{3/4} x^3+\frac{8}{135} \left (-3 x^2-2\right )^{3/4} x+\frac{32 \sqrt [4]{-3 x^2-2} x}{135 \left (\sqrt{-3 x^2-2}+\sqrt{2}\right )}+\frac{32 \sqrt [4]{2} \sqrt{-\frac{x^2}{\left (\sqrt{-3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{-3 x^2-2}+\sqrt{2}\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{-3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{135 \sqrt{3} x} \]
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Rubi [A] time = 0.110469, antiderivative size = 242, normalized size of antiderivative = 1., 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 321
Rule 230
Rule 305
Rule 220
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*}
Mathematica [C] time = 0.0141214, 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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Maple [C] time = 0.02, size = 48, normalized size = 0.2 \begin{align*}{\frac{2\,x \left ( 5\,{x}^{2}-4 \right ) \left ( 3\,{x}^{2}+2 \right ) }{135}{\frac{1}{\sqrt [4]{-3\,{x}^{2}-2}}}}-{\frac{8\, \left ( -1 \right ) ^{3/4}x{2}^{3/4}}{135}{\mbox{$_2$F$_1$}({\frac{1}{4}},{\frac{1}{2}};\,{\frac{3}{2}};\,-{\frac{3\,{x}^{2}}{2}})}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{{\left (-3 \, x^{2} - 2\right )}^{\frac{1}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.693486, size = 34, normalized size = 0.14 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{{\left (-3 \, x^{2} - 2\right )}^{\frac{1}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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