Optimal. Leaf size=63 \[ -\frac {81}{256} \sqrt {4 x^2+9} x+\frac {1}{6} \sqrt {4 x^2+9} x^5+\frac {3}{32} \sqrt {4 x^2+9} x^3+\frac {729}{512} \sinh ^{-1}\left (\frac {2 x}{3}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 63, 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 = {279, 321, 215} \[ \frac {1}{6} \sqrt {4 x^2+9} x^5+\frac {3}{32} \sqrt {4 x^2+9} x^3-\frac {81}{256} \sqrt {4 x^2+9} x+\frac {729}{512} \sinh ^{-1}\left (\frac {2 x}{3}\right ) \]
Antiderivative was successfully verified.
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Rule 215
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}{512} \sinh ^{-1}\left (\frac {2 x}{3}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 39, normalized size = 0.62 \[ \frac {1}{768} x \sqrt {4 x^2+9} \left (128 x^4+72 x^2-243\right )+\frac {729}{512} \sinh ^{-1}\left (\frac {2 x}{3}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 42, normalized size = 0.67 \[ \frac {1}{768} \, {\left (128 \, x^{5} + 72 \, x^{3} - 243 \, x\right )} \sqrt {4 \, x^{2} + 9} - \frac {729}{512} \, \log \left (-2 \, x + \sqrt {4 \, x^{2} + 9}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.17, size = 43, normalized size = 0.68 \[ \frac {1}{768} \, {\left (8 \, {\left (16 \, x^{2} + 9\right )} x^{2} - 243\right )} \sqrt {4 \, x^{2} + 9} x - \frac {729}{512} \, \log \left (-2 \, x + \sqrt {4 \, x^{2} + 9}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 46, normalized size = 0.73 \[ \frac {\left (4 x^{2}+9\right )^{\frac {3}{2}} x^{3}}{24}-\frac {9 \left (4 x^{2}+9\right )^{\frac {3}{2}} x}{128}+\frac {81 \sqrt {4 x^{2}+9}\, x}{256}+\frac {729 \arcsinh \left (\frac {2 x}{3}\right )}{512} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.94, size = 45, normalized size = 0.71 \[ \frac {1}{24} \, {\left (4 \, x^{2} + 9\right )}^{\frac {3}{2}} x^{3} - \frac {9}{128} \, {\left (4 \, x^{2} + 9\right )}^{\frac {3}{2}} x + \frac {81}{256} \, \sqrt {4 \, x^{2} + 9} x + \frac {729}{512} \, \operatorname {arsinh}\left (\frac {2}{3} \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 30, normalized size = 0.48 \[ \frac {729\,\mathrm {asinh}\left (\frac {2\,x}{3}\right )}{512}+\frac {\sqrt {x^2+\frac {9}{4}}\,\left (\frac {2\,x^5}{3}+\frac {3\,x^3}{8}-\frac {81\,x}{64}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.56, size = 75, normalized size = 1.19 \[ \frac {2 x^{7}}{3 \sqrt {4 x^{2} + 9}} + \frac {15 x^{5}}{8 \sqrt {4 x^{2} + 9}} - \frac {27 x^{3}}{64 \sqrt {4 x^{2} + 9}} - \frac {729 x}{256 \sqrt {4 x^{2} + 9}} + \frac {729 \operatorname {asinh}{\left (\frac {2 x}{3} \right )}}{512} \]
Verification of antiderivative is not currently implemented for this CAS.
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