3.2218 \(\int \frac{1}{(1+2 x) \left (2+3 x+5 x^2\right )^3} \, dx\)

Optimal. Leaf size=89 \[ \frac{20 x+37}{434 \left (5 x^2+3 x+2\right )^2}+\frac{2 (2290 x+2609)}{47089 \left (5 x^2+3 x+2\right )}-\frac{16}{343} \log \left (5 x^2+3 x+2\right )+\frac{32}{343} \log (2 x+1)+\frac{125624 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{329623 \sqrt{31}} \]

[Out]

(37 + 20*x)/(434*(2 + 3*x + 5*x^2)^2) + (2*(2609 + 2290*x))/(47089*(2 + 3*x + 5*
x^2)) + (125624*ArcTan[(3 + 10*x)/Sqrt[31]])/(329623*Sqrt[31]) + (32*Log[1 + 2*x
])/343 - (16*Log[2 + 3*x + 5*x^2])/343

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Rubi [A]  time = 0.191974, antiderivative size = 89, 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 \[ \frac{20 x+37}{434 \left (5 x^2+3 x+2\right )^2}+\frac{2 (2290 x+2609)}{47089 \left (5 x^2+3 x+2\right )}-\frac{16}{343} \log \left (5 x^2+3 x+2\right )+\frac{32}{343} \log (2 x+1)+\frac{125624 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{329623 \sqrt{31}} \]

Antiderivative was successfully verified.

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

[Out]

(37 + 20*x)/(434*(2 + 3*x + 5*x^2)^2) + (2*(2609 + 2290*x))/(47089*(2 + 3*x + 5*
x^2)) + (125624*ArcTan[(3 + 10*x)/Sqrt[31]])/(329623*Sqrt[31]) + (32*Log[1 + 2*x
])/343 - (16*Log[2 + 3*x + 5*x^2])/343

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Rubi in Sympy [A]  time = 26.1548, size = 82, normalized size = 0.92 \[ \frac{20 x + 37}{434 \left (5 x^{2} + 3 x + 2\right )^{2}} + \frac{9160 x + 10436}{94178 \left (5 x^{2} + 3 x + 2\right )} + \frac{32 \log{\left (2 x + 1 \right )}}{343} - \frac{16 \log{\left (5 x^{2} + 3 x + 2 \right )}}{343} + \frac{125624 \sqrt{31} \operatorname{atan}{\left (\sqrt{31} \left (\frac{10 x}{31} + \frac{3}{31}\right ) \right )}}{10218313} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(20*x + 37)/(434*(5*x**2 + 3*x + 2)**2) + (9160*x + 10436)/(94178*(5*x**2 + 3*x
+ 2)) + 32*log(2*x + 1)/343 - 16*log(5*x**2 + 3*x + 2)/343 + 125624*sqrt(31)*ata
n(sqrt(31)*(10*x/31 + 3/31))/10218313

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Mathematica [A]  time = 0.185468, size = 78, normalized size = 0.88 \[ \frac{8 \left (-59582 \log \left (4 \left (5 x^2+3 x+2\right )\right )+\frac{217 \left (45800 x^3+79660 x^2+53968 x+28901\right )}{16 \left (5 x^2+3 x+2\right )^2}+119164 \log (2 x+1)+15703 \sqrt{31} \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )\right )}{10218313} \]

Antiderivative was successfully verified.

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

[Out]

(8*((217*(28901 + 53968*x + 79660*x^2 + 45800*x^3))/(16*(2 + 3*x + 5*x^2)^2) + 1
5703*Sqrt[31]*ArcTan[(3 + 10*x)/Sqrt[31]] + 119164*Log[1 + 2*x] - 59582*Log[4*(2
+ 3*x + 5*x^2)]))/10218313

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Maple [A]  time = 0.017, size = 68, normalized size = 0.8 \[{\frac{32\,\ln \left ( 1+2\,x \right ) }{343}}-{\frac{25}{343\, \left ( 5\,{x}^{2}+3\,x+2 \right ) ^{2}} \left ( -{\frac{6412\,{x}^{3}}{961}}-{\frac{55762\,{x}^{2}}{4805}}-{\frac{188888\,x}{24025}}-{\frac{202307}{48050}} \right ) }-{\frac{16\,\ln \left ( 125\,{x}^{2}+75\,x+50 \right ) }{343}}+{\frac{125624\,\sqrt{31}}{10218313}\arctan \left ({\frac{ \left ( 250\,x+75 \right ) \sqrt{31}}{775}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

