Optimal. Leaf size=31 \[ \frac {1}{80} \left (4 x^2-9\right )^{5/2}+\frac {3}{16} \left (4 x^2-9\right )^{3/2} \]
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Rubi [A] time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac {1}{80} \left (4 x^2-9\right )^{5/2}+\frac {3}{16} \left (4 x^2-9\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^3 \sqrt {-9+4 x^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x \sqrt {-9+4 x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {9}{4} \sqrt {-9+4 x}+\frac {1}{4} (-9+4 x)^{3/2}\right ) \, dx,x,x^2\right )\\ &=\frac {3}{16} \left (-9+4 x^2\right )^{3/2}+\frac {1}{80} \left (-9+4 x^2\right )^{5/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 22, normalized size = 0.71 \[ \frac {1}{40} \left (2 x^2+3\right ) \left (4 x^2-9\right )^{3/2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 23, normalized size = 0.74 \[ \frac {1}{40} \, {\left (8 \, x^{4} - 6 \, x^{2} - 27\right )} \sqrt {4 \, x^{2} - 9} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.07, size = 23, normalized size = 0.74 \[ \frac {1}{80} \, {\left (4 \, x^{2} - 9\right )}^{\frac {5}{2}} + \frac {3}{16} \, {\left (4 \, x^{2} - 9\right )}^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 29, normalized size = 0.94 \[ \frac {\left (2 x -3\right ) \left (2 x +3\right ) \left (2 x^{2}+3\right ) \sqrt {4 x^{2}-9}}{40} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.91, size = 26, normalized size = 0.84 \[ \frac {1}{20} \, {\left (4 \, x^{2} - 9\right )}^{\frac {3}{2}} x^{2} + \frac {3}{40} \, {\left (4 \, x^{2} - 9\right )}^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.51, size = 23, normalized size = 0.74 \[ -\sqrt {4\,x^2-9}\,\left (-\frac {x^4}{5}+\frac {3\,x^2}{20}+\frac {27}{40}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.66, size = 44, normalized size = 1.42 \[ \frac {x^{4} \sqrt {4 x^{2} - 9}}{5} - \frac {3 x^{2} \sqrt {4 x^{2} - 9}}{20} - \frac {27 \sqrt {4 x^{2} - 9}}{40} \]
Verification of antiderivative is not currently implemented for this CAS.
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