Optimal. Leaf size=25 \[ -\frac {\sqrt {9+4 x^2}}{x}+2 \sinh ^{-1}\left (\frac {2 x}{3}\right ) \]
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Rubi [A]
time = 0.00, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {283, 221}
\begin {gather*} 2 \sinh ^{-1}\left (\frac {2 x}{3}\right )-\frac {\sqrt {4 x^2+9}}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 283
Rubi steps
\begin {align*} \int \frac {\sqrt {9+4 x^2}}{x^2} \, dx &=-\frac {\sqrt {9+4 x^2}}{x}+4 \int \frac {1}{\sqrt {9+4 x^2}} \, dx\\ &=-\frac {\sqrt {9+4 x^2}}{x}+2 \sinh ^{-1}\left (\frac {2 x}{3}\right )\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 35, normalized size = 1.40 \begin {gather*} -\frac {\sqrt {9+4 x^2}}{x}-2 \log \left (-2 x+\sqrt {9+4 x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 34, normalized size = 1.36
method | result | size |
risch | \(2 \arcsinh \left (\frac {2 x}{3}\right )-\frac {\sqrt {4 x^{2}+9}}{x}\) | \(22\) |
trager | \(-\frac {\sqrt {4 x^{2}+9}}{x}-2 \ln \left (\sqrt {4 x^{2}+9}-2 x \right )\) | \(32\) |
meijerg | \(-\frac {\frac {6 \sqrt {\pi }\, \sqrt {1+\frac {4 x^{2}}{9}}}{x}-4 \sqrt {\pi }\, \arcsinh \left (\frac {2 x}{3}\right )}{2 \sqrt {\pi }}\) | \(33\) |
default | \(-\frac {\left (4 x^{2}+9\right )^{\frac {3}{2}}}{9 x}+2 \arcsinh \left (\frac {2 x}{3}\right )+\frac {4 x \sqrt {4 x^{2}+9}}{9}\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 21, normalized size = 0.84 \begin {gather*} -\frac {\sqrt {4 \, x^{2} + 9}}{x} + 2 \, \operatorname {arsinh}\left (\frac {2}{3} \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.50, size = 35, normalized size = 1.40 \begin {gather*} -\frac {2 \, x \log \left (-2 \, x + \sqrt {4 \, x^{2} + 9}\right ) + 2 \, x + \sqrt {4 \, x^{2} + 9}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 19, normalized size = 0.76 \begin {gather*} 2 \operatorname {asinh}{\left (\frac {2 x}{3} \right )} - \frac {\sqrt {4 x^{2} + 9}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.51, size = 40, normalized size = 1.60 \begin {gather*} \frac {36}{{\left (2 \, x - \sqrt {4 \, x^{2} + 9}\right )}^{2} - 9} - 2 \, \log \left (-2 \, x + \sqrt {4 \, x^{2} + 9}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 19, normalized size = 0.76 \begin {gather*} 2\,\mathrm {asinh}\left (\frac {2\,x}{3}\right )-\frac {2\,\sqrt {x^2+\frac {9}{4}}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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