Optimal. Leaf size=31 \[ \frac {1}{48} \left (-4 x^2-9\right )^{3/2}+\frac {9}{16} \sqrt {-4 x^2-9} \]
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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} \[ \frac {1}{48} \left (-4 x^2-9\right )^{3/2}+\frac {9}{16} \sqrt {-4 x^2-9} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^3}{\sqrt {-9-4 x^2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {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} \sqrt {-9-4 x}\right ) \, dx,x,x^2\right )\\ &=\frac {9}{16} \sqrt {-9-4 x^2}+\frac {1}{48} \left (-9-4 x^2\right )^{3/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 22, normalized size = 0.71 \[ \frac {1}{24} \sqrt {-4 x^2-9} \left (9-2 x^2\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 18, normalized size = 0.58 \[ -\frac {1}{24} \, {\left (2 \, x^{2} - 9\right )} \sqrt {-4 \, x^{2} - 9} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.08, size = 24, normalized size = 0.77 \[ -\frac {1}{48} \, {\left (4 \, x^{2} + 9\right )}^{\frac {3}{2}} i + \frac {9}{16} \, \sqrt {-4 \, x^{2} - 9} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 19, normalized size = 0.61 \[ -\frac {\left (2 x^{2}-9\right ) \sqrt {-4 x^{2}-9}}{24} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.96, size = 26, normalized size = 0.84 \[ -\frac {1}{12} \, \sqrt {-4 \, x^{2} - 9} x^{2} + \frac {3}{8} \, \sqrt {-4 \, x^{2} - 9} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.07, size = 18, normalized size = 0.58 \[ -\frac {\left (2\,x^2-9\right )\,\sqrt {-4\,x^2-9}}{24} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.37, size = 31, normalized size = 1.00 \[ - \frac {x^{2} \sqrt {- 4 x^{2} - 9}}{12} + \frac {3 \sqrt {- 4 x^{2} - 9}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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