Optimal. Leaf size=59 \[ \frac{\tan ^{-1}\left (\frac{800 x^3-40 x^2+30 x+57}{6 \sqrt{11}}\right )}{2 \sqrt{11}}-\frac{\tan ^{-1}\left (\frac{7-40 x}{5 \sqrt{11}}\right )}{2 \sqrt{11}} \]
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Rubi [A] time = 0.031717, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.03, Rules used = {2090} \[ \frac{\tan ^{-1}\left (\frac{800 x^3-40 x^2+30 x+57}{6 \sqrt{11}}\right )}{2 \sqrt{11}}-\frac{\tan ^{-1}\left (\frac{7-40 x}{5 \sqrt{11}}\right )}{2 \sqrt{11}} \]
Antiderivative was successfully verified.
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Rule 2090
Rubi steps
\begin{align*} \int \frac{3+12 x+20 x^2}{9+24 x-12 x^2+80 x^3+320 x^4} \, dx &=-\frac{\tan ^{-1}\left (\frac{7-40 x}{5 \sqrt{11}}\right )}{2 \sqrt{11}}+\frac{\tan ^{-1}\left (\frac{57+30 x-40 x^2+800 x^3}{6 \sqrt{11}}\right )}{2 \sqrt{11}}\\ \end{align*}
Mathematica [C] time = 0.0184698, size = 86, normalized size = 1.46 \[ \frac{1}{8} \text{RootSum}\left [320 \text{$\#$1}^4+80 \text{$\#$1}^3-12 \text{$\#$1}^2+24 \text{$\#$1}+9\& ,\frac{20 \text{$\#$1}^2 \log (x-\text{$\#$1})+12 \text{$\#$1} \log (x-\text{$\#$1})+3 \log (x-\text{$\#$1})}{160 \text{$\#$1}^3+30 \text{$\#$1}^2-3 \text{$\#$1}+3}\& \right ] \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 52, normalized size = 0.9 \begin{align*}{\frac{\sqrt{11}}{22}\arctan \left ({\frac{ \left ( 40\,x-7 \right ) \sqrt{11}}{55}} \right ) }+{\frac{\sqrt{11}}{22}\arctan \left ( -{\frac{20\,\sqrt{11}{x}^{2}}{33}}+{\frac{5\,\sqrt{11}x}{11}}+{\frac{19\,\sqrt{11}}{22}}+{\frac{400\,\sqrt{11}{x}^{3}}{33}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{20 \, x^{2} + 12 \, x + 3}{320 \, x^{4} + 80 \, x^{3} - 12 \, x^{2} + 24 \, x + 9}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.9788, size = 158, normalized size = 2.68 \begin{align*} \frac{1}{22} \, \sqrt{11} \arctan \left (\frac{1}{66} \, \sqrt{11}{\left (800 \, x^{3} - 40 \, x^{2} + 30 \, x + 57\right )}\right ) + \frac{1}{22} \, \sqrt{11} \arctan \left (\frac{1}{55} \, \sqrt{11}{\left (40 \, x - 7\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.160272, size = 73, normalized size = 1.24 \begin{align*} \frac{\sqrt{11} \left (2 \operatorname{atan}{\left (\frac{8 \sqrt{11} x}{11} - \frac{7 \sqrt{11}}{55} \right )} + 2 \operatorname{atan}{\left (\frac{400 \sqrt{11} x^{3}}{33} - \frac{20 \sqrt{11} x^{2}}{33} + \frac{5 \sqrt{11} x}{11} + \frac{19 \sqrt{11}}{22} \right )}\right )}{44} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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