Optimal. Leaf size=63 \[ -\frac{423 x+1367}{3500 \left (5 x^2+2 x+3\right )}-\frac{41}{250} \log \left (5 x^2+2 x+3\right )+\frac{4 x}{25}+\frac{1313 \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{3500 \sqrt{14}} \]
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Rubi [A] time = 0.0773302, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {1660, 1657, 634, 618, 204, 628} \[ -\frac{423 x+1367}{3500 \left (5 x^2+2 x+3\right )}-\frac{41}{250} \log \left (5 x^2+2 x+3\right )+\frac{4 x}{25}+\frac{1313 \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{3500 \sqrt{14}} \]
Antiderivative was successfully verified.
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Rule 1660
Rule 1657
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{2+x+3 x^2-5 x^3+4 x^4}{\left (3+2 x+5 x^2\right )^2} \, dx &=-\frac{1367+423 x}{3500 \left (3+2 x+5 x^2\right )}+\frac{1}{56} \int \frac{\frac{738}{25}-\frac{1848 x}{25}+\frac{224 x^2}{5}}{3+2 x+5 x^2} \, dx\\ &=-\frac{1367+423 x}{3500 \left (3+2 x+5 x^2\right )}+\frac{1}{56} \int \left (\frac{224}{25}+\frac{2 (33-1148 x)}{25 \left (3+2 x+5 x^2\right )}\right ) \, dx\\ &=\frac{4 x}{25}-\frac{1367+423 x}{3500 \left (3+2 x+5 x^2\right )}+\frac{1}{700} \int \frac{33-1148 x}{3+2 x+5 x^2} \, dx\\ &=\frac{4 x}{25}-\frac{1367+423 x}{3500 \left (3+2 x+5 x^2\right )}-\frac{41}{250} \int \frac{2+10 x}{3+2 x+5 x^2} \, dx+\frac{1313 \int \frac{1}{3+2 x+5 x^2} \, dx}{3500}\\ &=\frac{4 x}{25}-\frac{1367+423 x}{3500 \left (3+2 x+5 x^2\right )}-\frac{41}{250} \log \left (3+2 x+5 x^2\right )-\frac{1313 \operatorname{Subst}\left (\int \frac{1}{-56-x^2} \, dx,x,2+10 x\right )}{1750}\\ &=\frac{4 x}{25}-\frac{1367+423 x}{3500 \left (3+2 x+5 x^2\right )}+\frac{1313 \tan ^{-1}\left (\frac{1+5 x}{\sqrt{14}}\right )}{3500 \sqrt{14}}-\frac{41}{250} \log \left (3+2 x+5 x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0367986, size = 59, normalized size = 0.94 \[ \frac{-\frac{14 (423 x+1367)}{5 x^2+2 x+3}-8036 \log \left (5 x^2+2 x+3\right )+7840 x+1313 \sqrt{14} \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{49000} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 51, normalized size = 0.8 \begin{align*}{\frac{4\,x}{25}}-{\frac{1}{25} \left ({\frac{423\,x}{700}}+{\frac{1367}{700}} \right ) \left ({x}^{2}+{\frac{2\,x}{5}}+{\frac{3}{5}} \right ) ^{-1}}-{\frac{41\,\ln \left ( 5\,{x}^{2}+2\,x+3 \right ) }{250}}+{\frac{1313\,\sqrt{14}}{49000}\arctan \left ({\frac{ \left ( 10\,x+2 \right ) \sqrt{14}}{28}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50011, size = 70, normalized size = 1.11 \begin{align*} \frac{1313}{49000} \, \sqrt{14} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) + \frac{4}{25} \, x - \frac{423 \, x + 1367}{3500 \,{\left (5 \, x^{2} + 2 \, x + 3\right )}} - \frac{41}{250} \, \log \left (5 \, x^{2} + 2 \, x + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.23975, size = 244, normalized size = 3.87 \begin{align*} \frac{39200 \, x^{3} + 1313 \, \sqrt{14}{\left (5 \, x^{2} + 2 \, x + 3\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) + 15680 \, x^{2} - 8036 \,{\left (5 \, x^{2} + 2 \, x + 3\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) + 17598 \, x - 19138}{49000 \,{\left (5 \, x^{2} + 2 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.171049, size = 63, normalized size = 1. \begin{align*} \frac{4 x}{25} - \frac{423 x + 1367}{17500 x^{2} + 7000 x + 10500} - \frac{41 \log{\left (x^{2} + \frac{2 x}{5} + \frac{3}{5} \right )}}{250} + \frac{1313 \sqrt{14} \operatorname{atan}{\left (\frac{5 \sqrt{14} x}{14} + \frac{\sqrt{14}}{14} \right )}}{49000} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16164, size = 70, normalized size = 1.11 \begin{align*} \frac{1313}{49000} \, \sqrt{14} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) + \frac{4}{25} \, x - \frac{423 \, x + 1367}{3500 \,{\left (5 \, x^{2} + 2 \, x + 3\right )}} - \frac{41}{250} \, \log \left (5 \, x^{2} + 2 \, x + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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