3.2221 \(\int \frac{1}{(1+2 x)^2 \left (2+3 x+5 x^2\right )^4} \, dx\)

Optimal. Leaf size=142 \[ \frac{20 x+37}{651 (2 x+1) \left (5 x^2+3 x+2\right )^3}+\frac{2 (603620 x+504757)}{10218313 (2 x+1) \left (5 x^2+3 x+2\right )}+\frac{2820 x+3047}{47089 (2 x+1) \left (5 x^2+3 x+2\right )^2}-\frac{1024 \log \left (5 x^2+3 x+2\right )}{16807}-\frac{6802312}{71528191 (2 x+1)}+\frac{2048 \log (2 x+1)}{16807}-\frac{116056984 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{500697337 \sqrt{31}} \]

[Out]

-6802312/(71528191*(1 + 2*x)) + (37 + 20*x)/(651*(1 + 2*x)*(2 + 3*x + 5*x^2)^3)
+ (3047 + 2820*x)/(47089*(1 + 2*x)*(2 + 3*x + 5*x^2)^2) + (2*(504757 + 603620*x)
)/(10218313*(1 + 2*x)*(2 + 3*x + 5*x^2)) - (116056984*ArcTan[(3 + 10*x)/Sqrt[31]
])/(500697337*Sqrt[31]) + (2048*Log[1 + 2*x])/16807 - (1024*Log[2 + 3*x + 5*x^2]
)/16807

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Rubi [A]  time = 0.252377, 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 \[ \frac{20 x+37}{651 (2 x+1) \left (5 x^2+3 x+2\right )^3}+\frac{2 (603620 x+504757)}{10218313 (2 x+1) \left (5 x^2+3 x+2\right )}+\frac{2820 x+3047}{47089 (2 x+1) \left (5 x^2+3 x+2\right )^2}-\frac{1024 \log \left (5 x^2+3 x+2\right )}{16807}-\frac{6802312}{71528191 (2 x+1)}+\frac{2048 \log (2 x+1)}{16807}-\frac{116056984 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{500697337 \sqrt{31}} \]

Antiderivative was successfully verified.

[In]  Int[1/((1 + 2*x)^2*(2 + 3*x + 5*x^2)^4),x]

[Out]

-6802312/(71528191*(1 + 2*x)) + (37 + 20*x)/(651*(1 + 2*x)*(2 + 3*x + 5*x^2)^3)
+ (3047 + 2820*x)/(47089*(1 + 2*x)*(2 + 3*x + 5*x^2)^2) + (2*(504757 + 603620*x)
)/(10218313*(1 + 2*x)*(2 + 3*x + 5*x^2)) - (116056984*ArcTan[(3 + 10*x)/Sqrt[31]
])/(500697337*Sqrt[31]) + (2048*Log[1 + 2*x])/16807 - (1024*Log[2 + 3*x + 5*x^2]
)/16807

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Rubi in Sympy [A]  time = 37.1936, size = 124, normalized size = 0.87 \[ \frac{2048 \log{\left (2 x + 1 \right )}}{16807} - \frac{1024 \log{\left (5 x^{2} + 3 x + 2 \right )}}{16807} - \frac{116056984 \sqrt{31} \operatorname{atan}{\left (\sqrt{31} \left (\frac{10 x}{31} + \frac{3}{31}\right ) \right )}}{15521617447} + \frac{20 x + 37}{651 \left (2 x + 1\right ) \left (5 x^{2} + 3 x + 2\right )^{3}} + \frac{16920 x + 18282}{282534 \left (2 x + 1\right ) \left (5 x^{2} + 3 x + 2\right )^{2}} + \frac{7243440 x + 6057084}{61309878 \left (2 x + 1\right ) \left (5 x^{2} + 3 x + 2\right )} - \frac{6802312}{71528191 \left (2 x + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1+2*x)**2/(5*x**2+3*x+2)**4,x)

[Out]

2048*log(2*x + 1)/16807 - 1024*log(5*x**2 + 3*x + 2)/16807 - 116056984*sqrt(31)*
atan(sqrt(31)*(10*x/31 + 3/31))/15521617447 + (20*x + 37)/(651*(2*x + 1)*(5*x**2
+ 3*x + 2)**3) + (16920*x + 18282)/(282534*(2*x + 1)*(5*x**2 + 3*x + 2)**2) + (
7243440*x + 6057084)/(61309878*(2*x + 1)*(5*x**2 + 3*x + 2)) - 6802312/(71528191
*(2*x + 1))

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Mathematica [A]  time = 0.179548, size = 119, normalized size = 0.84 \[ \frac{8 \left (-\frac{10218313 (270 x-43)}{8 \left (5 x^2+3 x+2\right )^3}-\frac{651 (3736330 x-1739037)}{4 \left (5 x^2+3 x+2\right )}-\frac{141267 (27530 x-7117)}{8 \left (5 x^2+3 x+2\right )^2}-354632064 \log \left (4 \left (5 x^2+3 x+2\right )\right )-\frac{310303056}{2 x+1}[phone] \log (2 x+1)-43521369 \sqrt{31} \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )\right )}{46564852341} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((1 + 2*x)^2*(2 + 3*x + 5*x^2)^4),x]

