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 = 0.03, antiderivative size = 27, normalized size of antiderivative = 1.00, 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 204
Rule 618
Rule 628
Rule 634
Rule 1657
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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.61, size = 21, normalized size = 0.78 \[ x + \frac {3}{8} \, \arctan \left (x + \frac {1}{2}\right ) + \frac {1}{8} \, \log \left (4 \, x^{2} + 4 \, x + 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 21, normalized size = 0.78 \[ x + \frac {3}{8} \, \arctan \left (x + \frac {1}{2}\right ) + \frac {1}{8} \, \log \left (4 \, x^{2} + 4 \, x + 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 22, normalized size = 0.81 \[ x +\frac {3 \arctan \left (x +\frac {1}{2}\right )}{8}+\frac {\ln \left (4 x^{2}+4 x +5\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.94, size = 21, normalized size = 0.78 \[ x + \frac {3}{8} \, \arctan \left (x + \frac {1}{2}\right ) + \frac {1}{8} \, \log \left (4 \, x^{2} + 4 \, x + 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.80, size = 17, normalized size = 0.63 \[ x+\frac {\ln \left (x^2+x+\frac {5}{4}\right )}{8}+\frac {3\,\mathrm {atan}\left (x+\frac {1}{2}\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 22, normalized size = 0.81 \[ x + \frac {\log {\left (x^{2} + x + \frac {5}{4} \right )}}{8} + \frac {3 \operatorname {atan}{\left (x + \frac {1}{2} \right )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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