Optimal. Leaf size=27 \[ \frac {1}{2} \sqrt {4 x^2+9} x+\frac {9}{4} \sinh ^{-1}\left (\frac {2 x}{3}\right ) \]
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Rubi [A] time = 0.00, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {195, 215} \[ \frac {1}{2} \sqrt {4 x^2+9} x+\frac {9}{4} \sinh ^{-1}\left (\frac {2 x}{3}\right ) \]
Antiderivative was successfully verified.
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Rule 195
Rule 215
Rubi steps
\begin {align*} \int \sqrt {9+4 x^2} \, dx &=\frac {1}{2} x \sqrt {9+4 x^2}+\frac {9}{2} \int \frac {1}{\sqrt {9+4 x^2}} \, dx\\ &=\frac {1}{2} x \sqrt {9+4 x^2}+\frac {9}{4} \sinh ^{-1}\left (\frac {2 x}{3}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 1.00 \[ \frac {1}{2} \sqrt {4 x^2+9} x+\frac {9}{4} \sinh ^{-1}\left (\frac {2 x}{3}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 29, normalized size = 1.07 \[ \frac {1}{2} \, \sqrt {4 \, x^{2} + 9} x - \frac {9}{4} \, \log \left (-2 \, x + \sqrt {4 \, x^{2} + 9}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.99, size = 29, normalized size = 1.07 \[ \frac {1}{2} \, \sqrt {4 \, x^{2} + 9} x - \frac {9}{4} \, \log \left (-2 \, x + \sqrt {4 \, x^{2} + 9}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 20, normalized size = 0.74 \[ \frac {\sqrt {4 x^{2}+9}\, x}{2}+\frac {9 \arcsinh \left (\frac {2 x}{3}\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.98, size = 19, normalized size = 0.70 \[ \frac {1}{2} \, \sqrt {4 \, x^{2} + 9} x + \frac {9}{4} \, \operatorname {arsinh}\left (\frac {2}{3} \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 16, normalized size = 0.59 \[ \frac {9\,\mathrm {asinh}\left (\frac {2\,x}{3}\right )}{4}+x\,\sqrt {x^2+\frac {9}{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 22, normalized size = 0.81 \[ \frac {x \sqrt {4 x^{2} + 9}}{2} + \frac {9 \operatorname {asinh}{\left (\frac {2 x}{3} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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