Optimal. Leaf size=43 \[ -\frac{11 (3 x+5)}{46 \left (2 x^2-x+3\right )}-\frac{82 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{23 \sqrt{23}} \]
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Rubi [A] time = 0.0260438, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {1660, 12, 618, 204} \[ -\frac{11 (3 x+5)}{46 \left (2 x^2-x+3\right )}-\frac{82 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{23 \sqrt{23}} \]
Antiderivative was successfully verified.
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Rule 1660
Rule 12
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{2+3 x+5 x^2}{\left (3-x+2 x^2\right )^2} \, dx &=-\frac{11 (5+3 x)}{46 \left (3-x+2 x^2\right )}+\frac{1}{23} \int \frac{41}{3-x+2 x^2} \, dx\\ &=-\frac{11 (5+3 x)}{46 \left (3-x+2 x^2\right )}+\frac{41}{23} \int \frac{1}{3-x+2 x^2} \, dx\\ &=-\frac{11 (5+3 x)}{46 \left (3-x+2 x^2\right )}-\frac{82}{23} \operatorname{Subst}\left (\int \frac{1}{-23-x^2} \, dx,x,-1+4 x\right )\\ &=-\frac{11 (5+3 x)}{46 \left (3-x+2 x^2\right )}-\frac{82 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{23 \sqrt{23}}\\ \end{align*}
Mathematica [A] time = 0.0143532, size = 43, normalized size = 1. \[ \frac{82 \tan ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{23 \sqrt{23}}-\frac{11 (3 x+5)}{46 \left (2 x^2-x+3\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 34, normalized size = 0.8 \begin{align*}{ \left ( -{\frac{33\,x}{92}}-{\frac{55}{92}} \right ) \left ({x}^{2}-{\frac{x}{2}}+{\frac{3}{2}} \right ) ^{-1}}+{\frac{82\,\sqrt{23}}{529}\arctan \left ({\frac{ \left ( -1+4\,x \right ) \sqrt{23}}{23}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.4319, size = 49, normalized size = 1.14 \begin{align*} \frac{82}{529} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{11 \,{\left (3 \, x + 5\right )}}{46 \,{\left (2 \, x^{2} - x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.988498, size = 138, normalized size = 3.21 \begin{align*} \frac{164 \, \sqrt{23}{\left (2 \, x^{2} - x + 3\right )} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - 759 \, x - 1265}{1058 \,{\left (2 \, x^{2} - x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.205915, size = 41, normalized size = 0.95 \begin{align*} - \frac{33 x + 55}{92 x^{2} - 46 x + 138} + \frac{82 \sqrt{23} \operatorname{atan}{\left (\frac{4 \sqrt{23} x}{23} - \frac{\sqrt{23}}{23} \right )}}{529} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13908, size = 49, normalized size = 1.14 \begin{align*} \frac{82}{529} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{11 \,{\left (3 \, x + 5\right )}}{46 \,{\left (2 \, x^{2} - x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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