Optimal. Leaf size=23 \[ 2 \sqrt {5+2 x+x^2}+\sinh ^{-1}\left (\frac {1+x}{2}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {654, 633, 221}
\begin {gather*} 2 \sqrt {x^2+2 x+5}+\sinh ^{-1}\left (\frac {x+1}{2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 633
Rule 654
Rubi steps
\begin {align*} \int \frac {3+2 x}{\sqrt {5+2 x+x^2}} \, dx &=2 \sqrt {5+2 x+x^2}+\int \frac {1}{\sqrt {5+2 x+x^2}} \, dx\\ &=2 \sqrt {5+2 x+x^2}+\frac {1}{4} \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{16}}} \, dx,x,2+2 x\right )\\ &=2 \sqrt {5+2 x+x^2}+\sinh ^{-1}\left (\frac {1+x}{2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 35, normalized size = 1.52 \begin {gather*} 2 \sqrt {5+2 x+x^2}-\log \left (-1-x+\sqrt {5+2 x+x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.72, size = 20, normalized size = 0.87
method | result | size |
default | \(\arcsinh \left (\frac {x}{2}+\frac {1}{2}\right )+2 \sqrt {x^{2}+2 x +5}\) | \(20\) |
risch | \(\arcsinh \left (\frac {x}{2}+\frac {1}{2}\right )+2 \sqrt {x^{2}+2 x +5}\) | \(20\) |
trager | \(2 \sqrt {x^{2}+2 x +5}-\ln \left (\sqrt {x^{2}+2 x +5}-1-x \right )\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 19, normalized size = 0.83 \begin {gather*} 2 \, \sqrt {x^{2} + 2 \, x + 5} + \operatorname {arsinh}\left (\frac {1}{2} \, x + \frac {1}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.20, size = 31, normalized size = 1.35 \begin {gather*} 2 \, \sqrt {x^{2} + 2 \, x + 5} - \log \left (-x + \sqrt {x^{2} + 2 \, x + 5} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 x + 3}{\sqrt {x^{2} + 2 x + 5}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.23, size = 31, normalized size = 1.35 \begin {gather*} 2 \, \sqrt {x^{2} + 2 \, x + 5} - \log \left (-x + \sqrt {x^{2} + 2 \, x + 5} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.30, size = 27, normalized size = 1.17 \begin {gather*} \ln \left (x+\sqrt {x^2+2\,x+5}+1\right )+2\,\sqrt {x^2+2\,x+5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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