Optimal. Leaf size=60 \[ -\frac{481 \log \left (x^2+x+1\right )}{5586}-\frac{79}{273 (x+5)}+\frac{200 \log (3-2 x)}{3211}+\frac{2731 \log (x+5)}{24843}+\frac{451 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{2793 \sqrt{3}} \]
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Rubi [A] time = 0.254217, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {6728, 634, 618, 204, 628} \[ -\frac{481 \log \left (x^2+x+1\right )}{5586}-\frac{79}{273 (x+5)}+\frac{200 \log (3-2 x)}{3211}+\frac{2731 \log (x+5)}{24843}+\frac{451 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{2793 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 6728
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{1+16 x}{(5+x)^2 (-3+2 x) \left (1+x+x^2\right )} \, dx &=\int \left (\frac{79}{273 (5+x)^2}+\frac{2731}{24843 (5+x)}+\frac{400}{3211 (-3+2 x)}+\frac{-15-481 x}{2793 \left (1+x+x^2\right )}\right ) \, dx\\ &=-\frac{79}{273 (5+x)}+\frac{200 \log (3-2 x)}{3211}+\frac{2731 \log (5+x)}{24843}+\frac{\int \frac{-15-481 x}{1+x+x^2} \, dx}{2793}\\ &=-\frac{79}{273 (5+x)}+\frac{200 \log (3-2 x)}{3211}+\frac{2731 \log (5+x)}{24843}+\frac{451 \int \frac{1}{1+x+x^2} \, dx}{5586}-\frac{481 \int \frac{1+2 x}{1+x+x^2} \, dx}{5586}\\ &=-\frac{79}{273 (5+x)}+\frac{200 \log (3-2 x)}{3211}+\frac{2731 \log (5+x)}{24843}-\frac{481 \log \left (1+x+x^2\right )}{5586}-\frac{451 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 x\right )}{2793}\\ &=-\frac{79}{273 (5+x)}+\frac{451 \tan ^{-1}\left (\frac{1+2 x}{\sqrt{3}}\right )}{2793 \sqrt{3}}+\frac{200 \log (3-2 x)}{3211}+\frac{2731 \log (5+x)}{24843}-\frac{481 \log \left (1+x+x^2\right )}{5586}\\ \end{align*}
Mathematica [A] time = 0.0522483, size = 54, normalized size = 0.9 \[ \frac{-243867 \log \left (x^2+x+1\right )-\frac{819546}{x+5}+176400 \log (3-2 x)+311334 \log (x+5)+152438 \sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{2832102} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 48, normalized size = 0.8 \begin{align*} -{\frac{79}{1365+273\,x}}+{\frac{2731\,\ln \left ( 5+x \right ) }{24843}}+{\frac{200\,\ln \left ( -3+2\,x \right ) }{3211}}-{\frac{481\,\ln \left ({x}^{2}+x+1 \right ) }{5586}}+{\frac{451\,\sqrt{3}}{8379}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.41567, size = 63, normalized size = 1.05 \begin{align*} \frac{451}{8379} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{79}{273 \,{\left (x + 5\right )}} - \frac{481}{5586} \, \log \left (x^{2} + x + 1\right ) + \frac{200}{3211} \, \log \left (2 \, x - 3\right ) + \frac{2731}{24843} \, \log \left (x + 5\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.00764, size = 236, normalized size = 3.93 \begin{align*} \frac{152438 \, \sqrt{3}{\left (x + 5\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - 243867 \,{\left (x + 5\right )} \log \left (x^{2} + x + 1\right ) + 176400 \,{\left (x + 5\right )} \log \left (2 \, x - 3\right ) + 311334 \,{\left (x + 5\right )} \log \left (x + 5\right ) - 819546}{2832102 \,{\left (x + 5\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.227584, size = 63, normalized size = 1.05 \begin{align*} \frac{200 \log{\left (x - \frac{3}{2} \right )}}{3211} + \frac{2731 \log{\left (x + 5 \right )}}{24843} - \frac{481 \log{\left (x^{2} + x + 1 \right )}}{5586} + \frac{451 \sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} + \frac{\sqrt{3}}{3} \right )}}{8379} - \frac{79}{273 x + 1365} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0654, size = 81, normalized size = 1.35 \begin{align*} \frac{451}{8379} \, \sqrt{3} \arctan \left (-\sqrt{3}{\left (\frac{14}{x + 5} - 3\right )}\right ) - \frac{79}{273 \,{\left (x + 5\right )}} - \frac{481}{5586} \, \log \left (-\frac{9}{x + 5} + \frac{21}{{\left (x + 5\right )}^{2}} + 1\right ) + \frac{200}{3211} \, \log \left ({\left | -\frac{13}{x + 5} + 2 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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