[Out]

(8*(-310303056/(1 + 2*x) - (10218313*(-43 + 270*x))/(8*(2 + 3*x + 5*x^2)^3) - (1
41267*(-7117 + 27530*x))/(8*(2 + 3*x + 5*x^2)^2) - (651*(-1739037 + 3736330*x))/
(4*(2 + 3*x + 5*x^2)) - 43521369*Sqrt[31]*ArcTan[(3 + 10*x)/Sqrt[31]] + 70926412
8*Log[1 + 2*x] - 354632064*Log[4*(2 + 3*x + 5*x^2)]))/46564852341

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Maple [A]  time = 0.017, size = 87, normalized size = 0.6 \[ -{\frac{128}{2401+4802\,x}}+{\frac{2048\,\ln \left ( 1+2\,x \right ) }{16807}}-{\frac{125}{16807\, \left ( 5\,{x}^{2}+3\,x+2 \right ) ^{3}} \left ({\frac{10461724\,{x}^{5}}{29791}}+{\frac{38423826\,{x}^{4}}{148955}}+{\frac{199128958\,{x}^{3}}{744775}}-{\frac{6944987\,{x}^{2}}{3723875}}-{\frac{410739\,x}{744775}}-{\frac{371196343}{11171625}} \right ) }-{\frac{1024\,\ln \left ( 625\,{x}^{2}+375\,x+250 \right ) }{16807}}-{\frac{116056984\,\sqrt{31}}{15521617447}\arctan \left ({\frac{ \left ( 1250\,x+375 \right ) \sqrt{31}}{3875}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1+2*x)^2/(5*x^2+3*x+2)^4,x)

[Out]

-128/2401/(1+2*x)+2048/16807*ln(1+2*x)-125/16807*(10461724/29791*x^5+38423826/14
8955*x^4+199128958/744775*x^3-6944987/3723875*x^2-410739/744775*x-371196343/1117
1625)/(5*x^2+3*x+2)^3-1024/16807*ln(625*x^2+375*x+250)-116056984/15521617447*31^
(1/2)*arctan(1/3875*(1250*x+375)*31^(1/2))

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Maxima [A]  time = 0.916736, size = 144, normalized size = 1.01 \[ -\frac{116056984}{15521617447} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) - \frac{2550867000 \, x^{6} + 3957759600 \, x^{5} + 4525420710 \, x^{4} + 2788779072 \, x^{3} + 1299394083 \, x^{2} + 304894531 \, x + 38489903}{214584573 \,{\left (250 \, x^{7} + 575 \, x^{6} + 795 \, x^{5} + 699 \, x^{4} + 435 \, x^{3} + 186 \, x^{2} + 52 \, x + 8\right )}} - \frac{1024}{16807} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) + \frac{2048}{16807} \, \log \left (2 \, x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((5*x^2 + 3*x + 2)^4*(2*x + 1)^2),x, algorithm="maxima")

[Out]

-116056984/15521617447*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) - 1/214584573*(
2550867000*x^6 + 3957759600*x^5 + 4525420710*x^4 + 2788779072*x^3 + 1299394083*x
^2 + 304894531*x + 38489903)/(250*x^7 + 575*x^6 + 795*x^5 + 699*x^4 + 435*x^3 +
186*x^2 + 52*x + 8) - 1024/16807*log(5*x^2 + 3*x + 2) + 2048/16807*log(2*x + 1)

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Fricas [A]  time = 0.233407, size = 301, normalized size = 2.12 \[ -\frac{\sqrt{31}{\left (91517952 \, \sqrt{31}{\left (250 \, x^{7} + 575 \, x^{6} + 795 \, x^{5} + 699 \, x^{4} + 435 \, x^{3} + 186 \, x^{2} + 52 \, x + 8\right )} \log \left (5 \, x^{2} + 3 \, x + 2\right ) - 183035904 \, \sqrt{31}{\left (250 \, x^{7} + 575 \, x^{6} + 795 \, x^{5} + 699 \, x^{4} + 435 \, x^{3} + 186 \, x^{2} + 52 \, x + 8\right )} \log \left (2 \, x + 1\right ) + 348170952 \,{\left (250 \, x^{7} + 575 \, x^{6} + 795 \, x^{5} + 699 \, x^{4} + 435 \, x^{3} + 186 \, x^{2} + 52 \, x + 8\right )} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + 7 \, \sqrt{31}{\left (2550867000 \, x^{6} + 3957759600 \, x^{5} + 4525420710 \, x^{4} + 2788779072 \, x^{3} + 1299394083 \, x^{2} + 304894531 \, x + 38489903\right )}\right )}}{46564852341 \,{\left (250 \, x^{7} + 575 \, x^{6} + 795 \, x^{5} + 699 \, x^{4} + 435 \, x^{3} + 186 \, x^{2} + 52 \, x + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((5*x^2 + 3*x + 2)^4*(2*x + 1)^2),x, algorithm="fricas")

