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 = 0.0992217, antiderivative size = 110, 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} \[ \frac{20 x+37}{651 \left (5 x^2+3 x+2\right )^3}+\frac{4 (203230 x+180133)}{10218313 \left (5 x^2+3 x+2\right )}+\frac{4 (1805 x+1983)}{141267 \left (5 x^2+3 x+2\right )^2}-\frac{64 \log \left (5 x^2+3 x+2\right )}{2401}+\frac{128 \log (2 x+1)}{2401}+\frac{19007376 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{71528191 \sqrt{31}} \]
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) \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*}
Mathematica [A] time = 0.112337, size = 88, normalized size = 0.8 \[ \frac{16 \left (\frac{217 \left (60969000 x^5+127202700 x^4+143405620 x^3+105257844 x^2+44933184 x+13831165\right )}{16 \left (5 x^2+3 x+2\right )^3}-11082252 \log \left (4 \left (5 x^2+3 x+2\right )\right )+22164504 \log (2 x+1)+3563883 \sqrt{31} \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )\right )}{6652121763} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 78, normalized size = 0.7 \begin{align*}{\frac{128\,\ln \left ( 1+2\,x \right ) }{2401}}-{\frac{125}{2401\, \left ( 5\,{x}^{2}+3\,x+2 \right ) ^{3}} \left ( -{\frac{1138088\,{x}^{5}}{29791}}-{\frac{11872252\,{x}^{4}}{148955}}-{\frac{200767868\,{x}^{3}}{2234325}}-{\frac{245601636\,{x}^{2}}{3723875}}-{\frac{104844096\,x}{3723875}}-{\frac{19363631}{2234325}} \right ) }-{\frac{64\,\ln \left ( 5\,{x}^{2}+3\,x+2 \right ) }{2401}}+{\frac{19007376\,\sqrt{31}}{2217373921}\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 = 1.51665, size = 131, normalized size = 1.19 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.38046, size = 633, normalized size = 5.75 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.274918, size = 110, normalized size = 1. \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07708, size = 105, normalized size = 0.95 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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