Optimal. Leaf size=42 \[ \frac{11}{8} \log \left (2 x^2-x+3\right )+\frac{5 x}{2}+\frac{33 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{4 \sqrt{23}} \]
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Rubi [A] time = 0.0348584, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {1657, 634, 618, 204, 628} \[ \frac{11}{8} \log \left (2 x^2-x+3\right )+\frac{5 x}{2}+\frac{33 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{4 \sqrt{23}} \]
Antiderivative was successfully verified.
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Rule 1657
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{2+3 x+5 x^2}{3-x+2 x^2} \, dx &=\int \left (\frac{5}{2}-\frac{11 (1-x)}{2 \left (3-x+2 x^2\right )}\right ) \, dx\\ &=\frac{5 x}{2}-\frac{11}{2} \int \frac{1-x}{3-x+2 x^2} \, dx\\ &=\frac{5 x}{2}+\frac{11}{8} \int \frac{-1+4 x}{3-x+2 x^2} \, dx-\frac{33}{8} \int \frac{1}{3-x+2 x^2} \, dx\\ &=\frac{5 x}{2}+\frac{11}{8} \log \left (3-x+2 x^2\right )+\frac{33}{4} \operatorname{Subst}\left (\int \frac{1}{-23-x^2} \, dx,x,-1+4 x\right )\\ &=\frac{5 x}{2}+\frac{33 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{4 \sqrt{23}}+\frac{11}{8} \log \left (3-x+2 x^2\right )\\ \end{align*}
Mathematica [A] time = 0.009824, size = 42, normalized size = 1. \[ \frac{11}{8} \log \left (2 x^2-x+3\right )+\frac{5 x}{2}-\frac{33 \tan ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{4 \sqrt{23}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 34, normalized size = 0.8 \begin{align*}{\frac{5\,x}{2}}+{\frac{11\,\ln \left ( 2\,{x}^{2}-x+3 \right ) }{8}}-{\frac{33\,\sqrt{23}}{92}\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 = 1.43148, size = 45, normalized size = 1.07 \begin{align*} -\frac{33}{92} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \frac{5}{2} \, x + \frac{11}{8} \, \log \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 = 1.05056, size = 112, normalized size = 2.67 \begin{align*} -\frac{33}{92} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \frac{5}{2} \, x + \frac{11}{8} \, \log \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 = 0.123831, size = 46, normalized size = 1.1 \begin{align*} \frac{5 x}{2} + \frac{11 \log{\left (x^{2} - \frac{x}{2} + \frac{3}{2} \right )}}{8} - \frac{33 \sqrt{23} \operatorname{atan}{\left (\frac{4 \sqrt{23} x}{23} - \frac{\sqrt{23}}{23} \right )}}{92} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19791, size = 45, normalized size = 1.07 \begin{align*} -\frac{33}{92} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \frac{5}{2} \, x + \frac{11}{8} \, \log \left (2 \, x^{2} - x + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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