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 = 0.83, 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 = 1.01, 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 = 19, normalized size = 0.61 \[ \frac {\left (4 x^{2}+9\right )^{\frac {3}{2}} \left (2 x^{2}-3\right )}{40} \]
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 \[ \frac {1}{20} \, {\left (4 \, x^{2} + 9\right )}^{\frac {3}{2}} x^{2} - \frac {3}{40} \, {\left (4 \, x^{2} + 9\right )}^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.02, size = 20, normalized size = 0.65 \[ \sqrt {x^2+\frac {9}{4}}\,\left (\frac {2\,x^4}{5}+\frac {3\,x^2}{10}-\frac {27}{20}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.63, 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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