Optimal. Leaf size=46 \[ \frac {81}{64} \sqrt {9+4 x^2}-\frac {3}{32} \left (9+4 x^2\right )^{3/2}+\frac {1}{320} \left (9+4 x^2\right )^{5/2} \]
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Rubi [A]
time = 0.01, 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 = {272, 45}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {x^5}{\sqrt {9+4 x^2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^2}{\sqrt {9+4 x}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {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 \begin {gather*} \frac {1}{40} \sqrt {9+4 x^2} \left (27-6 x^2+2 x^4\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 41, normalized size = 0.89
method | result | size |
trager | \(\sqrt {4 x^{2}+9}\, \left (\frac {1}{20} x^{4}-\frac {3}{20} x^{2}+\frac {27}{40}\right )\) | \(23\) |
gosper | \(\frac {\sqrt {4 x^{2}+9}\, \left (2 x^{4}-6 x^{2}+27\right )}{40}\) | \(24\) |
risch | \(\frac {\sqrt {4 x^{2}+9}\, \left (2 x^{4}-6 x^{2}+27\right )}{40}\) | \(24\) |
meijerg | \(\frac {-\frac {81 \sqrt {\pi }}{40}+\frac {81 \sqrt {\pi }\, \left (\frac {32}{27} x^{4}-\frac {32}{9} x^{2}+16\right ) \sqrt {1+\frac {4 x^{2}}{9}}}{640}}{\sqrt {\pi }}\) | \(38\) |
default | \(\frac {x^{4} \sqrt {4 x^{2}+9}}{20}-\frac {3 x^{2} \sqrt {4 x^{2}+9}}{20}+\frac {27 \sqrt {4 x^{2}+9}}{40}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 40, normalized size = 0.87 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.05, size = 23, normalized size = 0.50 \begin {gather*} \frac {1}{40} \, {\left (2 \, x^{4} - 6 \, x^{2} + 27\right )} \sqrt {4 \, x^{2} + 9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.23, size = 44, normalized size = 0.96 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.57, size = 34, normalized size = 0.74 \begin {gather*} \frac {1}{320} \, {\left (4 \, x^{2} + 9\right )}^{\frac {5}{2}} - \frac {3}{32} \, {\left (4 \, x^{2} + 9\right )}^{\frac {3}{2}} + \frac {81}{64} \, \sqrt {4 \, x^{2} + 9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.02, size = 21, normalized size = 0.46 \begin {gather*} \frac {\sqrt {x^2+\frac {9}{4}}\,\left (\frac {x^4}{5}-\frac {3\,x^2}{5}+\frac {27}{10}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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