Optimal. Leaf size=114 \[ \frac{20 x+37}{434 (2 x+1) \left (5 x^2+3 x+2\right )^2}+\frac{5820 x+6427}{47089 (2 x+1) \left (5 x^2+3 x+2\right )}-\frac{192 \log \left (5 x^2+3 x+2\right )}{2401}-\frac{51516}{329623 (2 x+1)}+\frac{384 \log (2 x+1)}{2401}-\frac{1065012 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{2307361 \sqrt{31}} \]
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Rubi [A] time = 0.0882789, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{20 x+37}{434 (2 x+1) \left (5 x^2+3 x+2\right )^2}+\frac{5820 x+6427}{47089 (2 x+1) \left (5 x^2+3 x+2\right )}-\frac{192 \log \left (5 x^2+3 x+2\right )}{2401}-\frac{51516}{329623 (2 x+1)}+\frac{384 \log (2 x+1)}{2401}-\frac{1065012 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{2307361 \sqrt{31}} \]
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 )^3} \, dx &=\frac{37+20 x}{434 (1+2 x) \left (2+3 x+5 x^2\right )^2}+\frac{1}{434} \int \frac{382+160 x}{(1+2 x)^2 \left (2+3 x+5 x^2\right )^2} \, dx\\ &=\frac{37+20 x}{434 (1+2 x) \left (2+3 x+5 x^2\right )^2}+\frac{6427+5820 x}{47089 (1+2 x) \left (2+3 x+5 x^2\right )}+\frac{\int \frac{74796+46560 x}{(1+2 x)^2 \left (2+3 x+5 x^2\right )} \, dx}{94178}\\ &=\frac{37+20 x}{434 (1+2 x) \left (2+3 x+5 x^2\right )^2}+\frac{6427+5820 x}{47089 (1+2 x) \left (2+3 x+5 x^2\right )}+\frac{\int \left (\frac{206064}{7 (1+2 x)^2}+\frac{1476096}{49 (1+2 x)}-\frac{12 (181007+307520 x)}{49 \left (2+3 x+5 x^2\right )}\right ) \, dx}{94178}\\ &=-\frac{51516}{329623 (1+2 x)}+\frac{37+20 x}{434 (1+2 x) \left (2+3 x+5 x^2\right )^2}+\frac{6427+5820 x}{47089 (1+2 x) \left (2+3 x+5 x^2\right )}+\frac{384 \log (1+2 x)}{2401}-\frac{6 \int \frac{181007+307520 x}{2+3 x+5 x^2} \, dx}{2307361}\\ &=-\frac{51516}{329623 (1+2 x)}+\frac{37+20 x}{434 (1+2 x) \left (2+3 x+5 x^2\right )^2}+\frac{6427+5820 x}{47089 (1+2 x) \left (2+3 x+5 x^2\right )}+\frac{384 \log (1+2 x)}{2401}-\frac{192 \int \frac{3+10 x}{2+3 x+5 x^2} \, dx}{2401}-\frac{532506 \int \frac{1}{2+3 x+5 x^2} \, dx}{2307361}\\ &=-\frac{51516}{329623 (1+2 x)}+\frac{37+20 x}{434 (1+2 x) \left (2+3 x+5 x^2\right )^2}+\frac{6427+5820 x}{47089 (1+2 x) \left (2+3 x+5 x^2\right )}+\frac{384 \log (1+2 x)}{2401}-\frac{192 \log \left (2+3 x+5 x^2\right )}{2401}+\frac{1065012 \operatorname{Subst}\left (\int \frac{1}{-31-x^2} \, dx,x,3+10 x\right )}{2307361}\\ &=-\frac{51516}{329623 (1+2 x)}+\frac{37+20 x}{434 (1+2 x) \left (2+3 x+5 x^2\right )^2}+\frac{6427+5820 x}{47089 (1+2 x) \left (2+3 x+5 x^2\right )}-\frac{1065012 \tan ^{-1}\left (\frac{3+10 x}{\sqrt{31}}\right )}{2307361 \sqrt{31}}+\frac{384 \log (1+2 x)}{2401}-\frac{192 \log \left (2+3 x+5 x^2\right )}{2401}\\ \end{align*}
