Optimal. Leaf size=14 \[ \log \left (x^2+4 x+5\right )+\tan ^{-1}(x+2) \]
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Rubi [A] time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {634, 618, 204, 628} \begin {gather*} \log \left (x^2+4 x+5\right )+\tan ^{-1}(x+2) \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {5+2 x}{5+4 x+x^2} \, dx &=\int \frac {1}{5+4 x+x^2} \, dx+\int \frac {4+2 x}{5+4 x+x^2} \, dx\\ &=\log \left (5+4 x+x^2\right )-2 \operatorname {Subst}\left (\int \frac {1}{-4-x^2} \, dx,x,4+2 x\right )\\ &=\tan ^{-1}(2+x)+\log \left (5+4 x+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} \log \left (x^2+4 x+5\right )+\tan ^{-1}(x+2) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {5+2 x}{5+4 x+x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.41, size = 14, normalized size = 1.00 \begin {gather*} \arctan \left (x + 2\right ) + \log \left (x^{2} + 4 \, x + 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 14, normalized size = 1.00 \begin {gather*} \arctan \left (x + 2\right ) + \log \left (x^{2} + 4 \, x + 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 15, normalized size = 1.07 \begin {gather*} \arctan \left (x +2\right )+\ln \left (x^{2}+4 x +5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.42, size = 14, normalized size = 1.00 \begin {gather*} \arctan \left (x + 2\right ) + \log \left (x^{2} + 4 \, x + 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.18, size = 14, normalized size = 1.00 \begin {gather*} \mathrm {atan}\left (x+2\right )+\ln \left (x^2+4\,x+5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 14, normalized size = 1.00 \begin {gather*} \log {\left (x^{2} + 4 x + 5 \right )} + \operatorname {atan}{\left (x + 2 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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