3.2220 \(\int \frac{1}{(1+2 x) \left (2+3 x+5 x^2\right )^4} \, dx\)

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]

(37 + 20*x)/(651*(2 + 3*x + 5*x^2)^3) + (4*(1983 + 1805*x))/(141267*(2 + 3*x + 5
*x^2)^2) + (4*(180133 + 203230*x))/(10218313*(2 + 3*x + 5*x^2)) + (19007376*ArcT
an[(3 + 10*x)/Sqrt[31]])/(71528191*Sqrt[31]) + (128*Log[1 + 2*x])/2401 - (64*Log
[2 + 3*x + 5*x^2])/2401

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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]

(37 + 20*x)/(651*(2 + 3*x + 5*x^2)^3) + (4*(1983 + 1805*x))/(141267*(2 + 3*x + 5
*x^2)^2) + (4*(180133 + 203230*x))/(10218313*(2 + 3*x + 5*x^2)) + (19007376*ArcT
an[(3 + 10*x)/Sqrt[31]])/(71528191*Sqrt[31]) + (128*Log[1 + 2*x])/2401 - (64*Log
[2 + 3*x + 5*x^2])/2401

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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]

(20*x + 37)/(651*(5*x**2 + 3*x + 2)**3) + (14440*x + 15864)/(282534*(5*x**2 + 3*
x + 2)**2) + (4877520*x + 4323192)/(61309878*(5*x**2 + 3*x + 2)) + 128*log(2*x +
1)/2401 - 64*log(5*x**2 + 3*x + 2)/2401 + 19007376*sqrt(31)*atan(sqrt(31)*(10*x
/31 + 3/31))/2217373921

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Mathematica [A]  time = 0.224697, size = 88, normalized size = 0.8 \[ \frac{16 \left (-11082252 \log \left (4 \left (5 x^2+3 x+2\right )\right )+\frac{217 \left (60969000 x^5+127202700 x^4+143405620 x^3+105257844 x^2+44933184 x+13831165\right )}{16 \left (5 x^2+3 x+2\right )^3}[phone] \log (2 x+1)+3563883 \sqrt{31} \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )\right )}{6652121763} \]

Antiderivative was successfully verified.

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

[Out]

(16*((217*(13831165 + 44933184*x + 105257844*x^2 + 143405620*x^3 + 127202700*x^4
+ 60969000*x^5))/(16*(2 + 3*x + 5*x^2)^3) + 3563883*Sqrt[31]*ArcTan[(3 + 10*x)/
Sqrt[31]] + 22164504*Log[1 + 2*x] - 11082252*Log[4*(2 + 3*x + 5*x^2)]))/66521217
63

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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]

128/2401*ln(1+2*x)-125/2401*(-1138088/29791*x^5-11872252/148955*x^4-200767868/22
34325*x^3-245601636/3723875*x^2-104844096/3723875*x-19363631/2234325)/(5*x^2+3*x
+2)^3-64/2401*ln(625*x^2+375*x+250)[phone]/2217373921*31^(1/2)*arctan(1/3875*(
1250*x+375)*31^(1/2))

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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]

19007376/2217373921*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 1/30654939*(6096
9000*x^5 + 127202700*x^4 + 143405620*x^3 + 105257844*x^2 + 44933184*x + 13831165
)/(125*x^6 + 225*x^5 + 285*x^4 + 207*x^3 + 114*x^2 + 36*x + 8) - 64/2401*log(5*x
^2 + 3*x + 2) + 128/2401*log(2*x + 1)

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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]

-1/6652121763*sqrt(31)*(5719872*sqrt(31)*(125*x^6 + 225*x^5 + 285*x^4 + 207*x^3
+ 114*x^2 + 36*x + 8)*log(5*x^2 + 3*x + 2) - 11439744*sqrt(31)*(125*x^6 + 225*x^
5 + 285*x^4 + 207*x^3 + 114*x^2 + 36*x + 8)*log(2*x + 1) - 57022128*(125*x^6 + 2
25*x^5 + 285*x^4 + 207*x^3 + 114*x^2 + 36*x + 8)*arctan(1/31*sqrt(31)*(10*x + 3)
) - 7*sqrt(31)*(60969000*x^5 + 127202700*x^4 + 143405620*x^3 + 105257844*x^2 + 4
4933184*x + 13831165))/(125*x^6 + 225*x^5 + 285*x^4 + 207*x^3 + 114*x^2 + 36*x +
8)

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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]

(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) + 128*log(x + 1/2)/2401 - 64*log
(x**2 + 3*x/5 + 2/5)/2401 + 19007376*sqrt(31)*atan(10*sqrt(31)*x/31 + 3*sqrt(31)
/31)/2217373921

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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]

19007376/2217373921*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 1/30654939*(6096
9000*x^5 + 127202700*x^4 + 143405620*x^3 + 105257844*x^2 + 44933184*x + 13831165
)/(5*x^2 + 3*x + 2)^3 - 64/2401*ln(5*x^2 + 3*x + 2) + 128/2401*ln(abs(2*x + 1))