Optimal. Leaf size=142 \[ \frac {20 x+37}{651 (2 x+1) \left (5 x^2+3 x+2\right )^3}+\frac {2 (603620 x+504757)}{10218313 (2 x+1) \left (5 x^2+3 x+2\right )}+\frac {2820 x+3047}{47089 (2 x+1) \left (5 x^2+3 x+2\right )^2}-\frac {1024 \log \left (5 x^2+3 x+2\right )}{16807}-\frac {6802312}{71528191 (2 x+1)}+\frac {2048 \log (2 x+1)}{16807}-\frac {116056984 \tan ^{-1}\left (\frac {10 x+3}{\sqrt {31}}\right )}{500697337 \sqrt {31}} \]
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Rubi [A] time = 0.11, antiderivative size = 142, 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} \[ \frac {20 x+37}{651 (2 x+1) \left (5 x^2+3 x+2\right )^3}+\frac {2 (603620 x+504757)}{10218313 (2 x+1) \left (5 x^2+3 x+2\right )}+\frac {2820 x+3047}{47089 (2 x+1) \left (5 x^2+3 x+2\right )^2}-\frac {1024 \log \left (5 x^2+3 x+2\right )}{16807}-\frac {6802312}{71528191 (2 x+1)}+\frac {2048 \log (2 x+1)}{16807}-\frac {116056984 \tan ^{-1}\left (\frac {10 x+3}{\sqrt {31}}\right )}{500697337 \sqrt {31}} \]
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)^2 \left (2+3 x+5 x^2\right )^4} \, dx &=\frac {37+20 x}{651 (1+2 x) \left (2+3 x+5 x^2\right )^3}+\frac {1}{651} \int \frac {546+240 x}{(1+2 x)^2 \left (2+3 x+5 x^2\right )^3} \, dx\\ &=\frac {37+20 x}{651 (1+2 x) \left (2+3 x+5 x^2\right )^3}+\frac {3047+2820 x}{47089 (1+2 x) \left (2+3 x+5 x^2\right )^2}+\frac {\int \frac {192972+135360 x}{(1+2 x)^2 \left (2+3 x+5 x^2\right )^2} \, dx}{282534}\\ &=\frac {37+20 x}{651 (1+2 x) \left (2+3 x+5 x^2\right )^3}+\frac {3047+2820 x}{47089 (1+2 x) \left (2+3 x+5 x^2\right )^2}+\frac {2 (504757+603620 x)}{10218313 (1+2 x) \left (2+3 x+5 x^2\right )}+\frac {\int \frac {34893816+28973760 x}{(1+2 x)^2 \left (2+3 x+5 x^2\right )} \, dx}{61309878}\\ &=\frac {37+20 x}{651 (1+2 x) \left (2+3 x+5 x^2\right )^3}+\frac {3047+2820 x}{47089 (1+2 x) \left (2+3 x+5 x^2\right )^2}+\frac {2 (504757+603620 x)}{10218313 (1+2 x) \left (2+3 x+5 x^2\right )}+\frac {\int \left (\frac {81627744}{7 (1+2 x)^2}+\frac {732143616}{49 (1+2 x)}-\frac {24 (37386611+76264960 x)}{49 \left (2+3 x+5 x^2\right )}\right ) \, dx}{61309878}\\ &=-\frac {6802312}{71528191 (1+2 x)}+\frac {37+20 x}{651 (1+2 x) \left (2+3 x+5 x^2\right )^3}+\frac {3047+2820 x}{47089 (1+2 x) \left (2+3 x+5 x^2\right )^2}+\frac {2 (504757+603620 x)}{10218313 (1+2 x) \left (2+3 x+5 x^2\right )}+\frac {2048 \log (1+2 x)}{16807}-\frac {4 \int \frac {37386611+76264960 x}{2+3 x+5 x^2} \, dx}{500697337}\\ &=-\frac {6802312}{71528191 (1+2 x)}+\frac {37+20 x}{651 (1+2 x) \left (2+3 x+5 x^2\right )^3}+\frac {3047+2820 x}{47089 (1+2 x) \left (2+3 x+5 x^2\right )^2}+\frac {2 (504757+603620 x)}{10218313 (1+2 x) \left (2+3 x+5 x^2\right )}+\frac {2048 \log (1+2 x)}{16807}-\frac {1024 \int \frac {3+10 x}{2+3 x+5 x^2} \, dx}{16807}-\frac {58028492 \int \frac {1}{2+3 x+5 x^2} \, dx}{500697337}\\ &=-\frac {6802312}{71528191 (1+2 