Optimal. Leaf size=67 \[ \frac {5 \sqrt {3} \operatorname {EllipticF}\left (\frac {1}{2} \sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right ),2\right )}{8 \sqrt [4]{2}}-\frac {5 \sqrt [4]{2-3 x^2}}{8 x}-\frac {\sqrt [4]{2-3 x^2}}{6 x^3} \]
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Rubi [A] time = 0.02, antiderivative size = 67, normalized size of antiderivative = 1.00, 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 = {325, 232} \[ -\frac {5 \sqrt [4]{2-3 x^2}}{8 x}-\frac {\sqrt [4]{2-3 x^2}}{6 x^3}+\frac {5 \sqrt {3} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{8 \sqrt [4]{2}} \]
Antiderivative was successfully verified.
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Rule 232
Rule 325
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (2-3 x^2\right )^{3/4}} \, dx &=-\frac {\sqrt [4]{2-3 x^2}}{6 x^3}+\frac {5}{4} \int \frac {1}{x^2 \left (2-3 x^2\right )^{3/4}} \, dx\\ &=-\frac {\sqrt [4]{2-3 x^2}}{6 x^3}-\frac {5 \sqrt [4]{2-3 x^2}}{8 x}+\frac {15}{16} \int \frac {1}{\left (2-3 x^2\right )^{3/4}} \, dx\\ &=-\frac {\sqrt [4]{2-3 x^2}}{6 x^3}-\frac {5 \sqrt [4]{2-3 x^2}}{8 x}+\frac {5 \sqrt {3} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{8 \sqrt [4]{2}}\\ \end {align*}
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Mathematica [C] time = 0.00, size = 29, normalized size = 0.43 \[ -\frac {\, _2F_1\left (-\frac {3}{2},\frac {3}{4};-\frac {1}{2};\frac {3 x^2}{2}\right )}{3\ 2^{3/4} x^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}}{3 \, x^{6} - 2 \, x^{4}}, x\right ) \]
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 \[ \int \frac {1}{{\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-3 x^{2}+2\right )^{\frac {3}{4}} x^{4}}\, dx \]
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 \[ \int \frac {1}{{\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}} x^{4}}\,{d x} \]
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 \[ \int \frac {1}{x^4\,{\left (2-3\,x^2\right )}^{3/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.95, size = 34, normalized size = 0.51 \[ - \frac {\sqrt [4]{2} {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {3}{4} \\ - \frac {1}{2} \end {matrix}\middle | {\frac {3 x^{2} e^{2 i \pi }}{2}} \right )}}{6 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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