Mathematica [A] time = 0.0665952, size = 98, normalized size = 0.86 \[ \frac{4 \left (-\frac{47089 (270 x-43)}{8 \left (5 x^2+3 x+2\right )^2}-\frac{217 (51910 x-15179)}{4 \left (5 x^2+3 x+2\right )}-1429968 \log \left (4 \left (5 x^2+3 x+2\right )\right )-\frac{1668296}{2 x+1}+2859936 \log (2 x+1)-266253 \sqrt{31} \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )\right )}{71528191} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 77, normalized size = 0.7 \begin{align*} -{\frac{32}{343+686\,x}}+{\frac{384\,\ln \left ( 1+2\,x \right ) }{2401}}-{\frac{25}{2401\, \left ( 5\,{x}^{2}+3\,x+2 \right ) ^{2}} \left ({\frac{72674\,{x}^{3}}{961}}+{\frac{111769\,{x}^{2}}{4805}}+{\frac{613046\,x}{24025}}-{\frac{490329}{48050}} \right ) }-{\frac{192\,\ln \left ( 5\,{x}^{2}+3\,x+2 \right ) }{2401}}-{\frac{1065012\,\sqrt{31}}{71528191}\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 = 1.51454, size = 117, normalized size = 1.03 \begin{align*} -\frac{1065012}{71528191} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) - \frac{2575800 \, x^{4} + 2683560 \, x^{3} + 2293598 \, x^{2} + 773110 \, x + 175969}{659246 \,{\left (50 \, x^{5} + 85 \, x^{4} + 88 \, x^{3} + 53 \, x^{2} + 20 \, x + 4\right )}} - \frac{192}{2401} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) + \frac{384}{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 = 2.97443, size = 517, normalized size = 4.54 \begin{align*} -\frac{558948600 \, x^{4} + 582332520 \, x^{3} + 2130024 \, \sqrt{31}{\left (50 \, x^{5} + 85 \, x^{4} + 88 \, x^{3} + 53 \, x^{2} + 20 \, x + 4\right )} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + 497710766 \, x^{2} + 11439744 \,{\left (50 \, x^{5} + 85 \, x^{4} + 88 \, x^{3} + 53 \, x^{2} + 20 \, x + 4\right )} \log \left (5 \, x^{2} + 3 \, x + 2\right ) - 22879488 \,{\left (50 \, x^{5} + 85 \, x^{4} + 88 \, x^{3} + 53 \, x^{2} + 20 \, x + 4\right )} \log \left (2 \, x + 1\right ) + 167764870 \, x + 38185273}{143056382 \,{\left (50 \, x^{5} + 85 \, x^{4} + 88 \, x^{3} + 53 \, x^{2} + 20 \, x + 4\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.250377, size = 100, normalized size = 0.88 \begin{align*} - \frac{2575800 x^{4} + 2683560 x^{3} + 2293598 x^{2} + 773110 x + 175969}{32962300 x^{5} + 56035910 x^{4} + 58013648 x^{3} + 34940038 x^{2} + 13184920 x + 2636984} + \frac{384 \log{\left (x + \frac{1}{2} \right )}}{2401} - \frac{192 \log{\left (x^{2} + \frac{3 x}{5} + \frac{2}{5} \right )}}{2401} - \frac{1065012 \sqrt{31} \operatorname{atan}{\left (\frac{10 \sqrt{31} x}{31} + \frac{3 \sqrt{31}}{31} \right )}}{71528191} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10521, size = 146, normalized size = 1.28 \begin{align*} -\frac{1065012}{71528191} \, \sqrt{31} \arctan \left (-\frac{1}{31} \, \sqrt{31}{\left (\frac{7}{2 \, x + 1} - 2\right )}\right ) - \frac{32}{343 \,{\left (2 \, x + 1\right )}} + \frac{4 \,{\left (\frac{1178375}{2 \, x + 1} - \frac{2320190}{{\left (2 \, x + 1\right )}^{2}} + \frac{87843}{{\left (2 \, x + 1\right )}^{3}} - 1304250\right )}}{2307361 \,{\left (\frac{4}{2 \, x + 1} - \frac{7}{{\left (2 \, x + 1\right )}^{2}} - 5\right )}^{2}} - \frac{192}{2401} \, \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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