Optimal. Leaf size=42 \[ -\frac{11}{50} \log \left (5 x^2+3 x+2\right )+\frac{2 x}{5}+\frac{143 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{25 \sqrt{31}} \]
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Rubi [A] time = 0.0402874, 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} \[ -\frac{11}{50} \log \left (5 x^2+3 x+2\right )+\frac{2 x}{5}+\frac{143 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{25 \sqrt{31}} \]
Antiderivative was successfully verified.
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Rule 1657
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{3-x+2 x^2}{2+3 x+5 x^2} \, dx &=\int \left (\frac{2}{5}+\frac{11 (1-x)}{5 \left (2+3 x+5 x^2\right )}\right ) \, dx\\ &=\frac{2 x}{5}+\frac{11}{5} \int \frac{1-x}{2+3 x+5 x^2} \, dx\\ &=\frac{2 x}{5}-\frac{11}{50} \int \frac{3+10 x}{2+3 x+5 x^2} \, dx+\frac{143}{50} \int \frac{1}{2+3 x+5 x^2} \, dx\\ &=\frac{2 x}{5}-\frac{11}{50} \log \left (2+3 x+5 x^2\right )-\frac{143}{25} \operatorname{Subst}\left (\int \frac{1}{-31-x^2} \, dx,x,3+10 x\right )\\ &=\frac{2 x}{5}+\frac{143 \tan ^{-1}\left (\frac{3+10 x}{\sqrt{31}}\right )}{25 \sqrt{31}}-\frac{11}{50} \log \left (2+3 x+5 x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0154453, size = 42, normalized size = 1. \[ -\frac{11}{50} \log \left (5 x^2+3 x+2\right )+\frac{2 x}{5}+\frac{143 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{25 \sqrt{31}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 34, normalized size = 0.8 \begin{align*}{\frac{2\,x}{5}}-{\frac{11\,\ln \left ( 5\,{x}^{2}+3\,x+2 \right ) }{50}}+{\frac{143\,\sqrt{31}}{775}\arctan \left ({\frac{ \left ( 3+10\,x \right ) \sqrt{31}}{31}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.4844, size = 45, normalized size = 1.07 \begin{align*} \frac{143}{775} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + \frac{2}{5} \, x - \frac{11}{50} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.83428, size = 119, normalized size = 2.83 \begin{align*} \frac{143}{775} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + \frac{2}{5} \, x - \frac{11}{50} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.15425, size = 49, normalized size = 1.17 \begin{align*} \frac{2 x}{5} - \frac{11 \log{\left (x^{2} + \frac{3 x}{5} + \frac{2}{5} \right )}}{50} + \frac{143 \sqrt{31} \operatorname{atan}{\left (\frac{10 \sqrt{31} x}{31} + \frac{3 \sqrt{31}}{31} \right )}}{775} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16842, size = 45, normalized size = 1.07 \begin{align*} \frac{143}{775} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + \frac{2}{5} \, x - \frac{11}{50} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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