x)}+\frac {37+20 x}{651 (1+2 x) \left (2+3 x+5 x^2\right )^3}+\frac {3047+2820 x}{47089 (1+2 x) \left (2+3 x+5 x^2\right )^2}+\frac {2 (504757+603620 x)}{10218313 (1+2 x) \left (2+3 x+5 x^2\right )}+\frac {2048 \log (1+2 x)}{16807}-\frac {1024 \log \left (2+3 x+5 x^2\right )}{16807}+\frac {116056984 \operatorname {Subst}\left (\int \frac {1}{-31-x^2} \, dx,x,3+10 x\right )}{500697337}\\ &=-\frac {6802312}{71528191 (1+2 x)}+\frac {37+20 x}{651 (1+2 x) \left (2+3 x+5 x^2\right )^3}+\frac {3047+2820 x}{47089 (1+2 x) \left (2+3 x+5 x^2\right )^2}+\frac {2 (504757+603620 x)}{10218313 (1+2 x) \left (2+3 x+5 x^2\right )}-\frac {116056984 \tan ^{-1}\left (\frac {3+10 x}{\sqrt {31}}\right )}{500697337 \sqrt {31}}+\frac {2048 \log (1+2 x)}{16807}-\frac {1024 \log \left (2+3 x+5 x^2\right )}{16807}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 119, normalized size = 0.84 \[ \frac {8 \left (-\frac {10218313 (270 x-43)}{8 \left (5 x^2+3 x+2\right )^3}-\frac {651 (3736330 x-1739037)}{4 \left (5 x^2+3 x+2\right )}-\frac {141267 (27530 x-7117)}{8 \left (5 x^2+3 x+2\right )^2}-354632064 \log \left (4 \left (5 x^2+3 x+2\right )\right )-\frac {310303056}{2 x+1}+709264128 \log (2 x+1)-43521369 \sqrt {31} \tan ^{-1}\left (\frac {10 x+3}{\sqrt {31}}\right )\right )}{46564852341} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 211, normalized size = 1.49 \[ -\frac {553538139000 \, x^{6} + 858833833200 \, x^{5} + 982016294070 \, x^{4} + 605165058624 \, x^{3} + 348170952 \, \sqrt {31} {\left (250 \, x^{7} + 575 \, x^{6} + 795 \, x^{5} + 699 \, x^{4} + 435 \, x^{3} + 186 \, x^{2} + 52 \, x + 8\right )} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + 281968516011 \, x^{2} + 2837056512 \, {\left (250 \, x^{7} + 575 \, x^{6} + 795 \, x^{5} + 699 \, x^{4} + 435 \, x^{3} + 186 \, x^{2} + 52 \, x + 8\right )} \log \left (5 \, x^{2} + 3 \, x + 2\right ) - 5674113024 \, {\left (250 \, x^{7} + 575 \, x^{6} + 795 \, x^{5} + 699 \, x^{4} + 435 \, x^{3} + 186 \, x^{2} + 52 \, x + 8\right )} \log \left (2 \, x + 1\right ) + 66162113227 \, x + 8352308951}{46564852341 \, {\left (250 \, x^{7} + 575 \, x^{6} + 795 \, x^{5} + 699 \, x^{4} + 435 \, x^{3} + 186 \, x^{2} + 52 \, x + 8\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 126, normalized size = 0.89 \[ -\frac {116056984}{15521617447} \, \sqrt {31} \arctan \left (-\frac {1}{31} \, \sqrt {31} {\left (\frac {7}{2 \, x + 1} - 2\right )}\right ) - \frac {128}{2401 \, {\left (2 \, x + 1\right )}} - \frac {8 \, {\left (\frac {3841449975}{2 \, x + 1} - \frac {8833663680}{{\left (2 \, x + 1\right )}^{2}} + \frac {7499779568}{{\left (2 \, x + 1\right )}^{3}} - \frac {7050406230}{{\left (2 \, x + 1\right )}^{4}} + \frac {1291725897}{{\left (2 \, x + 1\right )}^{5}} - 2009265250\right )}}{1502092011 \, {\left (\frac {4}{2 \, x + 1} - \frac {7}{{\left (2 \, x + 1\right )}^{2}} - 5\right )}^{3}} - \frac {1024}{16807} \, \log \left (-\frac {4}{2 \, x + 1} + \frac {7}{{\left (2 \, x + 1\right )}^{2}} + 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 87, normalized size = 0.61 \[ -\frac {116056984 \sqrt {31}\, \arctan \left (\frac {\left (10 x +3\right ) \sqrt {31}}{31}\right )}{15521617447}+\frac {2048 \ln \left (2 x +1\right )}{16807}-\frac {1024 \ln \left (5 x^{2}+3 x +2\right )}{16807}-\frac {128}{2401 \left (2 x +1\right )}-\frac {125 \left (\frac {10461724}{29791} x^{5}+\frac {38423826}{148955} x^{4}+\frac {199128958}{744775} x^{3}-\frac {6944987}{3723875} x^{2}-\frac {410739}{744775} x -\frac {371196343}{11171625}\right )}{16807 \left (5 x^{2}+3 x +2\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.00, size = 107, normalized size = 0.75 \[ -\frac {116056984}{15521617447} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) - \frac {2550867000 \, x^{6} + 3957759600 \, x^{5} + 4525420710 \, x^{4} + 2788779072 \, x^{3} + 1299394083 \, x^{2} + 304894531 \, x + 38489903}{214584573 \, {\left (250 \, x^{7} + 575 \, x^{6} + 795 \, x^{5} + 699 \, x^{4} + 435 \, x^{3} + 186 \, x^{2} + 52 \, x + 8\right )}} - \frac {1024}{16807} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) + \frac {2048}{16807} \, \log \left (2 \, x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 113, normalized size = 0.80 \[ \frac {2048\,\ln \left (x+\frac {1}{2}\right )}{16807}+\ln \left (x+\frac {3}{10}-\frac {\sqrt {31}\,1{}\mathrm {i}}{10}\right )\,\left (-\frac {1024}{16807}+\frac {\sqrt {31}\,58028492{}\mathrm {i}}{15521617447}\right )-\ln \left (x+\frac {3}{10}+\frac {\sqrt {31}\,1{}\mathrm {i}}{10}\right )\,\left (\frac {1024}{16807}+\frac {\sqrt {31}\,58028492{}\mathrm {i}}{15521617447}\right )-\frac {\frac {3401156\,x^6}{71528191}+\frac {26385064\,x^5}{357640955}+\frac {150847357\,x^4}{1788204775}+\frac {464796512\,x^3}{8941023875}+\frac {433131361\,x^2}{17882047750}+\frac {304894531\,x}{53646143250}+\frac {38489903}{53646143250}}{x^7+\frac {23\,x^6}{10}+\frac {159\,x^5}{50}+\frac {699\,x^4}{250}+\frac {87\,x^3}{50}+\frac {93\,x^2}{125}+\frac {26\,x}{125}+\frac {4}{125}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.31, size = 122, normalized size = 0.86 \[ \frac {- 2550867000 x^{6} - 3957759600 x^{5} - 4525420710 x^{4} - 2788779072 x^{3} - 1299394083 x^{2} - 304894531 x - 38489903}{53646143250 x^{7} + 123386129475 x^{6} + 170594735535 x^{5} + 149994616527 x^{4} + 93344289255 x^{3} + 39912730578 x^{2} + 11158397796 x + 1716676584} + \frac {2048 \log {\left (x + \frac {1}{2} \right )}}{16807} - \frac {1024 \log {\left (x^{2} + \frac {3 x}{5} + \frac {2}{5} \right )}}{16807} - \frac {116056984 \sqrt {31} \operatorname {atan}{\left (\frac {10 \sqrt {31} x}{31} + \frac {3 \sqrt {31}}{31} \right )}}{15521617447} \]
Verification of antiderivative is not currently implemented for this CAS.
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