Optimal. Leaf size=160 \[ \frac{9665-1446 x}{512072 \left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )}-\frac{252815 x+2328909}{174616552 \left (5 x^2+3 x+2\right )}+\frac{13-6 x}{1012 \left (2 x^2-x+3\right )^2 \left (5 x^2+3 x+2\right )}+\frac{181 \log \left (2 x^2-x+3\right )}{468512}-\frac{181 \log \left (5 x^2+3 x+2\right )}{468512}+\frac{2038497 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{123921424 \sqrt{23}}+\frac{246757 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{7261936 \sqrt{31}} \]
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Rubi [A] time = 0.160139, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 12, 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} \[ \frac{9665-1446 x}{512072 \left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )}-\frac{252815 x+2328909}{174616552 \left (5 x^2+3 x+2\right )}+\frac{13-6 x}{1012 \left (2 x^2-x+3\right )^2 \left (5 x^2+3 x+2\right )}+\frac{181 \log \left (2 x^2-x+3\right )}{468512}-\frac{181 \log \left (5 x^2+3 x+2\right )}{468512}+\frac{2038497 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{123921424 \sqrt{23}}+\frac{246757 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{7261936 \sqrt{31}} \]
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 )^2} \, dx &=\frac{13-6 x}{1012 \left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )}-\frac{\int \frac{-4081-3168 x+1650 x^2}{\left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )^2} \, dx}{11132}\\ &=\frac{13-6 x}{1012 \left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )}+\frac{9665-1446 x}{512072 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )}-\frac{\int \frac{-10650299-21902089 x+2624490 x^2}{\left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^2} \, dx}{61960712}\\ &=-\frac{2328909+252815 x}{174616552 \left (2+3 x+5 x^2\right )}+\frac{13-6 x}{1012 \left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )}+\frac{9665-1446 x}{512072 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )}-\frac{\int \frac{-14070251228+23317637266 x+1345987060 x^2}{\left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )} \, dx}{464829261424}\\ &=-\frac{2328909+252815 x}{174616552 \left (2+3 x+5 x^2\right )}+\frac{13-6 x}{1012 \left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )}+\frac{9665-1446 x}{512072 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )}-\frac{\int \frac{968672366266-173830777516 x}{3-x+2 x^2} \, dx}{112488681264608}-\frac{\int \frac{-1780781843236+434576943790 x}{2+3 x+5 x^2} \, dx}{112488681264608}\\ &=-\frac{2328909+252815 x}{174616552 \left (2+3 x+5 x^2\right )}+\frac{13-6 x}{1012 \left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )}+\frac{9665-1446 x}{512072 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )}+\frac{181 \int \frac{-1+4 x}{3-x+2 x^2} \, dx}{468512}-\frac{181 \int \frac{3+10 x}{2+3 x+5 x^2} \, dx}{468512}-\frac{2038497 \int \frac{1}{3-x+2 x^2} \, dx}{247842848}+\frac{246757 \int \frac{1}{2+3 x+5 x^2} \, dx}{14523872}\\ &=-\frac{2328909+252815 x}{174616552 \left (2+3 x+5 x^2\right )}+\frac{13-6 x}{1012 \left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )}+\frac{9665-1446 x}{512072 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )}+\frac{181 \log \left (3-x+2 x^2\right )}{468512}-\frac{181 \log \left (2+3 x+5 x^2\right )}{468512}+\frac{2038497 \operatorname{Subst}\left (\int \frac{1}{-23-x^2} \, dx,x,-1+4 x\right )}{123921424}-\frac{246757 \operatorname{Subst}\left (\int \frac{1}{-31-x^2} \, dx,x,3+10 x\right )}{7261936}\\ &=-\frac{2328909+252815 x}{174616552 \left (2+3 x+5 x^2\right )}+\frac{13-6 x}{1012 \left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )}+\frac{9665-1446 x}{512072 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )}+\frac{2038497 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{123921424 \sqrt{23}}+\frac{246757 \tan ^{-1}\left (\frac{3+10 x}{\sqrt{31}}\right )}{7261936 \sqrt{31}}+\frac{181 \log \left (3-x+2 x^2\right )}{468512}-\frac{181 \log \left (2+3 x+5 x^2\right )}{468512}\\ \end{align*}
