Optimal. Leaf size=23 \[ \sqrt {5+2 x+x^2}-\sinh ^{-1}\left (\frac {1+x}{2}\right ) \]
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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 = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {654, 633, 221}
\begin {gather*} \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 {x}{\sqrt {5+2 x+x^2}} \, dx &=\sqrt {5+2 x+x^2}-\int \frac {1}{\sqrt {5+2 x+x^2}} \, dx\\ &=\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 )\\ &=\sqrt {5+2 x+x^2}-\sinh ^{-1}\left (\frac {1+x}{2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 31, normalized size = 1.35 \begin {gather*} \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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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded while calling a Python object} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.13, size = 20, normalized size = 0.87
method | result | size |
default | \(-\arcsinh \left (\frac {1}{2}+\frac {x}{2}\right )+\sqrt {x^{2}+2 x +5}\) | \(20\) |
risch | \(-\arcsinh \left (\frac {1}{2}+\frac {x}{2}\right )+\sqrt {x^{2}+2 x +5}\) | \(20\) |
trager | \(\sqrt {x^{2}+2 x +5}-\ln \left (x +1+\sqrt {x^{2}+2 x +5}\right )\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.37, size = 19, normalized size = 0.83 \begin {gather*} \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 = 0.33, size = 27, normalized size = 1.17 \begin {gather*} \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 {x}{\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 = 0.00, size = 30, normalized size = 1.30 \begin {gather*} \sqrt {x^{2}+2 x+5}+\ln \left (\sqrt {x^{2}+2 x+5}-x-1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 27, normalized size = 1.17 \begin {gather*} \sqrt {x^2+2\,x+5}-\ln \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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