Optimal. Leaf size=54 \[ \frac {27}{128} \sqrt {-4 x^2-9} x+\frac {243}{256} \tan ^{-1}\left (\frac {2 x}{\sqrt {-4 x^2-9}}\right )-\frac {1}{16} \sqrt {-4 x^2-9} x^3 \]
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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, 203} \[ -\frac {1}{16} \sqrt {-4 x^2-9} x^3+\frac {27}{128} \sqrt {-4 x^2-9} x+\frac {243}{256} \tan ^{-1}\left (\frac {2 x}{\sqrt {-4 x^2-9}}\right ) \]
Antiderivative was successfully verified.
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Rule 203
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} \tan ^{-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 \[ \frac {1}{256} \left (2 x \sqrt {-4 x^2-9} \left (27-8 x^2\right )+243 \tan ^{-1}\left (\frac {2 x}{\sqrt {-4 x^2-9}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [C] time = 1.23, size = 67, normalized size = 1.24 \[ -\frac {1}{128} \, {\left (8 \, x^{3} - 27 \, x\right )} \sqrt {-4 \, x^{2} - 9} + \frac {243}{512} i \, \log \left (-\frac {8 \, x + 4 i \, \sqrt {-4 \, x^{2} - 9}}{x}\right ) - \frac {243}{512} i \, \log \left (-\frac {8 \, x - 4 i \, \sqrt {-4 \, x^{2} - 9}}{x}\right ) \]
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 \[ \int \frac {x^{4}}{\sqrt {-4 \, x^{2} - 9}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 43, normalized size = 0.80 \[ -\frac {\sqrt {-4 x^{2}-9}\, x^{3}}{16}+\frac {27 \sqrt {-4 x^{2}-9}\, x}{128}+\frac {243 \arctan \left (\frac {2 x}{\sqrt {-4 x^{2}-9}}\right )}{256} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 2.95, size = 33, normalized size = 0.61 \[ -\frac {1}{16} \, \sqrt {-4 \, x^{2} - 9} x^{3} + \frac {27}{128} \, \sqrt {-4 \, x^{2} - 9} x - \frac {243}{256} i \, \operatorname {arsinh}\left (\frac {2}{3} \, x\right ) \]
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 = 0.89, size = 53, normalized size = 0.98 \[ - \frac {x^{3} \sqrt {- 4 x^{2} - 9}}{16} + \frac {27 x \sqrt {- 4 x^{2} - 9}}{128} + \frac {243 \operatorname {atan}{\left (\frac {2 x}{\sqrt {- 4 x^{2} - 9}} \right )}}{256} \]
Verification of antiderivative is not currently implemented for this CAS.
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