Optimal. Leaf size=160 \[ \frac {-2328909-252815 x}{174616552 \left (2+3 x+5 x^2\right )}+\frac {13-6 x}{1012 \left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )}+\frac {9665-1446 x}{512072 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )}+\frac {2038497 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{123921424 \sqrt {23}}+\frac {246757 \tan ^{-1}\left (\frac {3+10 x}{\sqrt {31}}\right )}{7261936 \sqrt {31}}+\frac {181 \log \left (3-x+2 x^2\right )}{468512}-\frac {181 \log \left (2+3 x+5 x^2\right )}{468512} \]
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Rubi [A]
time = 0.10, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {988, 1074,
1086, 648, 632, 210, 642} \begin {gather*} \frac {2038497 \text {ArcTan}\left (\frac {1-4 x}{\sqrt {23}}\right )}{123921424 \sqrt {23}}+\frac {246757 \text {ArcTan}\left (\frac {10 x+3}{\sqrt {31}}\right )}{7261936 \sqrt {31}}+\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 988
Rule 1074
Rule 1086
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 \text {Subst}\left (\int \frac {1}{-23-x^2} \, dx,x,-1+4 x\right )}{123921424}-\frac {246757 \text {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*}
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Mathematica [A]
time = 0.09, size = 136, normalized size = 0.85 \begin {gather*} \frac {-31-14 x}{22264 \left (3-x+2 x^2\right )^2}+\frac {-1782-2923 x}{1408198 \left (3-x+2 x^2\right )}+\frac {-1474+1235 x}{330088 \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 {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 106, normalized size = 0.66
method | result | size |
default | \(\frac {-\frac {128612}{529} x^{3}-\frac {14102}{529} x^{2}-\frac {173195}{529} x -\frac {321497}{1058}}{58564 \left (2 x^{2}-x +3\right )^{2}}+\frac {181 \ln \left (2 x^{2}-x +3\right )}{468512}-\frac {2038497 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{2850192752}-\frac {-\frac {5434 x}{31}+\frac {32428}{155}}{234256 \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right )}-\frac {181 \ln \left (5 x^{2}+3 x +2\right )}{468512}+\frac {246757 \arctan \left (\frac {\left (3+10 x \right ) \sqrt {31}}{31}\right ) \sqrt {31}}{225120016}\) | \(106\) |
risch | \(\frac {-\frac {252815}{43654138} x^{5}-\frac {1038047}{21827069} x^{4}+\frac {5042869}{174616552} x^{3}-\frac {21674311}{174616552} x^{2}+\frac {1471955}{43654138} x -\frac {200677}{3968558}}{\left (2 x^{2}-x +3\right )^{2} \left (5 x^{2}+3 x +2\right )}+\frac {181 \ln \left (16 x^{2}-8 x +24\right )}{468512}-\frac {2038497 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{2850192752}-\frac {181 \ln \left (100 x^{2}+60 x +40\right )}{468512}+\frac {246757 \arctan \left (\frac {\left (3+10 x \right ) \sqrt {31}}{31}\right ) \sqrt {31}}{225120016}\) | \(111\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 116, normalized size = 0.72 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.39, size = 227, normalized size = 1.42 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.17, size = 143, normalized size = 0.89 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.03, size = 110, normalized size = 0.69 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.60, size = 136, normalized size = 0.85 \begin {gather*} -\frac {\frac {50563\,x^5}{174616552}+\frac {1038047\,x^4}{436541380}-\frac {5042869\,x^3}{3492331040}+\frac {21674311\,x^2}{3492331040}-\frac {294391\,x}{174616552}+\frac {200677}{79371160}}{x^6-\frac {2\,x^5}{5}+\frac {61\,x^4}{20}+\frac {x^3}{20}+\frac {53\,x^2}{20}+\frac {3\,x}{4}+\frac {9}{10}}-\ln \left (x+\frac {3}{10}-\frac {\sqrt {31}\,1{}\mathrm {i}}{10}\right )\,\left (\frac {181}{468512}+\frac {\sqrt {31}\,246757{}\mathrm {i}}{450240032}\right )+\ln \left (x+\frac {3}{10}+\frac {\sqrt {31}\,1{}\mathrm {i}}{10}\right )\,\left (-\frac {181}{468512}+\frac {\sqrt {31}\,246757{}\mathrm {i}}{450240032}\right )+\ln \left (x-\frac {1}{4}-\frac {\sqrt {23}\,1{}\mathrm {i}}{4}\right )\,\left (\frac {181}{468512}+\frac {\sqrt {23}\,2038497{}\mathrm {i}}{5700385504}\right )-\ln \left (x-\frac {1}{4}+\frac {\sqrt {23}\,1{}\mathrm {i}}{4}\right )\,\left (-\frac {181}{468512}+\frac {\sqrt {23}\,2038497{}\mathrm {i}}{5700385504}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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