Optimal. Leaf size=72 \[ -\frac {81}{256} \sqrt {4 x^2-9} x-\frac {729}{512} \tanh ^{-1}\left (\frac {2 x}{\sqrt {4 x^2-9}}\right )+\frac {1}{6} \sqrt {4 x^2-9} x^5-\frac {3}{32} \sqrt {4 x^2-9} x^3 \]
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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, 206} \[ \frac {1}{6} \sqrt {4 x^2-9} x^5-\frac {3}{32} \sqrt {4 x^2-9} x^3-\frac {81}{256} \sqrt {4 x^2-9} x-\frac {729}{512} \tanh ^{-1}\left (\frac {2 x}{\sqrt {4 x^2-9}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 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} \tanh ^{-1}\left (\frac {2 x}{\sqrt {-9+4 x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 49, normalized size = 0.68 \[ \frac {1}{768} x \sqrt {4 x^2-9} \left (128 x^4-72 x^2-243\right )-\frac {729}{512} \log \left (\sqrt {4 x^2-9}+2 x\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 42, normalized size = 0.58 \[ \frac {1}{768} \, {\left (128 \, x^{5} - 72 \, x^{3} - 243 \, x\right )} \sqrt {4 \, x^{2} - 9} + \frac {729}{512} \, \log \left (-2 \, x + \sqrt {4 \, x^{2} - 9}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.19, size = 44, normalized size = 0.61 \[ \frac {1}{768} \, {\left (8 \, {\left (16 \, x^{2} - 9\right )} x^{2} - 243\right )} \sqrt {4 \, x^{2} - 9} x + \frac {729}{512} \, \log \left ({\left | -2 \, x + \sqrt {4 \, x^{2} - 9} \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 61, normalized size = 0.85 \[ \frac {\left (4 x^{2}-9\right )^{\frac {3}{2}} x^{3}}{24}+\frac {9 \left (4 x^{2}-9\right )^{\frac {3}{2}} x}{128}+\frac {81 \sqrt {4 x^{2}-9}\, x}{256}-\frac {729 \sqrt {4}\, \ln \left (\sqrt {4}\, x +\sqrt {4 x^{2}-9}\right )}{1024} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.92, size = 57, normalized size = 0.79 \[ \frac {1}{24} \, {\left (4 \, x^{2} - 9\right )}^{\frac {3}{2}} x^{3} + \frac {9}{128} \, {\left (4 \, x^{2} - 9\right )}^{\frac {3}{2}} x + \frac {81}{256} \, \sqrt {4 \, x^{2} - 9} x - \frac {729}{512} \, \log \left (8 \, x + 4 \, \sqrt {4 \, x^{2} - 9}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^4\,\sqrt {4\,x^2-9} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.56, size = 167, normalized size = 2.32 \[ \begin {cases} \frac {2 x^{7}}{3 \sqrt {4 x^{2} - 9}} - \frac {15 x^{5}}{8 \sqrt {4 x^{2} - 9}} - \frac {27 x^{3}}{64 \sqrt {4 x^{2} - 9}} + \frac {729 x}{256 \sqrt {4 x^{2} - 9}} - \frac {729 \operatorname {acosh}{\left (\frac {2 x}{3} \right )}}{512} & \text {for}\: \frac {4 \left |{x^{2}}\right |}{9} > 1 \\- \frac {2 i x^{7}}{3 \sqrt {9 - 4 x^{2}}} + \frac {15 i x^{5}}{8 \sqrt {9 - 4 x^{2}}} + \frac {27 i x^{3}}{64 \sqrt {9 - 4 x^{2}}} - \frac {729 i x}{256 \sqrt {9 - 4 x^{2}}} + \frac {729 i \operatorname {asin}{\left (\frac {2 x}{3} \right )}}{512} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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