Optimal. Leaf size=24 \[ -\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {5+x+x^2}}{\sqrt {2}}\right ) \]
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Rubi [A]
time = 0.01, 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 = {1038, 212}
\begin {gather*} -\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {x^2+x+5}}{\sqrt {2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 1038
Rubi steps
\begin {align*} \int \frac {1+2 x}{\left (3+x+x^2\right ) \sqrt {5+x+x^2}} \, dx &=-\left (2 \text {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 [A]
time = 0.20, size = 24, normalized size = 1.00 \begin {gather*} -\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {5+x+x^2}}{\sqrt {2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.32, size = 20, normalized size = 0.83
method | result | size |
default | \(-\arctanh \left (\frac {\sqrt {x^{2}+x +5}\, \sqrt {2}}{2}\right ) \sqrt {2}\) | \(20\) |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}-2\right ) x^{2}-\RootOf \left (\textit {\_Z}^{2}-2\right ) x +4 \sqrt {x^{2}+x +5}-7 \RootOf \left (\textit {\_Z}^{2}-2\right )}{x^{2}+x +3}\right )}{2}\) | \(58\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 34, normalized size = 1.42 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.13, size = 68, normalized size = 2.83 \begin {gather*} 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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 39 vs.
\(2 (19) = 38\).
time = 4.46, size = 39, normalized size = 1.62 \begin {gather*} -\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 ) \end {gather*}
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 \begin {gather*} -\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {x^2+x+5}}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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