Optimal. Leaf size=23 \[ 2 \sqrt {x^2+2 x+5}+\sinh ^{-1}\left (\frac {x+1}{2}\right ) \]
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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 = {640, 619, 215} \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 215
Rule 619
Rule 640
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} \operatorname {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 = 0.01, size = 25, normalized size = 1.09 \begin {gather*} 2 \sqrt {x^2+2 x+5}+\sinh ^{-1}\left (\frac {1}{4} (2 x+2)\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.15, size = 35, normalized size = 1.52 \begin {gather*} 2 \sqrt {x^2+2 x+5}-\log \left (\sqrt {x^2+2 x+5}-x-1\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, 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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giac [A] time = 0.17, 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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maple [A] time = 0.06, size = 20, normalized size = 0.87 \begin {gather*} \arcsinh \left (\frac {x}{2}+\frac {1}{2}\right )+2 \sqrt {x^{2}+2 x +5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.89, 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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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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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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