32/343*ln(1+2*x)-25/343*(-6412/961*x^3-55762/4805*x^2-188888/24025*x-202307/4805
0)/(5*x^2+3*x+2)^2-16/343*ln(125*x^2+75*x+50)+125624/10218313*31^(1/2)*arctan(1/
775*(250*x+75)*31^(1/2))

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Maxima [A]  time = 0.917749, size = 104, normalized size = 1.17 \[ \frac{125624}{10218313} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + \frac{45800 \, x^{3} + 79660 \, x^{2} + 53968 \, x + 28901}{94178 \,{\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} - \frac{16}{343} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) + \frac{32}{343} \, \log \left (2 \, x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

125624/10218313*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 1/94178*(45800*x^3 +
79660*x^2 + 53968*x + 28901)/(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4) - 16/343*log
(5*x^2 + 3*x + 2) + 32/343*log(2*x + 1)

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Fricas [A]  time = 0.229912, size = 200, normalized size = 2.25 \[ -\frac{\sqrt{31}{\left (30752 \, \sqrt{31}{\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \log \left (5 \, x^{2} + 3 \, x + 2\right ) - 61504 \, \sqrt{31}{\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \log \left (2 \, x + 1\right ) - 251248 \,{\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) - 7 \, \sqrt{31}{\left (45800 \, x^{3} + 79660 \, x^{2} + 53968 \, x + 28901\right )}\right )}}{20436626 \,{\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/20436626*sqrt(31)*(30752*sqrt(31)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*log(5
*x^2 + 3*x + 2) - 61504*sqrt(31)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*log(2*x +
1) - 251248*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*arctan(1/31*sqrt(31)*(10*x +
3)) - 7*sqrt(31)*(45800*x^3 + 79660*x^2 + 53968*x + 28901))/(25*x^4 + 30*x^3 + 2
9*x^2 + 12*x + 4)

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Sympy [A]  time = 0.732131, size = 90, normalized size = 1.01 \[ \frac{45800 x^{3} + 79660 x^{2} + 53968 x + 28901}{2354450 x^{4} + 2825340 x^{3} + 2731162 x^{2} + 1130136 x + 376712} + \frac{32 \log{\left (x + \frac{1}{2} \right )}}{343} - \frac{16 \log{\left (x^{2} + \frac{3 x}{5} + \frac{2}{5} \right )}}{343} + \frac{125624 \sqrt{31} \operatorname{atan}{\left (\frac{10 \sqrt{31} x}{31} + \frac{3 \sqrt{31}}{31} \right )}}{10218313} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(45800*x**3 + 79660*x**2 + 53968*x + 28901)/(2354450*x**4 + 2825340*x**3 + 27311
62*x**2 + 1130136*x + 376712) + 32*log(x + 1/2)/343 - 16*log(x**2 + 3*x/5 + 2/5)
/343 + 125624*sqrt(31)*atan(10*sqrt(31)*x/31 + 3*sqrt(31)/31)/10218313

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GIAC/XCAS [A]  time = 0.223026, size = 92, normalized size = 1.03 \[ \frac{125624}{10218313} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + \frac{45800 \, x^{3} + 79660 \, x^{2} + 53968 \, x + 28901}{94178 \,{\left (5 \, x^{2} + 3 \, x + 2\right )}^{2}} - \frac{16}{343} \,{\rm ln}\left (5 \, x^{2} + 3 \, x + 2\right ) + \frac{32}{343} \,{\rm ln}\left ({\left | 2 \, x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

125624/10218313*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 1/94178*(45800*x^3 +
79660*x^2 + 53968*x + 28901)/(5*x^2 + 3*x + 2)^2 - 16/343*ln(5*x^2 + 3*x + 2) +
32/343*ln(abs(2*x + 1))