Optimal. Leaf size=77 \[ \frac{8 x^3}{75}-\frac{54 x^2}{125}+\frac{1331 (247 x+443)}{96875 \left (5 x^2+3 x+2\right )}-\frac{10769 \log \left (5 x^2+3 x+2\right )}{6250}+\frac{1466 x}{625}+\frac{3819607 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{96875 \sqrt{31}} \]
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Rubi [A] time = 0.0722482, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {1660, 1657, 634, 618, 204, 628} \[ \frac{8 x^3}{75}-\frac{54 x^2}{125}+\frac{1331 (247 x+443)}{96875 \left (5 x^2+3 x+2\right )}-\frac{10769 \log \left (5 x^2+3 x+2\right )}{6250}+\frac{1466 x}{625}+\frac{3819607 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{96875 \sqrt{31}} \]
Antiderivative was successfully verified.
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Rule 1660
Rule 1657
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{\left (3-x+2 x^2\right )^3}{\left (2+3 x+5 x^2\right )^2} \, dx &=\frac{1331 (443+247 x)}{96875 \left (2+3 x+5 x^2\right )}+\frac{1}{31} \int \frac{\frac{372701}{625}-\frac{230981 x}{625}+\frac{37882 x^2}{125}-\frac{2604 x^3}{25}+\frac{248 x^4}{5}}{2+3 x+5 x^2} \, dx\\ &=\frac{1331 (443+247 x)}{96875 \left (2+3 x+5 x^2\right )}+\frac{1}{31} \int \left (\frac{45446}{625}-\frac{3348 x}{125}+\frac{248 x^2}{25}+\frac{121 (2329-2759 x)}{625 \left (2+3 x+5 x^2\right )}\right ) \, dx\\ &=\frac{1466 x}{625}-\frac{54 x^2}{125}+\frac{8 x^3}{75}+\frac{1331 (443+247 x)}{96875 \left (2+3 x+5 x^2\right )}+\frac{121 \int \frac{2329-2759 x}{2+3 x+5 x^2} \, dx}{19375}\\ &=\frac{1466 x}{625}-\frac{54 x^2}{125}+\frac{8 x^3}{75}+\frac{1331 (443+247 x)}{96875 \left (2+3 x+5 x^2\right )}-\frac{10769 \int \frac{3+10 x}{2+3 x+5 x^2} \, dx}{6250}+\frac{3819607 \int \frac{1}{2+3 x+5 x^2} \, dx}{193750}\\ &=\frac{1466 x}{625}-\frac{54 x^2}{125}+\frac{8 x^3}{75}+\frac{1331 (443+247 x)}{96875 \left (2+3 x+5 x^2\right )}-\frac{10769 \log \left (2+3 x+5 x^2\right )}{6250}-\frac{3819607 \operatorname{Subst}\left (\int \frac{1}{-31-x^2} \, dx,x,3+10 x\right )}{96875}\\ &=\frac{1466 x}{625}-\frac{54 x^2}{125}+\frac{8 x^3}{75}+\frac{1331 (443+247 x)}{96875 \left (2+3 x+5 x^2\right )}+\frac{3819607 \tan ^{-1}\left (\frac{3+10 x}{\sqrt{31}}\right )}{96875 \sqrt{31}}-\frac{10769 \log \left (2+3 x+5 x^2\right )}{6250}\\ \end{align*}
Mathematica [A] time = 0.026893, size = 77, normalized size = 1. \[ \frac{8 x^3}{75}-\frac{54 x^2}{125}+\frac{1331 (247 x+443)}{96875 \left (5 x^2+3 x+2\right )}-\frac{10769 \log \left (5 x^2+3 x+2\right )}{6250}+\frac{1466 x}{625}+\frac{3819607 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{96875 \sqrt{31}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 61, normalized size = 0.8 \begin{align*}{\frac{8\,{x}^{3}}{75}}-{\frac{54\,{x}^{2}}{125}}+{\frac{1466\,x}{625}}-{\frac{121}{625} \left ( -{\frac{2717\,x}{775}}-{\frac{4873}{775}} \right ) \left ({x}^{2}+{\frac{3\,x}{5}}+{\frac{2}{5}} \right ) ^{-1}}-{\frac{10769\,\ln \left ( 5\,{x}^{2}+3\,x+2 \right ) }{6250}}+{\frac{3819607\,\sqrt{31}}{3003125}\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 = 1.47863, size = 84, normalized size = 1.09 \begin{align*} \frac{8}{75} \, x^{3} - \frac{54}{125} \, x^{2} + \frac{3819607}{3003125} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + \frac{1466}{625} \, x + \frac{1331 \,{\left (247 \, x + 443\right )}}{96875 \,{\left (5 \, x^{2} + 3 \, x + 2\right )}} - \frac{10769}{6250} \, \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 = 1.03024, size = 321, normalized size = 4.17 \begin{align*} \frac{9610000 \, x^{5} - 33154500 \, x^{4} + 191815600 \, x^{3} + 22917642 \, \sqrt{31}{\left (5 \, x^{2} + 3 \, x + 2\right )} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + 111226140 \, x^{2} - 31047027 \,{\left (5 \, x^{2} + 3 \, x + 2\right )} \log \left (5 \, x^{2} + 3 \, x + 2\right ) + 145678362 \, x + 109671738}{18018750 \,{\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 = 0.17165, size = 78, normalized size = 1.01 \begin{align*} \frac{8 x^{3}}{75} - \frac{54 x^{2}}{125} + \frac{1466 x}{625} + \frac{328757 x + 589633}{484375 x^{2} + 290625 x + 193750} - \frac{10769 \log{\left (x^{2} + \frac{3 x}{5} + \frac{2}{5} \right )}}{6250} + \frac{3819607 \sqrt{31} \operatorname{atan}{\left (\frac{10 \sqrt{31} x}{31} + \frac{3 \sqrt{31}}{31} \right )}}{3003125} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20751, size = 84, normalized size = 1.09 \begin{align*} \frac{8}{75} \, x^{3} - \frac{54}{125} \, x^{2} + \frac{3819607}{3003125} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + \frac{1466}{625} \, x + \frac{1331 \,{\left (247 \, x + 443\right )}}{96875 \,{\left (5 \, x^{2} + 3 \, x + 2\right )}} - \frac{10769}{6250} \, \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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