Optimal. Leaf size=17 \[ \frac {2 \sinh ^{-1}\left (\frac {\sqrt {b x}}{2}\right )}{b} \]
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Rubi [A]
time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {65, 221}
\begin {gather*} \frac {2 \sinh ^{-1}\left (\frac {\sqrt {b x}}{2}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 221
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {b x} \sqrt {4+b x}} \, dx &=\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {4+x^2}} \, dx,x,\sqrt {b x}\right )}{b}\\ &=\frac {2 \sinh ^{-1}\left (\frac {\sqrt {b x}}{2}\right )}{b}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(42\) vs. \(2(17)=34\).
time = 0.04, size = 42, normalized size = 2.47 \begin {gather*} -\frac {2 \sqrt {x} \log \left (-\sqrt {b} \sqrt {x}+\sqrt {4+b x}\right )}{\sqrt {b} \sqrt {b x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(59\) vs.
\(2(13)=26\).
time = 0.14, size = 60, normalized size = 3.53
method | result | size |
meijerg | \(\frac {2 \sqrt {x}\, \arcsinh \left (\frac {\sqrt {b}\, \sqrt {x}}{2}\right )}{\sqrt {b}\, \sqrt {b x}}\) | \(23\) |
default | \(\frac {\sqrt {b x \left (b x +4\right )}\, \ln \left (\frac {b^{2} x +2 b}{\sqrt {b^{2}}}+\sqrt {x^{2} b^{2}+4 b x}\right )}{\sqrt {b x}\, \sqrt {b x +4}\, \sqrt {b^{2}}}\) | \(60\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 32 vs.
\(2 (13) = 26\).
time = 0.29, size = 32, normalized size = 1.88 \begin {gather*} \frac {\log \left (2 \, b^{2} x + 2 \, \sqrt {b^{2} x^{2} + 4 \, b x} b + 4 \, b\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.80, size = 25, normalized size = 1.47 \begin {gather*} -\frac {\log \left (-b x + \sqrt {b x + 4} \sqrt {b x} - 2\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.57, size = 15, normalized size = 0.88 \begin {gather*} \frac {2 \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{2} \right )}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.60, size = 21, normalized size = 1.24 \begin {gather*} -\frac {2 \, \log \left (\sqrt {b x + 4} - \sqrt {b x}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.31, size = 33, normalized size = 1.94 \begin {gather*} -\frac {4\,\mathrm {atan}\left (\frac {b\,\left (\sqrt {b\,x+4}-2\right )}{\sqrt {b\,x}\,\sqrt {-b^2}}\right )}{\sqrt {-b^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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