[Out]

-1/46564852341*sqrt(31)*(91517952*sqrt(31)*(250*x^7 + 575*x^6 + 795*x^5 + 699*x^
4 + 435*x^3 + 186*x^2 + 52*x + 8)*log(5*x^2 + 3*x + 2) - 183035904*sqrt(31)*(250
*x^7 + 575*x^6 + 795*x^5 + 699*x^4 + 435*x^3 + 186*x^2 + 52*x + 8)*log(2*x + 1)
+ 348170952*(250*x^7 + 575*x^6 + 795*x^5 + 699*x^4 + 435*x^3 + 186*x^2 + 52*x +
8)*arctan(1/31*sqrt(31)*(10*x + 3)) + 7*sqrt(31)*(2550867000*x^6 + 3957759600*x^
5 + 4525420710*x^4 + 2788779072*x^3 + 1299394083*x^2 + 304894531*x + 38489903))/
(250*x^7 + 575*x^6 + 795*x^5 + 699*x^4 + 435*x^3 + 186*x^2 + 52*x + 8)

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Sympy [A]  time = 0.884029, size = 121, normalized size = 0.85 \[ - \frac{2550867000 x^{6} + 3957759600 x^{5} + 4525420710 x^{4} + 2788779072 x^{3} + 1299394083 x^{2} + 304894531 x + 38489903}{53646143250 x^{7} + 123386129475 x^{6} + 170594735535 x^{5} + 149994616527 x^{4} + 93344289255 x^{3} + 39912730578 x^{2} + 11158397796 x + 1716676584} + \frac{2048 \log{\left (x + \frac{1}{2} \right )}}{16807} - \frac{1024 \log{\left (x^{2} + \frac{3 x}{5} + \frac{2}{5} \right )}}{16807} - \frac{116056984 \sqrt{31} \operatorname{atan}{\left (\frac{10 \sqrt{31} x}{31} + \frac{3 \sqrt{31}}{31} \right )}}{15521617447} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(1+2*x)**2/(5*x**2+3*x+2)**4,x)

[Out]

-(2550867000*x**6 + 3957759600*x**5 + 4525420710*x**4 + 2788779072*x**3 + 129939
4083*x**2 + 304894531*x + 38489903)/(53646143250*x**7 + 123386129475*x**6 + 1705
94735535*x**5 + 149994616527*x**4 + 93344289255*x**3 + 39912730578*x**2 + 111583
97796*x + 1716676584) + 2048*log(x + 1/2)/16807 - 1024*log(x**2 + 3*x/5 + 2/5)/1
6807 - 116056984*sqrt(31)*atan(10*sqrt(31)*x/31 + 3*sqrt(31)/31)/15521617447

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GIAC/XCAS [A]  time = 0.207785, size = 170, normalized size = 1.2 \[ -\frac{116056984}{15521617447} \, \sqrt{31} \arctan \left (-\frac{1}{31} \, \sqrt{31}{\left (\frac{7}{2 \, x + 1} - 2\right )}\right ) - \frac{128}{2401 \,{\left (2 \, x + 1\right )}} - \frac{8 \,{\left (\frac{3841449975}{2 \, x + 1} - \frac{8833663680}{{\left (2 \, x + 1\right )}^{2}} + \frac{7499779568}{{\left (2 \, x + 1\right )}^{3}} - \frac{7050406230}{{\left (2 \, x + 1\right )}^{4}} + \frac{1291725897}{{\left (2 \, x + 1\right )}^{5}} - 2009265250\right )}}{1502092011 \,{\left (\frac{4}{2 \, x + 1} - \frac{7}{{\left (2 \, x + 1\right )}^{2}} - 5\right )}^{3}} - \frac{1024}{16807} \,{\rm ln}\left (-\frac{4}{2 \, x + 1} + \frac{7}{{\left (2 \, x + 1\right )}^{2}} + 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((5*x^2 + 3*x + 2)^4*(2*x + 1)^2),x, algorithm="giac")

[Out]

-116056984/15521617447*sqrt(31)*arctan(-1/31*sqrt(31)*(7/(2*x + 1) - 2)) - 128/2
401/(2*x + 1) - 8/1502092011*(3841449975/(2*x + 1) - 8833663680/(2*x + 1)^2 + 74
99779568/(2*x + 1)^3 - 7050406230/(2*x + 1)^4 + 1291725897/(2*x + 1)^5 - 2009265
250)/(4/(2*x + 1) - 7/(2*x + 1)^2 - 5)^3 - 1024/16807*ln(-4/(2*x + 1) + 7/(2*x +
1)^2 + 5)