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 = 0.108502, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, 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 740
Rule 822
Rule 800
Rule 634
Rule 618
Rule 204
Rule 628
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*}
Mathematica [A] time = 0.0901292, 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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Maple [A] time = 0.05, size = 87, normalized size = 0.6 \begin{align*} -{\frac{128}{2401+4802\,x}}+{\frac{2048\,\ln \left ( 1+2\,x \right ) }{16807}}-{\frac{125}{16807\, \left ( 5\,{x}^{2}+3\,x+2 \right ) ^{3}} \left ({\frac{10461724\,{x}^{5}}{29791}}+{\frac{38423826\,{x}^{4}}{148955}}+{\frac{199128958\,{x}^{3}}{744775}}-{\frac{6944987\,{x}^{2}}{3723875}}-{\frac{410739\,x}{744775}}-{\frac{371196343}{11171625}} \right ) }-{\frac{1024\,\ln \left ( 5\,{x}^{2}+3\,x+2 \right ) }{16807}}-{\frac{116056984\,\sqrt{31}}{15521617447}\arctan \left ({\frac{ \left ( 3+10\,x \right ) \sqrt{31}}{31}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.56445, size = 144, normalized size = 1.01 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.43371, size = 726, normalized size = 5.11 \begin{align*} -\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.299741, size = 121, normalized size = 0.85 \begin{align*} - \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09174, size = 170, normalized size = 1.2 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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