Optimal. Leaf size=54 \[ \frac {27}{128} \sqrt {4 x^2-9} x+\frac {243}{256} \tanh ^{-1}\left (\frac {2 x}{\sqrt {4 x^2-9}}\right )+\frac {1}{16} \sqrt {4 x^2-9} x^3 \]
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Rubi [A] time = 0.01, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {321, 217, 206} \[ \frac {1}{16} \sqrt {4 x^2-9} x^3+\frac {27}{128} \sqrt {4 x^2-9} x+\frac {243}{256} \tanh ^{-1}\left (\frac {2 x}{\sqrt {4 x^2-9}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 321
Rubi steps
\begin {align*} \int \frac {x^4}{\sqrt {-9+4 x^2}} \, dx &=\frac {1}{16} x^3 \sqrt {-9+4 x^2}+\frac {27}{16} \int \frac {x^2}{\sqrt {-9+4 x^2}} \, dx\\ &=\frac {27}{128} x \sqrt {-9+4 x^2}+\frac {1}{16} x^3 \sqrt {-9+4 x^2}+\frac {243}{128} \int \frac {1}{\sqrt {-9+4 x^2}} \, dx\\ &=\frac {27}{128} x \sqrt {-9+4 x^2}+\frac {1}{16} x^3 \sqrt {-9+4 x^2}+\frac {243}{128} \operatorname {Subst}\left (\int \frac {1}{1-4 x^2} \, dx,x,\frac {x}{\sqrt {-9+4 x^2}}\right )\\ &=\frac {27}{128} x \sqrt {-9+4 x^2}+\frac {1}{16} x^3 \sqrt {-9+4 x^2}+\frac {243}{256} \tanh ^{-1}\left (\frac {2 x}{\sqrt {-9+4 x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 43, normalized size = 0.80 \[ \frac {1}{256} \left (2 x \sqrt {4 x^2-9} \left (8 x^2+27\right )+243 \tanh ^{-1}\left (\frac {2 x}{\sqrt {4 x^2-9}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 37, normalized size = 0.69 \[ \frac {1}{128} \, {\left (8 \, x^{3} + 27 \, x\right )} \sqrt {4 \, x^{2} - 9} - \frac {243}{256} \, \log \left (-2 \, x + \sqrt {4 \, x^{2} - 9}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.24, size = 37, normalized size = 0.69 \[ \frac {1}{128} \, {\left (8 \, x^{2} + 27\right )} \sqrt {4 \, x^{2} - 9} x - \frac {243}{256} \, \log \left ({\left | -2 \, x + \sqrt {4 \, x^{2} - 9} \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 49, normalized size = 0.91 \[ \frac {\sqrt {4 x^{2}-9}\, x^{3}}{16}+\frac {27 \sqrt {4 x^{2}-9}\, x}{128}+\frac {243 \sqrt {4}\, \ln \left (\sqrt {4}\, x +\sqrt {4 x^{2}-9}\right )}{512} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.81, size = 45, normalized size = 0.83 \[ \frac {1}{16} \, \sqrt {4 \, x^{2} - 9} x^{3} + \frac {27}{128} \, \sqrt {4 \, x^{2} - 9} x + \frac {243}{256} \, \log \left (8 \, x + 4 \, \sqrt {4 \, x^{2} - 9}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^4}{\sqrt {4\,x^2-9}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.72, size = 39, normalized size = 0.72 \[ \frac {x^{3} \sqrt {4 x^{2} - 9}}{16} + \frac {27 x \sqrt {4 x^{2} - 9}}{128} + \frac {243 \operatorname {acosh}{\left (\frac {2 x}{3} \right )}}{256} \]
Verification of antiderivative is not currently implemented for this CAS.
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