Optimal. Leaf size=127 \[ -\frac {25 (117-137 x)}{172546 \left (2+3 x+5 x^2\right )}+\frac {13-6 x}{506 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )}+\frac {2769 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{122452 \sqrt {23}}+\frac {12643 \tan ^{-1}\left (\frac {3+10 x}{\sqrt {31}}\right )}{165044 \sqrt {31}}+\frac {19 \log \left (3-x+2 x^2\right )}{10648}-\frac {19 \log \left (2+3 x+5 x^2\right )}{10648} \]
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Rubi [A]
time = 0.08, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps
used = 11, 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 {2769 \text {ArcTan}\left (\frac {1-4 x}{\sqrt {23}}\right )}{122452 \sqrt {23}}+\frac {12643 \text {ArcTan}\left (\frac {10 x+3}{\sqrt {31}}\right )}{165044 \sqrt {31}}-\frac {25 (117-137 x)}{172546 \left (5 x^2+3 x+2\right )}+\frac {13-6 x}{506 \left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )}+\frac {19 \log \left (2 x^2-x+3\right )}{10648}-\frac {19 \log \left (5 x^2+3 x+2\right )}{10648} \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 )^2 \left (2+3 x+5 x^2\right )^2} \, dx &=\frac {13-6 x}{506 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )}-\frac {\int \frac {-2321-2299 x+990 x^2}{\left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^2} \, dx}{5566}\\ &=-\frac {25 (117-137 x)}{172546 \left (2+3 x+5 x^2\right )}+\frac {13-6 x}{506 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )}-\frac {\int \frac {-3034196+4654870 x-1657700 x^2}{\left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )} \, dx}{41756132}\\ &=-\frac {25 (117-137 x)}{172546 \left (2+3 x+5 x^2\right )}+\frac {13-6 x}{506 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )}-\frac {\int \frac {132282766-72124228 x}{3-x+2 x^2} \, dx}{10104983944}-\frac {\int \frac {-332946988+180310570 x}{2+3 x+5 x^2} \, dx}{10104983944}\\ &=-\frac {25 (117-137 x)}{172546 \left (2+3 x+5 x^2\right )}+\frac {13-6 x}{506 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )}+\frac {19 \int \frac {-1+4 x}{3-x+2 x^2} \, dx}{10648}-\frac {19 \int \frac {3+10 x}{2+3 x+5 x^2} \, dx}{10648}-\frac {2769 \int \frac {1}{3-x+2 x^2} \, dx}{244904}+\frac {12643 \int \frac {1}{2+3 x+5 x^2} \, dx}{330088}\\ &=-\frac {25 (117-137 x)}{172546 \left (2+3 x+5 x^2\right )}+\frac {13-6 x}{506 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )}+\frac {19 \log \left (3-x+2 x^2\right )}{10648}-\frac {19 \log \left (2+3 x+5 x^2\right )}{10648}+\frac {2769 \text {Subst}\left (\int \frac {1}{-23-x^2} \, dx,x,-1+4 x\right )}{122452}-\frac {12643 \text {Subst}\left (\int \frac {1}{-31-x^2} \, dx,x,3+10 x\right )}{165044}\\ &=-\frac {25 (117-137 x)}{172546 \left (2+3 x+5 x^2\right )}+\frac {13-6 x}{506 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )}+\frac {2769 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{122452 \sqrt {23}}+\frac {12643 \tan ^{-1}\left (\frac {3+10 x}{\sqrt {31}}\right )}{165044 \sqrt {31}}+\frac {19 \log \left (3-x+2 x^2\right )}{10648}-\frac {19 \log \left (2+3 x+5 x^2\right )}{10648}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 106, normalized size = 0.83 \begin {gather*} \frac {\frac {31372 \left (-4342+11154 x-9275 x^2+6850 x^3\right )}{6+7 x+16 x^2+x^3+10 x^4}-5322018 \sqrt {23} \tan ^{-1}\left (\frac {-1+4 x}{\sqrt {23}}\right )+13376294 \sqrt {31} \tan ^{-1}\left (\frac {3+10 x}{\sqrt {31}}\right )+9659011 \log \left (3-x+2 x^2\right )-9659011 \log \left (2+3 x+5 x^2\right )}{5413113112} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 94, normalized size = 0.74
method | result | size |
default | \(\frac {-\frac {77 x}{23}-\frac {341}{46}}{5324 x^{2}-2662 x +7986}+\frac {19 \ln \left (2 x^{2}-x +3\right )}{10648}-\frac {2769 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{2816396}-\frac {-\frac {759 x}{31}+\frac {1078}{155}}{5324 \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right )}-\frac {19 \ln \left (5 x^{2}+3 x +2\right )}{10648}+\frac {12643 \arctan \left (\frac {\left (3+10 x \right ) \sqrt {31}}{31}\right ) \sqrt {31}}{5116364}\) | \(94\) |
