Optimal. Leaf size=72 \[ -\frac {81}{256} \sqrt {-4 x^2-9} x-\frac {729}{512} \tan ^{-1}\left (\frac {2 x}{\sqrt {-4 x^2-9}}\right )+\frac {1}{6} \sqrt {-4 x^2-9} x^5+\frac {3}{32} \sqrt {-4 x^2-9} x^3 \]
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Rubi [A] time = 0.02, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {279, 321, 217, 203} \[ \frac {1}{6} \sqrt {-4 x^2-9} x^5+\frac {3}{32} \sqrt {-4 x^2-9} x^3-\frac {81}{256} \sqrt {-4 x^2-9} x-\frac {729}{512} \tan ^{-1}\left (\frac {2 x}{\sqrt {-4 x^2-9}}\right ) \]
Antiderivative was successfully verified.
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Rule 203
Rule 217
Rule 279
Rule 321
Rubi steps
\begin {align*} \int x^4 \sqrt {-9-4 x^2} \, dx &=\frac {1}{6} x^5 \sqrt {-9-4 x^2}-\frac {3}{2} \int \frac {x^4}{\sqrt {-9-4 x^2}} \, dx\\ &=\frac {3}{32} x^3 \sqrt {-9-4 x^2}+\frac {1}{6} x^5 \sqrt {-9-4 x^2}+\frac {81}{32} \int \frac {x^2}{\sqrt {-9-4 x^2}} \, dx\\ &=-\frac {81}{256} x \sqrt {-9-4 x^2}+\frac {3}{32} x^3 \sqrt {-9-4 x^2}+\frac {1}{6} x^5 \sqrt {-9-4 x^2}-\frac {729}{256} \int \frac {1}{\sqrt {-9-4 x^2}} \, dx\\ &=-\frac {81}{256} x \sqrt {-9-4 x^2}+\frac {3}{32} x^3 \sqrt {-9-4 x^2}+\frac {1}{6} x^5 \sqrt {-9-4 x^2}-\frac {729}{256} \operatorname {Subst}\left (\int \frac {1}{1+4 x^2} \, dx,x,\frac {x}{\sqrt {-9-4 x^2}}\right )\\ &=-\frac {81}{256} x \sqrt {-9-4 x^2}+\frac {3}{32} x^3 \sqrt {-9-4 x^2}+\frac {1}{6} x^5 \sqrt {-9-4 x^2}-\frac {729}{512} \tan ^{-1}\left (\frac {2 x}{\sqrt {-9-4 x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 48, normalized size = 0.67 \[ \frac {1}{768} x \sqrt {-4 x^2-9} \left (128 x^4+72 x^2-243\right )-\frac {729}{512} \tan ^{-1}\left (\frac {2 x}{\sqrt {-4 x^2-9}}\right ) \]
Antiderivative was successfully verified.
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fricas [C] time = 0.63, size = 72, normalized size = 1.00 \[ \frac {1}{768} \, {\left (128 \, x^{5} + 72 \, x^{3} - 243 \, x\right )} \sqrt {-4 \, x^{2} - 9} - \frac {729}{1024} i \, \log \left (-\frac {8 \, x + 4 i \, \sqrt {-4 \, x^{2} - 9}}{x}\right ) + \frac {729}{1024} i \, \log \left (-\frac {8 \, x - 4 i \, \sqrt {-4 \, x^{2} - 9}}{x}\right ) \]
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 \[ \int \sqrt {-4 \, x^{2} - 9} x^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 55, normalized size = 0.76 \[ -\frac {\left (-4 x^{2}-9\right )^{\frac {3}{2}} x^{3}}{24}+\frac {9 \left (-4 x^{2}-9\right )^{\frac {3}{2}} x}{128}+\frac {81 \sqrt {-4 x^{2}-9}\, x}{256}-\frac {729 \arctan \left (\frac {2 x}{\sqrt {-4 x^{2}-9}}\right )}{512} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 2.94, size = 45, normalized size = 0.62 \[ -\frac {1}{24} \, {\left (-4 \, x^{2} - 9\right )}^{\frac {3}{2}} x^{3} + \frac {9}{128} \, {\left (-4 \, x^{2} - 9\right )}^{\frac {3}{2}} x + \frac {81}{256} \, \sqrt {-4 \, x^{2} - 9} x + \frac {729}{512} i \, \operatorname {arsinh}\left (\frac {2}{3} \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^4\,\sqrt {-4\,x^2-9} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 4.67, size = 83, normalized size = 1.15 \[ \frac {2 i x^{7}}{3 \sqrt {4 x^{2} + 9}} + \frac {15 i x^{5}}{8 \sqrt {4 x^{2} + 9}} - \frac {27 i x^{3}}{64 \sqrt {4 x^{2} + 9}} - \frac {729 i x}{256 \sqrt {4 x^{2} + 9}} + \frac {729 i \operatorname {asinh}{\left (\frac {2 x}{3} \right )}}{512} \]
Verification of antiderivative is not currently implemented for this CAS.
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