Mathematica [A] time = 0.101724, size = 136, normalized size = 0.85 \[ \frac{-2923 x-1782}{1408198 \left (2 x^2-x+3\right )}+\frac{1235 x-1474}{330088 \left (5 x^2+3 x+2\right )}+\frac{-14 x-31}{22264 \left (2 x^2-x+3\right )^2}+\frac{181 \log \left (2 x^2-x+3\right )}{468512}-\frac{181 \log \left (5 x^2+3 x+2\right )}{468512}-\frac{2038497 \tan ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{123921424 \sqrt{23}}+\frac{246757 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{7261936 \sqrt{31}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 106, normalized size = 0.7 \begin{align*} -{\frac{1}{234256} \left ( -{\frac{5434\,x}{31}}+{\frac{32428}{155}} \right ) \left ({x}^{2}+{\frac{3\,x}{5}}+{\frac{2}{5}} \right ) ^{-1}}-{\frac{181\,\ln \left ( 5\,{x}^{2}+3\,x+2 \right ) }{468512}}+{\frac{246757\,\sqrt{31}}{225120016}\arctan \left ({\frac{ \left ( 3+10\,x \right ) \sqrt{31}}{31}} \right ) }+{\frac{1}{58564\, \left ( 2\,{x}^{2}-x+3 \right ) ^{2}} \left ( -{\frac{128612\,{x}^{3}}{529}}-{\frac{14102\,{x}^{2}}{529}}-{\frac{173195\,x}{529}}-{\frac{321497}{1058}} \right ) }+{\frac{181\,\ln \left ( 2\,{x}^{2}-x+3 \right ) }{468512}}-{\frac{2038497\,\sqrt{23}}{2850192752}\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 = 1.46593, size = 157, normalized size = 0.98 \begin{align*} \frac{246757}{225120016} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) - \frac{2038497}{2850192752} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{1011260 \, x^{5} + 8304376 \, x^{4} - 5042869 \, x^{3} + 21674311 \, x^{2} - 5887820 \, x + 8829788}{174616552 \,{\left (20 \, x^{6} - 8 \, x^{5} + 61 \, x^{4} + x^{3} + 53 \, x^{2} + 15 \, x + 18\right )}} - \frac{181}{468512} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) + \frac{181}{468512} \, \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 = 1.03044, size = 763, normalized size = 4.77 \begin{align*} -\frac{31725248720 \, x^{5} + 260524883872 \, x^{4} - 158204886268 \, x^{3} - 6004584838 \, \sqrt{31}{\left (20 \, x^{6} - 8 \, x^{5} + 61 \, x^{4} + x^{3} + 53 \, x^{2} + 15 \, x + 18\right )} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + 3917991234 \, \sqrt{23}{\left (20 \, x^{6} - 8 \, x^{5} + 61 \, x^{4} + x^{3} + 53 \, x^{2} + 15 \, x + 18\right )} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + 679966484692 \, x^{2} + 2116340147 \,{\left (20 \, x^{6} - 8 \, x^{5} + 61 \, x^{4} + x^{3} + 53 \, x^{2} + 15 \, x + 18\right )} \log \left (5 \, x^{2} + 3 \, x + 2\right ) - 2116340147 \,{\left (20 \, x^{6} - 8 \, x^{5} + 61 \, x^{4} + x^{3} + 53 \, x^{2} + 15 \, x + 18\right )} \log \left (2 \, x^{2} - x + 3\right ) - 184712689040 \, x + 277008109136}{5478070469344 \,{\left (20 \, x^{6} - 8 \, x^{5} + 61 \, x^{4} + x^{3} + 53 \, x^{2} + 15 \, x + 18\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.534726, size = 143, normalized size = 0.89 \begin{align*} - \frac{1011260 x^{5} + 8304376 x^{4} - 5042869 x^{3} + 21674311 x^{2} - 5887820 x + 8829788}{3492331040 x^{6} - 1396932416 x^{5} + 10651609672 x^{4} + 174616552 x^{3} + 9254677256 x^{2} + 2619248280 x + 3143097936} + \frac{181 \log{\left (x^{2} - \frac{x}{2} + \frac{3}{2} \right )}}{468512} - \frac{181 \log{\left (x^{2} + \frac{3 x}{5} + \frac{2}{5} \right )}}{468512} - \frac{2038497 \sqrt{23} \operatorname{atan}{\left (\frac{4 \sqrt{23} x}{23} - \frac{\sqrt{23}}{23} \right )}}{2850192752} + \frac{246757 \sqrt{31} \operatorname{atan}{\left (\frac{10 \sqrt{31} x}{31} + \frac{3 \sqrt{31}}{31} \right )}}{225120016} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1564, size = 149, normalized size = 0.93 \begin{align*} \frac{246757}{225120016} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) - \frac{2038497}{2850192752} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{1011260 \, x^{5} + 8304376 \, x^{4} - 5042869 \, x^{3} + 21674311 \, x^{2} - 5887820 \, x + 8829788}{174616552 \,{\left (5 \, x^{2} + 3 \, x + 2\right )}{\left (2 \, x^{2} - x + 3\right )}^{2}} - \frac{181}{468512} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) + \frac{181}{468512} \, \log \left (2 \, x^{2} - x + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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