Optimal. Leaf size=64 \[ -\frac{423 x+1367}{7000 \left (5 x^2+2 x+3\right )^2}+\frac{11015 x+34347}{196000 \left (5 x^2+2 x+3\right )}+\frac{339 \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{1568 \sqrt{14}} \]
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Rubi [A] time = 0.0497837, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {1660, 12, 618, 204} \[ -\frac{423 x+1367}{7000 \left (5 x^2+2 x+3\right )^2}+\frac{11015 x+34347}{196000 \left (5 x^2+2 x+3\right )}+\frac{339 \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{1568 \sqrt{14}} \]
Antiderivative was successfully verified.
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Rule 1660
Rule 12
Rule 618
Rule 204
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 )^3} \, dx &=-\frac{1367+423 x}{7000 \left (3+2 x+5 x^2\right )^2}+\frac{1}{112} \int \frac{\frac{6534}{125}-\frac{3696 x}{25}+\frac{448 x^2}{5}}{\left (3+2 x+5 x^2\right )^2} \, dx\\ &=-\frac{1367+423 x}{7000 \left (3+2 x+5 x^2\right )^2}+\frac{34347+11015 x}{196000 \left (3+2 x+5 x^2\right )}+\frac{\int \frac{1356}{3+2 x+5 x^2} \, dx}{6272}\\ &=-\frac{1367+423 x}{7000 \left (3+2 x+5 x^2\right )^2}+\frac{34347+11015 x}{196000 \left (3+2 x+5 x^2\right )}+\frac{339 \int \frac{1}{3+2 x+5 x^2} \, dx}{1568}\\ &=-\frac{1367+423 x}{7000 \left (3+2 x+5 x^2\right )^2}+\frac{34347+11015 x}{196000 \left (3+2 x+5 x^2\right )}-\frac{339}{784} \operatorname{Subst}\left (\int \frac{1}{-56-x^2} \, dx,x,2+10 x\right )\\ &=-\frac{1367+423 x}{7000 \left (3+2 x+5 x^2\right )^2}+\frac{34347+11015 x}{196000 \left (3+2 x+5 x^2\right )}+\frac{339 \tan ^{-1}\left (\frac{1+5 x}{\sqrt{14}}\right )}{1568 \sqrt{14}}\\ \end{align*}
Mathematica [A] time = 0.0382764, size = 53, normalized size = 0.83 \[ \frac{\frac{14 \left (11015 x^3+38753 x^2+17979 x+12953\right )}{\left (5 x^2+2 x+3\right )^2}+8475 \sqrt{14} \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{548800} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 47, normalized size = 0.7 \begin{align*} 25\,{\frac{1}{ \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{2}} \left ({\frac{2203\,{x}^{3}}{196000}}+{\frac{38753\,{x}^{2}}{980000}}+{\frac{17979\,x}{980000}}+{\frac{12953}{980000}} \right ) }+{\frac{339\,\sqrt{14}}{21952}\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 = 1.50018, size = 76, normalized size = 1.19 \begin{align*} \frac{339}{21952} \, \sqrt{14} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) + \frac{11015 \, x^{3} + 38753 \, x^{2} + 17979 \, x + 12953}{39200 \,{\left (25 \, x^{4} + 20 \, x^{3} + 34 \, x^{2} + 12 \, x + 9\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.29036, size = 243, normalized size = 3.8 \begin{align*} \frac{154210 \, x^{3} + 8475 \, \sqrt{14}{\left (25 \, x^{4} + 20 \, x^{3} + 34 \, x^{2} + 12 \, x + 9\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) + 542542 \, x^{2} + 251706 \, x + 181342}{548800 \,{\left (25 \, x^{4} + 20 \, x^{3} + 34 \, x^{2} + 12 \, x + 9\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.184545, size = 61, normalized size = 0.95 \begin{align*} \frac{11015 x^{3} + 38753 x^{2} + 17979 x + 12953}{980000 x^{4} + 784000 x^{3} + 1332800 x^{2} + 470400 x + 352800} + \frac{339 \sqrt{14} \operatorname{atan}{\left (\frac{5 \sqrt{14} x}{14} + \frac{\sqrt{14}}{14} \right )}}{21952} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14812, size = 62, normalized size = 0.97 \begin{align*} \frac{339}{21952} \, \sqrt{14} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) + \frac{11015 \, x^{3} + 38753 \, x^{2} + 17979 \, x + 12953}{39200 \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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