Optimal. Leaf size=46 \[ \frac {1}{448} \left (4 x^2+9\right )^{7/2}-\frac {9}{160} \left (4 x^2+9\right )^{5/2}+\frac {27}{64} \left (4 x^2+9\right )^{3/2} \]
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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} \[ \frac {1}{448} \left (4 x^2+9\right )^{7/2}-\frac {9}{160} \left (4 x^2+9\right )^{5/2}+\frac {27}{64} \left (4 x^2+9\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^5 \sqrt {9+4 x^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int 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} (9+4 x)^{3/2}+\frac {1}{16} (9+4 x)^{5/2}\right ) \, dx,x,x^2\right )\\ &=\frac {27}{64} \left (9+4 x^2\right )^{3/2}-\frac {9}{160} \left (9+4 x^2\right )^{5/2}+\frac {1}{448} \left (9+4 x^2\right )^{7/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 0.59 \[ \frac {1}{280} \left (4 x^2+9\right )^{3/2} \left (10 x^4-18 x^2+27\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 28, normalized size = 0.61 \[ \frac {1}{280} \, {\left (40 \, x^{6} + 18 \, x^{4} - 54 \, x^{2} + 243\right )} \sqrt {4 \, x^{2} + 9} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.03, size = 34, normalized size = 0.74 \[ \frac {1}{448} \, {\left (4 \, x^{2} + 9\right )}^{\frac {7}{2}} - \frac {9}{160} \, {\left (4 \, x^{2} + 9\right )}^{\frac {5}{2}} + \frac {27}{64} \, {\left (4 \, x^{2} + 9\right )}^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 24, normalized size = 0.52 \[ \frac {\left (4 x^{2}+9\right )^{\frac {3}{2}} \left (10 x^{4}-18 x^{2}+27\right )}{280} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.97, size = 40, normalized size = 0.87 \[ \frac {1}{28} \, {\left (4 \, x^{2} + 9\right )}^{\frac {3}{2}} x^{4} - \frac {9}{140} \, {\left (4 \, x^{2} + 9\right )}^{\frac {3}{2}} x^{2} + \frac {27}{280} \, {\left (4 \, x^{2} + 9\right )}^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.55, size = 25, normalized size = 0.54 \[ \sqrt {x^2+\frac {9}{4}}\,\left (\frac {2\,x^6}{7}+\frac {9\,x^4}{70}-\frac {27\,x^2}{70}+\frac {243}{140}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.97, size = 61, normalized size = 1.33 \[ \frac {x^{6} \sqrt {4 x^{2} + 9}}{7} + \frac {9 x^{4} \sqrt {4 x^{2} + 9}}{140} - \frac {27 x^{2} \sqrt {4 x^{2} + 9}}{140} + \frac {243 \sqrt {4 x^{2} + 9}}{280} \]
Verification of antiderivative is not currently implemented for this CAS.
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