Optimal. Leaf size=27 \[ \frac{1}{8} \log \left (4 x^2+4 x+5\right )+x+\frac{3}{8} \tan ^{-1}\left (x+\frac{1}{2}\right ) \]
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Rubi [A] time = 0.0276561, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {1657, 634, 618, 204, 628} \[ \frac{1}{8} \log \left (4 x^2+4 x+5\right )+x+\frac{3}{8} \tan ^{-1}\left (x+\frac{1}{2}\right ) \]
Antiderivative was successfully verified.
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Rule 1657
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{7+5 x+4 x^2}{5+4 x+4 x^2} \, dx &=\int \left (1+\frac{2+x}{5+4 x+4 x^2}\right ) \, dx\\ &=x+\int \frac{2+x}{5+4 x+4 x^2} \, dx\\ &=x+\frac{1}{8} \int \frac{4+8 x}{5+4 x+4 x^2} \, dx+\frac{3}{2} \int \frac{1}{5+4 x+4 x^2} \, dx\\ &=x+\frac{1}{8} \log \left (5+4 x+4 x^2\right )-3 \operatorname{Subst}\left (\int \frac{1}{-64-x^2} \, dx,x,4+8 x\right )\\ &=x+\frac{3}{8} \tan ^{-1}\left (\frac{1}{2}+x\right )+\frac{1}{8} \log \left (5+4 x+4 x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0054358, size = 31, normalized size = 1.15 \[ \frac{1}{8} \log \left (4 x^2+4 x+5\right )+x+\frac{3}{8} \tan ^{-1}\left (\frac{1}{2} (2 x+1)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 22, normalized size = 0.8 \begin{align*} x+{\frac{3}{8}\arctan \left ( x+{\frac{1}{2}} \right ) }+{\frac{\ln \left ( 4\,{x}^{2}+4\,x+5 \right ) }{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.52614, size = 28, normalized size = 1.04 \begin{align*} x + \frac{3}{8} \, \arctan \left (x + \frac{1}{2}\right ) + \frac{1}{8} \, \log \left (4 \, x^{2} + 4 \, x + 5\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60817, size = 70, normalized size = 2.59 \begin{align*} x + \frac{3}{8} \, \arctan \left (x + \frac{1}{2}\right ) + \frac{1}{8} \, \log \left (4 \, x^{2} + 4 \, x + 5\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.10131, size = 22, normalized size = 0.81 \begin{align*} x + \frac{\log{\left (x^{2} + x + \frac{5}{4} \right )}}{8} + \frac{3 \operatorname{atan}{\left (x + \frac{1}{2} \right )}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24282, size = 28, normalized size = 1.04 \begin{align*} x + \frac{3}{8} \, \arctan \left (x + \frac{1}{2}\right ) + \frac{1}{8} \, \log \left (4 \, x^{2} + 4 \, x + 5\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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