Optimal. Leaf size=115 \[ \frac{3625-746 x}{256036 \left (2 x^2-x+3\right )}+\frac{13-6 x}{1012 \left (2 x^2-x+3\right )^2}-\frac{119 \log \left (2 x^2-x+3\right )}{21296}+\frac{119 \log \left (5 x^2+3 x+2\right )}{21296}-\frac{53403 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{5632792 \sqrt{23}}+\frac{247 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{10648 \sqrt{31}} \]
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Rubi [A] time = 0.1235, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {974, 1060, 1072, 634, 618, 204, 628} \[ \frac{3625-746 x}{256036 \left (2 x^2-x+3\right )}+\frac{13-6 x}{1012 \left (2 x^2-x+3\right )^2}-\frac{119 \log \left (2 x^2-x+3\right )}{21296}+\frac{119 \log \left (5 x^2+3 x+2\right )}{21296}-\frac{53403 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{5632792 \sqrt{23}}+\frac{247 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{10648 \sqrt{31}} \]
Antiderivative was successfully verified.
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Rule 974
Rule 1060
Rule 1072
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{\left (3-x+2 x^2\right )^3 \left (2+3 x+5 x^2\right )} \, dx &=\frac{13-6 x}{1012 \left (3-x+2 x^2\right )^2}-\frac{\int \frac{-3652-1936 x+990 x^2}{\left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )} \, dx}{11132}\\ &=\frac{13-6 x}{1012 \left (3-x+2 x^2\right )^2}+\frac{3625-746 x}{256036 \left (3-x+2 x^2\right )}-\frac{\int \frac{-6551908-7779574 x+902660 x^2}{\left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )} \, dx}{61960712}\\ &=\frac{13-6 x}{1012 \left (3-x+2 x^2\right )^2}+\frac{3625-746 x}{256036 \left (3-x+2 x^2\right )}-\frac{\int \frac{-154867174+335151124 x}{3-x+2 x^2} \, dx}{14994492304}-\frac{\int \frac{-425275796-837877810 x}{2+3 x+5 x^2} \, dx}{14994492304}\\ &=\frac{13-6 x}{1012 \left (3-x+2 x^2\right )^2}+\frac{3625-746 x}{256036 \left (3-x+2 x^2\right )}+\frac{53403 \int \frac{1}{3-x+2 x^2} \, dx}{11265584}-\frac{119 \int \frac{-1+4 x}{3-x+2 x^2} \, dx}{21296}+\frac{119 \int \frac{3+10 x}{2+3 x+5 x^2} \, dx}{21296}+\frac{247 \int \frac{1}{2+3 x+5 x^2} \, dx}{21296}\\ &=\frac{13-6 x}{1012 \left (3-x+2 x^2\right )^2}+\frac{3625-746 x}{256036 \left (3-x+2 x^2\right )}-\frac{119 \log \left (3-x+2 x^2\right )}{21296}+\frac{119 \log \left (2+3 x+5 x^2\right )}{21296}-\frac{53403 \operatorname{Subst}\left (\int \frac{1}{-23-x^2} \, dx,x,-1+4 x\right )}{5632792}-\frac{247 \operatorname{Subst}\left (\int \frac{1}{-31-x^2} \, dx,x,3+10 x\right )}{10648}\\ &=\frac{13-6 x}{1012 \left (3-x+2 x^2\right )^2}+\frac{3625-746 x}{256036 \left (3-x+2 x^2\right )}-\frac{53403 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{5632792 \sqrt{23}}+\frac{247 \tan ^{-1}\left (\frac{3+10 x}{\sqrt{31}}\right )}{10648 \sqrt{31}}-\frac{119 \log \left (3-x+2 x^2\right )}{21296}+\frac{119 \log \left (2+3 x+5 x^2\right )}{21296}\\ \end{align*}
Mathematica [A] time = 0.154692, size = 99, normalized size = 0.86 \[ \frac{713 \left (-\frac{44 \left (1492 x^3-7996 x^2+7381 x-14164\right )}{\left (-2 x^2+x-3\right )^2}-62951 \log \left (2 x^2-x+3\right )+62951 \log \left (5 x^2+3 x+2\right )\right )+3310986 \sqrt{23} \tan ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )+6010498 \sqrt{31} \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{8032361392} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 89, normalized size = 0.8 \begin{align*}{\frac{119\,\ln \left ( 5\,{x}^{2}+3\,x+2 \right ) }{21296}}+{\frac{247\,\sqrt{31}}{330088}\arctan \left ({\frac{ \left ( 3+10\,x \right ) \sqrt{31}}{31}} \right ) }-{\frac{1}{2662\, \left ( 2\,{x}^{2}-x+3 \right ) ^{2}} \left ({\frac{8206\,{x}^{3}}{529}}-{\frac{43978\,{x}^{2}}{529}}+{\frac{81191\,x}{1058}}-{\frac{77902}{529}} \right ) }-{\frac{119\,\ln \left ( 2\,{x}^{2}-x+3 \right ) }{21296}}+{\frac{53403\,\sqrt{23}}{129554216}\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 = 1.43151, size = 132, normalized size = 1.15 \begin{align*} \frac{247}{330088} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + \frac{53403}{129554216} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{1492 \, x^{3} - 7996 \, x^{2} + 7381 \, x - 14164}{256036 \,{\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} + \frac{119}{21296} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) - \frac{119}{21296} \, \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 = 1.02491, size = 556, normalized size = 4.83 \begin{align*} -\frac{46807024 \, x^{3} - 6010498 \, \sqrt{31}{\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) - 3310986 \, \sqrt{23}{\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - 250850512 \, x^{2} - 44884063 \,{\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \log \left (5 \, x^{2} + 3 \, x + 2\right ) + 44884063 \,{\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \log \left (2 \, x^{2} - x + 3\right ) + 231556732 \, x - 444353008}{8032361392 \,{\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.396684, size = 122, normalized size = 1.06 \begin{align*} - \frac{1492 x^{3} - 7996 x^{2} + 7381 x - 14164}{1024144 x^{4} - 1024144 x^{3} + 3328468 x^{2} - 1536216 x + 2304324} - \frac{119 \log{\left (x^{2} - \frac{x}{2} + \frac{3}{2} \right )}}{21296} + \frac{119 \log{\left (x^{2} + \frac{3 x}{5} + \frac{2}{5} \right )}}{21296} + \frac{53403 \sqrt{23} \operatorname{atan}{\left (\frac{4 \sqrt{23} x}{23} - \frac{\sqrt{23}}{23} \right )}}{129554216} + \frac{247 \sqrt{31} \operatorname{atan}{\left (\frac{10 \sqrt{31} x}{31} + \frac{3 \sqrt{31}}{31} \right )}}{330088} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15984, size = 119, normalized size = 1.03 \begin{align*} \frac{247}{330088} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + \frac{53403}{129554216} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{1492 \, x^{3} - 7996 \, x^{2} + 7381 \, x - 14164}{256036 \,{\left (2 \, x^{2} - x + 3\right )}^{2}} + \frac{119}{21296} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) - \frac{119}{21296} \, \log \left (2 \, x^{2} - x + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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