risch | \(\frac {\frac {3425}{86273} x^{3}-\frac {9275}{172546} x^{2}+\frac {507}{7843} x -\frac {2171}{86273}}{\left (2 x^{2}-x +3\right ) \left (5 x^{2}+3 x +2\right )}+\frac {19 \ln \left (16 x^{2}-8 x +24\right )}{10648}-\frac {2769 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{2816396}-\frac {19 \ln \left (100 x^{2}+60 x +40\right )}{10648}+\frac {12643 \arctan \left (\frac {\left (3+10 x \right ) \sqrt {31}}{31}\right ) \sqrt {31}}{5116364}\) | \(101\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 96, normalized size = 0.76 \begin {gather*} \frac {12643}{5116364} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) - \frac {2769}{2816396} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {6850 \, x^{3} - 9275 \, x^{2} + 11154 \, x - 4342}{172546 \, {\left (10 \, x^{4} + x^{3} + 16 \, x^{2} + 7 \, x + 6\right )}} - \frac {19}{10648} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) + \frac {19}{10648} \, \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 = 1.45, size = 167, normalized size = 1.31 \begin {gather*} \frac {214898200 \, x^{3} + 13376294 \, \sqrt {31} {\left (10 \, x^{4} + x^{3} + 16 \, x^{2} + 7 \, x + 6\right )} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) - 5322018 \, \sqrt {23} {\left (10 \, x^{4} + x^{3} + 16 \, x^{2} + 7 \, x + 6\right )} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - 290975300 \, x^{2} - 9659011 \, {\left (10 \, x^{4} + x^{3} + 16 \, x^{2} + 7 \, x + 6\right )} \log \left (5 \, x^{2} + 3 \, x + 2\right ) + 9659011 \, {\left (10 \, x^{4} + x^{3} + 16 \, x^{2} + 7 \, x + 6\right )} \log \left (2 \, x^{2} - x + 3\right ) + 349923288 \, x - 136217224}{5413113112 \, {\left (10 \, x^{4} + x^{3} + 16 \, x^{2} + 7 \, x + 6\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.15, size = 122, normalized size = 0.96 \begin {gather*} \frac {6850 x^{3} - 9275 x^{2} + 11154 x - 4342}{1725460 x^{4} + 172546 x^{3} + 2760736 x^{2} + 1207822 x + 1035276} + \frac {19 \log {\left (x^{2} - \frac {x}{2} + \frac {3}{2} \right )}}{10648} - \frac {19 \log {\left (x^{2} + \frac {3 x}{5} + \frac {2}{5} \right )}}{10648} - \frac {2769 \sqrt {23} \operatorname {atan}{\left (\frac {4 \sqrt {23} x}{23} - \frac {\sqrt {23}}{23} \right )}}{2816396} + \frac {12643 \sqrt {31} \operatorname {atan}{\left (\frac {10 \sqrt {31} x}{31} + \frac {3 \sqrt {31}}{31} \right )}}{5116364} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.65, size = 96, normalized size = 0.76 \begin {gather*} \frac {12643}{5116364} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) - \frac {2769}{2816396} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {6850 \, x^{3} - 9275 \, x^{2} + 11154 \, x - 4342}{172546 \, {\left (10 \, x^{4} + x^{3} + 16 \, x^{2} + 7 \, x + 6\right )}} - \frac {19}{10648} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) + \frac {19}{10648} \, \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 = 0.18, size = 115, normalized size = 0.91 \begin {gather*} \ln \left (x-\frac {1}{4}-\frac {\sqrt {23}\,1{}\mathrm {i}}{4}\right )\,\left (\frac {19}{10648}+\frac {\sqrt {23}\,2769{}\mathrm {i}}{5632792}\right )-\ln \left (x-\frac {1}{4}+\frac {\sqrt {23}\,1{}\mathrm {i}}{4}\right )\,\left (-\frac {19}{10648}+\frac {\sqrt {23}\,2769{}\mathrm {i}}{5632792}\right )-\ln \left (x+\frac {3}{10}-\frac {\sqrt {31}\,1{}\mathrm {i}}{10}\right )\,\left (\frac {19}{10648}+\frac {\sqrt {31}\,12643{}\mathrm {i}}{10232728}\right )+\ln \left (x+\frac {3}{10}+\frac {\sqrt {31}\,1{}\mathrm {i}}{10}\right )\,\left (-\frac {19}{10648}+\frac {\sqrt {31}\,12643{}\mathrm {i}}{10232728}\right )+\frac {\frac {685\,x^3}{172546}-\frac {1855\,x^2}{345092}+\frac {507\,x}{78430}-\frac {2171}{862730}}{x^4+\frac {x^3}{10}+\frac {8\,x^2}{5}+\frac {7\,x}{10}+\frac {3}{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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