Optimal. Leaf size=242 \[ \frac {\left (-2+3 x^2\right )^{3/4}}{6 x^3}+\frac {3 \left (-2+3 x^2\right )^{3/4}}{8 x}-\frac {9 x \sqrt [4]{-2+3 x^2}}{8 \left (\sqrt {2}+\sqrt {-2+3 x^2}\right )}+\frac {3 \sqrt {3} \sqrt {\frac {x^2}{\left (\sqrt {2}+\sqrt {-2+3 x^2}\right )^2}} \left (\sqrt {2}+\sqrt {-2+3 x^2}\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-2+3 x^2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{4\ 2^{3/4} x}-\frac {3 \sqrt {3} \sqrt {\frac {x^2}{\left (\sqrt {2}+\sqrt {-2+3 x^2}\right )^2}} \left (\sqrt {2}+\sqrt {-2+3 x^2}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-2+3 x^2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{8\ 2^{3/4} x} \]
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Rubi [A]
time = 0.08, 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 = {331, 236, 311,
226, 1210} \begin {gather*} -\frac {3 \sqrt {3} \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 \text {ArcTan}\left (\frac {\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{8\ 2^{3/4} x}+\frac {3 \sqrt {3} \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 \text {ArcTan}\left (\frac {\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{4\ 2^{3/4} x}-\frac {9 \sqrt [4]{3 x^2-2} x}{8 \left (\sqrt {3 x^2-2}+\sqrt {2}\right )}+\frac {3 \left (3 x^2-2\right )^{3/4}}{8 x}+\frac {\left (3 x^2-2\right )^{3/4}}{6 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 236
Rule 311
Rule 331
Rule 1210
Rubi steps
\begin {align*} \int \frac {1}{x^4 \sqrt [4]{-2+3 x^2}} \, dx &=\frac {\left (-2+3 x^2\right )^{3/4}}{6 x^3}+\frac {3}{4} \int \frac {1}{x^2 \sqrt [4]{-2+3 x^2}} \, dx\\ &=\frac {\left (-2+3 x^2\right )^{3/4}}{6 x^3}+\frac {3 \left (-2+3 x^2\right )^{3/4}}{8 x}-\frac {9}{16} \int \frac {1}{\sqrt [4]{-2+3 x^2}} \, dx\\ &=\frac {\left (-2+3 x^2\right )^{3/4}}{6 x^3}+\frac {3 \left (-2+3 x^2\right )^{3/4}}{8 x}-\frac {\left (3 \sqrt {\frac {3}{2}} \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1+\frac {x^4}{2}}} \, dx,x,\sqrt [4]{-2+3 x^2}\right )}{8 x}\\ &=\frac {\left (-2+3 x^2\right )^{3/4}}{6 x^3}+\frac {3 \left (-2+3 x^2\right )^{3/4}}{8 x}-\frac {\left (3 \sqrt {3} \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{2}}} \, dx,x,\sqrt [4]{-2+3 x^2}\right )}{8 x}+\frac {\left (3 \sqrt {3} \sqrt {x^2}\right ) \text {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 )}{8 x}\\ &=\frac {\left (-2+3 x^2\right )^{3/4}}{6 x^3}+\frac {3 \left (-2+3 x^2\right )^{3/4}}{8 x}-\frac {9 x \sqrt [4]{-2+3 x^2}}{8 \left (\sqrt {2}+\sqrt {-2+3 x^2}\right )}+\frac {3 \sqrt {3} \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 )}{4\ 2^{3/4} x}-\frac {3 \sqrt {3} \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 )}{8\ 2^{3/4} x}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.02, size = 48, normalized size = 0.20 \begin {gather*} -\frac {\sqrt [4]{1-\frac {3 x^2}{2}} \, _2F_1\left (-\frac {3}{2},\frac {1}{4};-\frac {1}{2};\frac {3 x^2}{2}\right )}{3 x^3 \sqrt [4]{-2+3 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.15, size = 42, normalized size = 0.17
method | result | size |
meijerg | \(-\frac {2^{\frac {3}{4}} \left (-\mathrm {signum}\left (-1+\frac {3 x^{2}}{2}\right )\right )^{\frac {1}{4}} \hypergeom \left (\left [-\frac {3}{2}, \frac {1}{4}\right ], \left [-\frac {1}{2}\right ], \frac {3 x^{2}}{2}\right )}{6 \mathrm {signum}\left (-1+\frac {3 x^{2}}{2}\right )^{\frac {1}{4}} x^{3}}\) | \(42\) |
risch | \(\frac {27 x^{4}-6 x^{2}-8}{24 x^{3} \left (3 x^{2}-2\right )^{\frac {1}{4}}}-\frac {9 \,2^{\frac {3}{4}} \left (-\mathrm {signum}\left (-1+\frac {3 x^{2}}{2}\right )\right )^{\frac {1}{4}} x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], \frac {3 x^{2}}{2}\right )}{32 \mathrm {signum}\left (-1+\frac {3 x^{2}}{2}\right )^{\frac {1}{4}}}\) | \(67\) |
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 = 0.49, size = 34, normalized size = 0.14 \begin {gather*} \frac {2^{\frac {3}{4}} e^{\frac {3 i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {1}{4} \\ - \frac {1}{2} \end {matrix}\middle | {\frac {3 x^{2}}{2}} \right )}}{6 x^{3}} \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 = -0.00 \begin {gather*} \int \frac {1}{x^4\,{\left (3\,x^2-2\right )}^{1/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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