Optimal. Leaf size=110 \[ \frac {20 x+37}{651 \left (5 x^2+3 x+2\right )^3}+\frac {4 (203230 x+180133)}{10218313 \left (5 x^2+3 x+2\right )}+\frac {4 (1805 x+1983)}{141267 \left (5 x^2+3 x+2\right )^2}-\frac {64 \log \left (5 x^2+3 x+2\right )}{2401}+\frac {128 \log (2 x+1)}{2401}+\frac {19007376 \tan ^{-1}\left (\frac {10 x+3}{\sqrt {31}}\right )}{71528191 \sqrt {31}} \]
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Rubi [A] time = 0.10, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {740, 822, 800, 634, 618, 204, 628} \begin {gather*} \frac {20 x+37}{651 \left (5 x^2+3 x+2\right )^3}+\frac {4 (203230 x+180133)}{10218313 \left (5 x^2+3 x+2\right )}+\frac {4 (1805 x+1983)}{141267 \left (5 x^2+3 x+2\right )^2}-\frac {64 \log \left (5 x^2+3 x+2\right )}{2401}+\frac {128 \log (2 x+1)}{2401}+\frac {19007376 \tan ^{-1}\left (\frac {10 x+3}{\sqrt {31}}\right )}{71528191 \sqrt {31}} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 740
Rule 800
Rule 822
Rubi steps
\begin {align*} \int \frac {1}{(1+2 x) \left (2+3 x+5 x^2\right )^4} \, dx &=\frac {37+20 x}{651 \left (2+3 x+5 x^2\right )^3}+\frac {1}{651} \int \frac {472+200 x}{(1+2 x) \left (2+3 x+5 x^2\right )^3} \, dx\\ &=\frac {37+20 x}{651 \left (2+3 x+5 x^2\right )^3}+\frac {4 (1983+1805 x)}{141267 \left (2+3 x+5 x^2\right )^2}+\frac {\int \frac {135576+86640 x}{(1+2 x) \left (2+3 x+5 x^2\right )^2} \, dx}{282534}\\ &=\frac {37+20 x}{651 \left (2+3 x+5 x^2\right )^3}+\frac {4 (1983+1805 x)}{141267 \left (2+3 x+5 x^2\right )^2}+\frac {4 (180133+203230 x)}{10218313 \left (2+3 x+5 x^2\right )}+\frac {\int \frac {16317264+9755040 x}{(1+2 x) \left (2+3 x+5 x^2\right )} \, dx}{61309878}\\ &=\frac {37+20 x}{651 \left (2+3 x+5 x^2\right )^3}+\frac {4 (1983+1805 x)}{141267 \left (2+3 x+5 x^2\right )^2}+\frac {4 (180133+203230 x)}{10218313 \left (2+3 x+5 x^2\right )}+\frac {\int \left (\frac {45758976}{7 (1+2 x)}-\frac {48 (-472977+2383280 x)}{7 \left (2+3 x+5 x^2\right )}\right ) \, dx}{61309878}\\ &=\frac {37+20 x}{651 \left (2+3 x+5 x^2\right )^3}+\frac {4 (1983+1805 x)}{141267 \left (2+3 x+5 x^2\right )^2}+\frac {4 (180133+203230 x)}{10218313 \left (2+3 x+5 x^2\right )}+\frac {128 \log (1+2 x)}{2401}-\frac {8 \int \frac {-472977+2383280 x}{2+3 x+5 x^2} \, dx}{71528191}\\ &=\frac {37+20 x}{651 \left (2+3 x+5 x^2\right )^3}+\frac {4 (1983+1805 x)}{141267 \left (2+3 x+5 x^2\right )^2}+\frac {4 (180133+203230 x)}{10218313 \left (2+3 x+5 x^2\right )}+\frac {128 \log (1+2 x)}{2401}-\frac {64 \int \frac {3+10 x}{2+3 x+5 x^2} \, dx}{2401}+\frac {9503688 \int \frac {1}{2+3 x+5 x^2} \, dx}{71528191}\\ &=\frac {37+20 x}{651 \left (2+3 x+5 x^2\right )^3}+\frac {4 (1983+1805 x)}{141267 \left (2+3 x+5 x^2\right )^2}+\frac {4 (180133+203230 x)}{10218313 \left (2+3 x+5 x^2\right )}+\frac {128 \log (1+2 x)}{2401}-\frac {64 \log \left (2+3 x+5 x^2\right )}{2401}-\frac {19007376 \operatorname {Subst}\left (\int \frac {1}{-31-x^2} \, dx,x,3+10 x\right )}{71528191}\\ &=\frac {37+20 x}{651 \left (2+3 x+5 x^2\right )^3}+\frac {4 (1983+1805 x)}{141267 \left (2+3 x+5 x^2\right )^2}+\frac {4 (180133+203230 x)}{10218313 \left (2+3 x+5 x^2\right )}+\frac {19007376 \tan ^{-1}\left (\frac {3+10 x}{\sqrt {31}}\right )}{71528191 \sqrt {31}}+\frac {128 \log (1+2 x)}{2401}-\frac {64 \log \left (2+3 x+5 x^2\right )}{2401}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 88, normalized size = 0.80 \begin {gather*} \frac {16 \left (-11082252 \log \left (4 \left (5 x^2+3 x+2\right )\right )+\frac {217 \left (60969000 x^5+127202700 x^4+143405620 x^3+105257844 x^2+44933184 x+13831165\right )}{16 \left (5 x^2+3 x+2\right )^3}+22164504 \log (2 x+1)+3563883 \sqrt {31} \tan ^{-1}\left (\frac {10 x+3}{\sqrt {31}}\right )\right )}{6652121763} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(1+2 x) \left (2+3 x+5 x^2\right )^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.43, size = 186, normalized size = 1.69 \begin {gather*} \frac {13230273000 \, x^{5} + 27602985900 \, x^{4} + 31119019540 \, x^{3} + 57022128 \, \sqrt {31} {\left (125 \, x^{6} + 225 \, x^{5} + 285 \, x^{4} + 207 \, x^{3} + 114 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + 22840952148 \, x^{2} - 177316032 \, {\left (125 \, x^{6} + 225 \, x^{5} + 285 \, x^{4} + 207 \, x^{3} + 114 \, x^{2} + 36 \, x + 8\right )} \log \left (5 \, x^{2} + 3 \, x + 2\right ) + 354632064 \, {\left (125 \, x^{6} + 225 \, x^{5} + 285 \, x^{4} + 207 \, x^{3} + 114 \, x^{2} + 36 \, x + 8\right )} \log \left (2 \, x + 1\right ) + 9750500928 \, x + 3001362805}{6652121763 \, {\left (125 \, x^{6} + 225 \, x^{5} + 285 \, x^{4} + 207 \, x^{3} + 114 \, x^{2} + 36 \, x + 8\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 78, normalized size = 0.71 \begin {gather*} \frac {19007376}{2217373921} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {60969000 \, x^{5} + 127202700 \, x^{4} + 143405620 \, x^{3} + 105257844 \, x^{2} + 44933184 \, x + 13831165}{30654939 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}^{3}} - \frac {64}{2401} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) + \frac {128}{2401} \, \log \left ({\left | 2 \, x + 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 78, normalized size = 0.71 \begin {gather*} \frac {19007376 \sqrt {31}\, \arctan \left (\frac {\left (10 x +3\right ) \sqrt {31}}{31}\right )}{2217373921}+\frac {128 \ln \left (2 x +1\right )}{2401}-\frac {64 \ln \left (5 x^{2}+3 x +2\right )}{2401}-\frac {125 \left (-\frac {1138088}{29791} x^{5}-\frac {11872252}{148955} x^{4}-\frac {200767868}{2234325} x^{3}-\frac {245601636}{3723875} x^{2}-\frac {104844096}{3723875} x -\frac {19363631}{2234325}\right )}{2401 \left (5 x^{2}+3 x +2\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.90, size = 97, normalized size = 0.88 \begin {gather*} \frac {19007376}{2217373921} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {60969000 \, x^{5} + 127202700 \, x^{4} + 143405620 \, x^{3} + 105257844 \, x^{2} + 44933184 \, x + 13831165}{30654939 \, {\left (125 \, x^{6} + 225 \, x^{5} + 285 \, x^{4} + 207 \, x^{3} + 114 \, x^{2} + 36 \, x + 8\right )}} - \frac {64}{2401} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) + \frac {128}{2401} \, \log \left (2 \, x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 102, normalized size = 0.93 \begin {gather*} \frac {128\,\ln \left (x+\frac {1}{2}\right )}{2401}-\ln \left (x+\frac {3}{10}-\frac {\sqrt {31}\,1{}\mathrm {i}}{10}\right )\,\left (\frac {64}{2401}+\frac {\sqrt {31}\,9503688{}\mathrm {i}}{2217373921}\right )+\ln \left (x+\frac {3}{10}+\frac {\sqrt {31}\,1{}\mathrm {i}}{10}\right )\,\left (-\frac {64}{2401}+\frac {\sqrt {31}\,9503688{}\mathrm {i}}{2217373921}\right )+\frac {\frac {162584\,x^5}{10218313}+\frac {1696036\,x^4}{51091565}+\frac {28681124\,x^3}{766373475}+\frac {35085948\,x^2}{1277289125}+\frac {14977728\,x}{1277289125}+\frac {2766233}{766373475}}{x^6+\frac {9\,x^5}{5}+\frac {57\,x^4}{25}+\frac {207\,x^3}{125}+\frac {114\,x^2}{125}+\frac {36\,x}{125}+\frac {8}{125}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.29, size = 110, normalized size = 1.00 \begin {gather*} \frac {60969000 x^{5} + 127202700 x^{4} + 143405620 x^{3} + 105257844 x^{2} + 44933184 x + 13831165}{3831867375 x^{6} + 6897361275 x^{5} + 8736657615 x^{4} + 6345572373 x^{3} + 3494663046 x^{2} + 1103577804 x + 245239512} + \frac {128 \log {\left (x + \frac {1}{2} \right )}}{2401} - \frac {64 \log {\left (x^{2} + \frac {3 x}{5} + \frac {2}{5} \right )}}{2401} + \frac {19007376 \sqrt {31} \operatorname {atan}{\left (\frac {10 \sqrt {31} x}{31} + \frac {3 \sqrt {31}}{31} \right )}}{2217373921} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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