Optimal. Leaf size=46 \[ -\frac {1}{320} \left (-4 x^2-9\right )^{5/2}-\frac {3}{32} \left (-4 x^2-9\right )^{3/2}-\frac {81}{64} \sqrt {-4 x^2-9} \]
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Rubi [A] time = 0.02, antiderivative size = 46, 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} \[ -\frac {1}{320} \left (-4 x^2-9\right )^{5/2}-\frac {3}{32} \left (-4 x^2-9\right )^{3/2}-\frac {81}{64} \sqrt {-4 x^2-9} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^5}{\sqrt {-9-4 x^2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {-9-4 x}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {81}{16 \sqrt {-9-4 x}}+\frac {9}{8} \sqrt {-9-4 x}+\frac {1}{16} (-9-4 x)^{3/2}\right ) \, dx,x,x^2\right )\\ &=-\frac {81}{64} \sqrt {-9-4 x^2}-\frac {3}{32} \left (-9-4 x^2\right )^{3/2}-\frac {1}{320} \left (-9-4 x^2\right )^{5/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 0.59 \[ -\frac {1}{40} \sqrt {-4 x^2-9} \left (2 x^4-6 x^2+27\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 23, normalized size = 0.50 \[ -\frac {1}{40} \, {\left (2 \, x^{4} - 6 \, x^{2} + 27\right )} \sqrt {-4 \, x^{2} - 9} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.00, size = 36, normalized size = 0.78 \[ -\frac {1}{320} \, {\left (4 \, x^{2} + 9\right )}^{\frac {5}{2}} i + \frac {3}{32} \, {\left (4 \, x^{2} + 9\right )}^{\frac {3}{2}} i - \frac {81}{64} \, \sqrt {-4 \, x^{2} - 9} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 24, normalized size = 0.52 \[ -\frac {\left (2 x^{4}-6 x^{2}+27\right ) \sqrt {-4 x^{2}-9}}{40} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.91, size = 40, normalized size = 0.87 \[ -\frac {1}{20} \, \sqrt {-4 \, x^{2} - 9} x^{4} + \frac {3}{20} \, \sqrt {-4 \, x^{2} - 9} x^{2} - \frac {27}{40} \, \sqrt {-4 \, x^{2} - 9} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.29, size = 23, normalized size = 0.50 \[ -\sqrt {-4\,x^2-9}\,\left (\frac {x^4}{20}-\frac {3\,x^2}{20}+\frac {27}{40}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.21, size = 49, normalized size = 1.07 \[ - \frac {x^{4} \sqrt {- 4 x^{2} - 9}}{20} + \frac {3 x^{2} \sqrt {- 4 x^{2} - 9}}{20} - \frac {27 \sqrt {- 4 x^{2} - 9}}{40} \]
Verification of antiderivative is not currently implemented for this CAS.
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