Optimal. Leaf size=54 \[ \frac {9}{32} \sqrt {-4 x^2-9} x+\frac {81}{64} \tan ^{-1}\left (\frac {2 x}{\sqrt {-4 x^2-9}}\right )+\frac {1}{4} \sqrt {-4 x^2-9} x^3 \]
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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 = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {279, 321, 217, 203} \[ \frac {1}{4} \sqrt {-4 x^2-9} x^3+\frac {9}{32} \sqrt {-4 x^2-9} x+\frac {81}{64} \tan ^{-1}\left (\frac {2 x}{\sqrt {-4 x^2-9}}\right ) \]
Antiderivative was successfully verified.
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Rule 203
Rule 217
Rule 279
Rule 321
Rubi steps
\begin {align*} \int x^2 \sqrt {-9-4 x^2} \, dx &=\frac {1}{4} x^3 \sqrt {-9-4 x^2}-\frac {9}{4} \int \frac {x^2}{\sqrt {-9-4 x^2}} \, dx\\ &=\frac {9}{32} x \sqrt {-9-4 x^2}+\frac {1}{4} x^3 \sqrt {-9-4 x^2}+\frac {81}{32} \int \frac {1}{\sqrt {-9-4 x^2}} \, dx\\ &=\frac {9}{32} x \sqrt {-9-4 x^2}+\frac {1}{4} x^3 \sqrt {-9-4 x^2}+\frac {81}{32} \operatorname {Subst}\left (\int \frac {1}{1+4 x^2} \, dx,x,\frac {x}{\sqrt {-9-4 x^2}}\right )\\ &=\frac {9}{32} x \sqrt {-9-4 x^2}+\frac {1}{4} x^3 \sqrt {-9-4 x^2}+\frac {81}{64} \tan ^{-1}\left (\frac {2 x}{\sqrt {-9-4 x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 43, normalized size = 0.80 \[ \frac {1}{64} \left (2 x \sqrt {-4 x^2-9} \left (8 x^2+9\right )+81 \tan ^{-1}\left (\frac {2 x}{\sqrt {-4 x^2-9}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [C] time = 0.79, size = 67, normalized size = 1.24 \[ \frac {1}{32} \, {\left (8 \, x^{3} + 9 \, x\right )} \sqrt {-4 \, x^{2} - 9} + \frac {81}{128} i \, \log \left (-\frac {8 \, x + 4 i \, \sqrt {-4 \, x^{2} - 9}}{x}\right ) - \frac {81}{128} i \, \log \left (-\frac {8 \, x - 4 i \, \sqrt {-4 \, x^{2} - 9}}{x}\right ) \]
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 \[ \int \sqrt {-4 \, x^{2} - 9} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 41, normalized size = 0.76 \[ -\frac {\left (-4 x^{2}-9\right )^{\frac {3}{2}} x}{16}-\frac {9 \sqrt {-4 x^{2}-9}\, x}{32}+\frac {81 \arctan \left (\frac {2 x}{\sqrt {-4 x^{2}-9}}\right )}{64} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 2.98, size = 31, normalized size = 0.57 \[ -\frac {1}{16} \, {\left (-4 \, x^{2} - 9\right )}^{\frac {3}{2}} x - \frac {9}{32} \, \sqrt {-4 \, x^{2} - 9} x - \frac {81}{64} i \, \operatorname {arsinh}\left (\frac {2}{3} \, x\right ) \]
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 \[ \int x^2\,\sqrt {-4\,x^2-9} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.76, size = 61, normalized size = 1.13 \[ \frac {i x^{5}}{\sqrt {4 x^{2} + 9}} + \frac {27 i x^{3}}{8 \sqrt {4 x^{2} + 9}} + \frac {81 i x}{32 \sqrt {4 x^{2} + 9}} - \frac {81 i \operatorname {asinh}{\left (\frac {2 x}{3} \right )}}{64} \]
Verification of antiderivative is not currently implemented for this CAS.
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