Optimal. Leaf size=65 \[ \frac{16\ 2^{3/4} \text{EllipticF}\left (\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right ),2\right )}{63 \sqrt{3}}-\frac{2}{21} \sqrt [4]{2-3 x^2} x^3-\frac{8}{63} \sqrt [4]{2-3 x^2} x \]
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Rubi [A] time = 0.0155594, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {321, 232} \[ -\frac{2}{21} \sqrt [4]{2-3 x^2} x^3-\frac{8}{63} \sqrt [4]{2-3 x^2} x+\frac{16\ 2^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{63 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 321
Rule 232
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{16\ 2^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{63 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.024523, size = 54, normalized size = 0.83 \[ -\frac{2}{189} \left (3 x \sqrt [4]{2-3 x^2} \left (3 x^2+4\right )-8\ 2^{3/4} \sqrt{3} \text{EllipticF}\left (\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right ),2\right )\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.011, size = 0, normalized size = 0. \begin{align*} \int{{x}^{4} \left ( -3\,{x}^{2}+2 \right ) ^{-{\frac{3}{4}}}}\, dx \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{3}{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*}{\rm integral}\left (-\frac{{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} x^{4}}{3 \, x^{2} - 2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.682182, size = 29, normalized size = 0.45 \begin{align*} \frac{\sqrt [4]{2} x^{5}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{3 x^{2} e^{2 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{3}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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