Optimal. Leaf size=110 \[ \frac{20 x+37}{651 \left (5 x^2+3 x+2\right )^3}+\frac{4 (203230 x+180133)}{10218313 \left (5 x^2+3 x+2\right )}+\frac{4 (1805 x+1983)}{141267 \left (5 x^2+3 x+2\right )^2}-\frac{64 \log \left (5 x^2+3 x+2\right )}{2401}+\frac{128 \log (2 x+1)}{2401}+\frac{19007376 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{71528191 \sqrt{31}} \]
[Out]
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Rubi [A] time = 0.228245, 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 \[ \frac{20 x+37}{651 \left (5 x^2+3 x+2\right )^3}+\frac{4 (203230 x+180133)}{10218313 \left (5 x^2+3 x+2\right )}+\frac{4 (1805 x+1983)}{141267 \left (5 x^2+3 x+2\right )^2}-\frac{64 \log \left (5 x^2+3 x+2\right )}{2401}+\frac{128 \log (2 x+1)}{2401}+\frac{19007376 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{71528191 \sqrt{31}} \]
Antiderivative was successfully verified.
[In] Int[1/((1 + 2*x)*(2 + 3*x + 5*x^2)^4),x]
[Out]
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Rubi in Sympy [A] time = 33.8428, size = 100, normalized size = 0.91 \[ \frac{20 x + 37}{651 \left (5 x^{2} + 3 x + 2\right )^{3}} + \frac{14440 x + 15864}{282534 \left (5 x^{2} + 3 x + 2\right )^{2}} + \frac{4877520 x + 4323192}{61309878 \left (5 x^{2} + 3 x + 2\right )} + \frac{128 \log{\left (2 x + 1 \right )}}{2401} - \frac{64 \log{\left (5 x^{2} + 3 x + 2 \right )}}{2401} + \frac{19007376 \sqrt{31} \operatorname{atan}{\left (\sqrt{31} \left (\frac{10 x}{31} + \frac{3}{31}\right ) \right )}}{2217373921} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(1+2*x)/(5*x**2+3*x+2)**4,x)
[Out]
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Mathematica [A] time = 0.224697, size = 88, normalized size = 0.8 \[ \frac{16 \left (-11082252 \log \left (4 \left (5 x^2+3 x+2\right )\right )+\frac{217 \left (60969000 x^5+127202700 x^4+143405620 x^3+105257844 x^2+44933184 x+13831165\right )}{16 \left (5 x^2+3 x+2\right )^3}+22164504 \log (2 x+1)+3563883 \sqrt{31} \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )\right )}{6652121763} \]
Antiderivative was successfully verified.
[In] Integrate[1/((1 + 2*x)*(2 + 3*x + 5*x^2)^4),x]
[Out]
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Maple [A] time = 0.018, size = 78, normalized size = 0.7 \[{\frac{128\,\ln \left ( 1+2\,x \right ) }{2401}}-{\frac{125}{2401\, \left ( 5\,{x}^{2}+3\,x+2 \right ) ^{3}} \left ( -{\frac{1138088\,{x}^{5}}{29791}}-{\frac{11872252\,{x}^{4}}{148955}}-{\frac{200767868\,{x}^{3}}{2234325}}-{\frac{245601636\,{x}^{2}}{3723875}}-{\frac{104844096\,x}{3723875}}-{\frac{19363631}{2234325}} \right ) }-{\frac{64\,\ln \left ( 625\,{x}^{2}+375\,x+250 \right ) }{2401}}+{\frac{19007376\,\sqrt{31}}{2217373921}\arctan \left ({\frac{ \left ( 1250\,x+375 \right ) \sqrt{31}}{3875}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(1+2*x)/(5*x^2+3*x+2)^4,x)
[Out]
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Maxima [A] time = 0.91807, size = 131, normalized size = 1.19 \[ \frac{19007376}{2217373921} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + \frac{60969000 \, x^{5} + 127202700 \, x^{4} + 143405620 \, x^{3} + 105257844 \, x^{2} + 44933184 \, x + 13831165}{30654939 \,{\left (125 \, x^{6} + 225 \, x^{5} + 285 \, x^{4} + 207 \, x^{3} + 114 \, x^{2} + 36 \, x + 8\right )}} - \frac{64}{2401} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) + \frac{128}{2401} \, \log \left (2 \, x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x^2 + 3*x + 2)^4*(2*x + 1)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.228728, size = 267, normalized size = 2.43 \[ -\frac{\sqrt{31}{\left (5719872 \, \sqrt{31}{\left (125 \, x^{6} + 225 \, x^{5} + 285 \, x^{4} + 207 \, x^{3} + 114 \, x^{2} + 36 \, x + 8\right )} \log \left (5 \, x^{2} + 3 \, x + 2\right ) - 11439744 \, \sqrt{31}{\left (125 \, x^{6} + 225 \, x^{5} + 285 \, x^{4} + 207 \, x^{3} + 114 \, x^{2} + 36 \, x + 8\right )} \log \left (2 \, x + 1\right ) - 57022128 \,{\left (125 \, x^{6} + 225 \, x^{5} + 285 \, x^{4} + 207 \, x^{3} + 114 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) - 7 \, \sqrt{31}{\left (60969000 \, x^{5} + 127202700 \, x^{4} + 143405620 \, x^{3} + 105257844 \, x^{2} + 44933184 \, x + 13831165\right )}\right )}}{6652121763 \,{\left (125 \, x^{6} + 225 \, x^{5} + 285 \, x^{4} + 207 \, x^{3} + 114 \, x^{2} + 36 \, x + 8\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x^2 + 3*x + 2)^4*(2*x + 1)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.829551, size = 110, normalized size = 1. \[ \frac{60969000 x^{5} + 127202700 x^{4} + 143405620 x^{3} + 105257844 x^{2} + 44933184 x + 13831165}{3831867375 x^{6} + 6897361275 x^{5} + 8736657615 x^{4} + 6345572373 x^{3} + 3494663046 x^{2} + 1103577804 x + 245239512} + \frac{128 \log{\left (x + \frac{1}{2} \right )}}{2401} - \frac{64 \log{\left (x^{2} + \frac{3 x}{5} + \frac{2}{5} \right )}}{2401} + \frac{19007376 \sqrt{31} \operatorname{atan}{\left (\frac{10 \sqrt{31} x}{31} + \frac{3 \sqrt{31}}{31} \right )}}{2217373921} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(1+2*x)/(5*x**2+3*x+2)**4,x)
[Out]
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GIAC/XCAS [A] time = 0.205981, size = 105, normalized size = 0.95 \[ \frac{19007376}{2217373921} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + \frac{60969000 \, x^{5} + 127202700 \, x^{4} + 143405620 \, x^{3} + 105257844 \, x^{2} + 44933184 \, x + 13831165}{30654939 \,{\left (5 \, x^{2} + 3 \, x + 2\right )}^{3}} - \frac{64}{2401} \,{\rm ln}\left (5 \, x^{2} + 3 \, x + 2\right ) + \frac{128}{2401} \,{\rm ln}\left ({\left | 2 \, x + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x^2 + 3*x + 2)^4*(2*x + 1)),x, algorithm="giac")
[Out]