Optimal. Leaf size=85 \[ -\frac {\sqrt [4]{2-3 x^2}}{10 x^5}-\frac {9 \sqrt [4]{2-3 x^2}}{40 x^3}-\frac {27 \sqrt [4]{2-3 x^2}}{32 x}+\frac {27 \sqrt {3} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{32 \sqrt [4]{2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {331, 238}
\begin {gather*} \frac {27 \sqrt {3} F\left (\left .\frac {1}{2} \text {ArcSin}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{32 \sqrt [4]{2}}-\frac {27 \sqrt [4]{2-3 x^2}}{32 x}-\frac {\sqrt [4]{2-3 x^2}}{10 x^5}-\frac {9 \sqrt [4]{2-3 x^2}}{40 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 238
Rule 331
Rubi steps
\begin {align*} \int \frac {1}{x^6 \left (2-3 x^2\right )^{3/4}} \, dx &=-\frac {\sqrt [4]{2-3 x^2}}{10 x^5}+\frac {27}{20} \int \frac {1}{x^4 \left (2-3 x^2\right )^{3/4}} \, dx\\ &=-\frac {\sqrt [4]{2-3 x^2}}{10 x^5}-\frac {9 \sqrt [4]{2-3 x^2}}{40 x^3}+\frac {27}{16} \int \frac {1}{x^2 \left (2-3 x^2\right )^{3/4}} \, dx\\ &=-\frac {\sqrt [4]{2-3 x^2}}{10 x^5}-\frac {9 \sqrt [4]{2-3 x^2}}{40 x^3}-\frac {27 \sqrt [4]{2-3 x^2}}{32 x}+\frac {81}{64} \int \frac {1}{\left (2-3 x^2\right )^{3/4}} \, dx\\ &=-\frac {\sqrt [4]{2-3 x^2}}{10 x^5}-\frac {9 \sqrt [4]{2-3 x^2}}{40 x^3}-\frac {27 \sqrt [4]{2-3 x^2}}{32 x}+\frac {27 \sqrt {3} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{32 \sqrt [4]{2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.01, size = 29, normalized size = 0.34 \begin {gather*} -\frac {\, _2F_1\left (-\frac {5}{2},\frac {3}{4};-\frac {3}{2};\frac {3 x^2}{2}\right )}{5\ 2^{3/4} x^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order
4.
time = 0.09, size = 20, normalized size = 0.24
method | result | size |
meijerg | \(-\frac {2^{\frac {1}{4}} \hypergeom \left (\left [-\frac {5}{2}, \frac {3}{4}\right ], \left [-\frac {3}{2}\right ], \frac {3 x^{2}}{2}\right )}{10 x^{5}}\) | \(20\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.58, size = 34, normalized size = 0.40 \begin {gather*} - \frac {\sqrt [4]{2} {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{2}, \frac {3}{4} \\ - \frac {3}{2} \end {matrix}\middle | {\frac {3 x^{2} e^{2 i \pi }}{2}} \right )}}{10 x^{5}} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{x^6\,{\left (2-3\,x^2\right )}^{3/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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