Optimal. Leaf size=70 \[ \frac{8 x^5}{25}-\frac{21 x^4}{25}+\frac{1222 x^3}{375}-\frac{7451 x^2}{1250}-\frac{158389 \log \left (5 x^2+3 x+2\right )}{31250}+\frac{49508 x}{3125}+\frac{328757 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{15625 \sqrt{31}} \]
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Rubi [A] time = 0.0529201, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1657, 634, 618, 204, 628} \[ \frac{8 x^5}{25}-\frac{21 x^4}{25}+\frac{1222 x^3}{375}-\frac{7451 x^2}{1250}-\frac{158389 \log \left (5 x^2+3 x+2\right )}{31250}+\frac{49508 x}{3125}+\frac{328757 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{15625 \sqrt{31}} \]
Antiderivative was successfully verified.
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Rule 1657
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{\left (3-x+2 x^2\right )^3}{2+3 x+5 x^2} \, dx &=\int \left (\frac{49508}{3125}-\frac{7451 x}{625}+\frac{1222 x^2}{125}-\frac{84 x^3}{25}+\frac{8 x^4}{5}-\frac{1331 (11+119 x)}{3125 \left (2+3 x+5 x^2\right )}\right ) \, dx\\ &=\frac{49508 x}{3125}-\frac{7451 x^2}{1250}+\frac{1222 x^3}{375}-\frac{21 x^4}{25}+\frac{8 x^5}{25}-\frac{1331 \int \frac{11+119 x}{2+3 x+5 x^2} \, dx}{3125}\\ &=\frac{49508 x}{3125}-\frac{7451 x^2}{1250}+\frac{1222 x^3}{375}-\frac{21 x^4}{25}+\frac{8 x^5}{25}-\frac{158389 \int \frac{3+10 x}{2+3 x+5 x^2} \, dx}{31250}+\frac{328757 \int \frac{1}{2+3 x+5 x^2} \, dx}{31250}\\ &=\frac{49508 x}{3125}-\frac{7451 x^2}{1250}+\frac{1222 x^3}{375}-\frac{21 x^4}{25}+\frac{8 x^5}{25}-\frac{158389 \log \left (2+3 x+5 x^2\right )}{31250}-\frac{328757 \operatorname{Subst}\left (\int \frac{1}{-31-x^2} \, dx,x,3+10 x\right )}{15625}\\ &=\frac{49508 x}{3125}-\frac{7451 x^2}{1250}+\frac{1222 x^3}{375}-\frac{21 x^4}{25}+\frac{8 x^5}{25}+\frac{328757 \tan ^{-1}\left (\frac{3+10 x}{\sqrt{31}}\right )}{15625 \sqrt{31}}-\frac{158389 \log \left (2+3 x+5 x^2\right )}{31250}\\ \end{align*}
Mathematica [A] time = 0.0224648, size = 63, normalized size = 0.9 \[ \frac{31 \left (5 x \left (6000 x^4-15750 x^3+61100 x^2-111765 x+297048\right )-475167 \log \left (5 x^2+3 x+2\right )\right )+1972542 \sqrt{31} \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{2906250} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 54, normalized size = 0.8 \begin{align*}{\frac{49508\,x}{3125}}-{\frac{7451\,{x}^{2}}{1250}}+{\frac{1222\,{x}^{3}}{375}}-{\frac{21\,{x}^{4}}{25}}+{\frac{8\,{x}^{5}}{25}}-{\frac{158389\,\ln \left ( 5\,{x}^{2}+3\,x+2 \right ) }{31250}}+{\frac{328757\,\sqrt{31}}{484375}\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 = 1.46638, size = 72, normalized size = 1.03 \begin{align*} \frac{8}{25} \, x^{5} - \frac{21}{25} \, x^{4} + \frac{1222}{375} \, x^{3} - \frac{7451}{1250} \, x^{2} + \frac{328757}{484375} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + \frac{49508}{3125} \, x - \frac{158389}{31250} \, \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 = 0.991387, size = 219, normalized size = 3.13 \begin{align*} \frac{8}{25} \, x^{5} - \frac{21}{25} \, x^{4} + \frac{1222}{375} \, x^{3} - \frac{7451}{1250} \, x^{2} + \frac{328757}{484375} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + \frac{49508}{3125} \, x - \frac{158389}{31250} \, \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 = 0.149317, size = 76, normalized size = 1.09 \begin{align*} \frac{8 x^{5}}{25} - \frac{21 x^{4}}{25} + \frac{1222 x^{3}}{375} - \frac{7451 x^{2}}{1250} + \frac{49508 x}{3125} - \frac{158389 \log{\left (x^{2} + \frac{3 x}{5} + \frac{2}{5} \right )}}{31250} + \frac{328757 \sqrt{31} \operatorname{atan}{\left (\frac{10 \sqrt{31} x}{31} + \frac{3 \sqrt{31}}{31} \right )}}{484375} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.187, size = 72, normalized size = 1.03 \begin{align*} \frac{8}{25} \, x^{5} - \frac{21}{25} \, x^{4} + \frac{1222}{375} \, x^{3} - \frac{7451}{1250} \, x^{2} + \frac{328757}{484375} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + \frac{49508}{3125} \, x - \frac{158389}{31250} \, \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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