Optimal. Leaf size=24 \[ -\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {x^2+x+5}}{\sqrt {2}}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1024, 206} \[ -\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {x^2+x+5}}{\sqrt {2}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 1024
Rubi steps
\begin {align*} \int \frac {1+2 x}{\left (3+x+x^2\right ) \sqrt {5+x+x^2}} \, dx &=-\left (2 \operatorname {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {5+x+x^2}\right )\right )\\ &=-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {5+x+x^2}}{\sqrt {2}}\right )\\ \end {align*}
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Mathematica [C] time = 0.06, size = 90, normalized size = 3.75 \[ -\frac {\tanh ^{-1}\left (\frac {-2 i \sqrt {11} x-i \sqrt {11}+19}{4 \sqrt {2} \sqrt {x^2+x+5}}\right )+\tanh ^{-1}\left (\frac {2 i \sqrt {11} x+i \sqrt {11}+19}{4 \sqrt {2} \sqrt {x^2+x+5}}\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.10, size = 34, normalized size = 1.42 \[ \frac {1}{2} \, \sqrt {2} \log \left (\frac {x^{2} - 2 \, \sqrt {2} \sqrt {x^{2} + x + 5} + x + 7}{x^{2} + x + 3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.36, size = 39, normalized size = 1.62 \[ -\frac {1}{2} \, \sqrt {2} \log \left (\sqrt {2} + \sqrt {x^{2} + x + 5}\right ) + \frac {1}{2} \, \sqrt {2} \log \left (-\sqrt {2} + \sqrt {x^{2} + x + 5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 20, normalized size = 0.83 \[ -\sqrt {2}\, \arctanh \left (\frac {\sqrt {x^{2}+x +5}\, \sqrt {2}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {2 \, x + 1}{\sqrt {x^{2} + x + 5} {\left (x^{2} + x + 3\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.78, size = 19, normalized size = 0.79 \[ -\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {x^2+x+5}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.81, size = 68, normalized size = 2.83 \[ 2 \left (\begin {cases} - \frac {\sqrt {2} \operatorname {acoth}{\left (\frac {\sqrt {2}}{\sqrt {x^{2} + x + 5}} \right )}}{2} & \text {for}\: \frac {1}{x^{2} + x + 5} > \frac {1}{2} \\- \frac {\sqrt {2} \operatorname {atanh}{\left (\frac {\sqrt {2}}{\sqrt {x^{2} + x + 5}} \right )}}{2} & \text {for}\: \frac {1}{x^{2} + x + 5} < \frac {1}{2} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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