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

Optimal. Leaf size=115 \[ \frac{3625-746 x}{256036 \left (2 x^2-x+3\right )}+\frac{13-6 x}{1012 \left (2 x^2-x+3\right )^2}-\frac{119 \log \left (2 x^2-x+3\right )}{21296}+\frac{119 \log \left (5 x^2+3 x+2\right )}{21296}-\frac{53403 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{5632792 \sqrt{23}}+\frac{247 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{10648 \sqrt{31}} \]

[Out]

(13 - 6*x)/(1012*(3 - x + 2*x^2)^2) + (3625 - 746*x)/(256036*(3 - x + 2*x^2)) -
(53403*ArcTan[(1 - 4*x)/Sqrt[23]])/(5632792*Sqrt[23]) + (247*ArcTan[(3 + 10*x)/S
qrt[31]])/(10648*Sqrt[31]) - (119*Log[3 - x + 2*x^2])/21296 + (119*Log[2 + 3*x +
5*x^2])/21296

_______________________________________________________________________________________

Rubi [A]  time = 0.280958, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28 \[ \frac{3625-746 x}{256036 \left (2 x^2-x+3\right )}+\frac{13-6 x}{1012 \left (2 x^2-x+3\right )^2}-\frac{119 \log \left (2 x^2-x+3\right )}{21296}+\frac{119 \log \left (5 x^2+3 x+2\right )}{21296}-\frac{53403 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{5632792 \sqrt{23}}+\frac{247 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{10648 \sqrt{31}} \]

Antiderivative was successfully verified.

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

[Out]

(13 - 6*x)/(1012*(3 - x + 2*x^2)^2) + (3625 - 746*x)/(256036*(3 - x + 2*x^2)) -
(53403*ArcTan[(1 - 4*x)/Sqrt[23]])/(5632792*Sqrt[23]) + (247*ArcTan[(3 + 10*x)/S
qrt[31]])/(10648*Sqrt[31]) - (119*Log[3 - x + 2*x^2])/21296 + (119*Log[2 + 3*x +
5*x^2])/21296

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 77.8536, size = 107, normalized size = 0.93 \[ \frac{- 180532 x + 877250}{61960712 \left (2 x^{2} - x + 3\right )} + \frac{- 66 x + 143}{11132 \left (2 x^{2} - x + 3\right )^{2}} - \frac{119 \log{\left (2 x^{2} - x + 3 \right )}}{21296} + \frac{119 \log{\left (5 x^{2} + 3 x + 2 \right )}}{21296} + \frac{53403 \sqrt{23} \operatorname{atan}{\left (\sqrt{23} \left (\frac{4 x}{23} - \frac{1}{23}\right ) \right )}}{129554216} + \frac{247 \sqrt{31} \operatorname{atan}{\left (\sqrt{31} \left (\frac{10 x}{31} + \frac{3}{31}\right ) \right )}}{330088} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-180532*x + 877250)/(61960712*(2*x**2 - x + 3)) + (-66*x + 143)/(11132*(2*x**2
- x + 3)**2) - 119*log(2*x**2 - x + 3)/21296 + 119*log(5*x**2 + 3*x + 2)/21296 +
53403*sqrt(23)*atan(sqrt(23)*(4*x/23 - 1/23))/129554216 + 247*sqrt(31)*atan(sqr
t(31)*(10*x/31 + 3/31))/330088

_______________________________________________________________________________________

Mathematica [A]  time = 0.33811, size = 99, normalized size = 0.86 \[ \frac{713 \left (-62951 \log \left (2 x^2-x+3\right )+62951 \log \left (5 x^2+3 x+2\right )-\frac{44 \left (1492 x^3-7996 x^2+7381 x-14164\right )}{\left (-2 x^2+x-3\right )^2}\right )+3310986 \sqrt{23} \tan ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )+6010498 \sqrt{31} \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{8032361392} \]

Antiderivative was successfully verified.

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

[Out]

(3310986*Sqrt[23]*ArcTan[(-1 + 4*x)/Sqrt[23]] + 6010498*Sqrt[31]*ArcTan[(3 + 10*
x)/Sqrt[31]] + 713*((-44*(-14164 + 7381*x - 7996*x^2 + 1492*x^3))/(-3 + x - 2*x^
2)^2 - 62951*Log[3 - x + 2*x^2] + 62951*Log[2 + 3*x + 5*x^2]))/8032361392

_______________________________________________________________________________________

Maple [A]  time = 0.01, size = 89, normalized size = 0.8 \[{\frac{119\,\ln \left ( 5\,{x}^{2}+3\,x+2 \right ) }{21296}}+{\frac{247\,\sqrt{31}}{330088}\arctan \left ({\frac{ \left ( 3+10\,x \right ) \sqrt{31}}{31}} \right ) }-{\frac{1}{2662\, \left ( 2\,{x}^{2}-x+3 \right ) ^{2}} \left ({\frac{8206\,{x}^{3}}{529}}-{\frac{43978\,{x}^{2}}{529}}+{\frac{81191\,x}{1058}}-{\frac{77902}{529}} \right ) }-{\frac{119\,\ln \left ( 8\,{x}^{2}-4\,x+12 \right ) }{21296}}+{\frac{53403\,\sqrt{23}}{129554216}\arctan \left ({\frac{ \left ( 16\,x-4 \right ) \sqrt{23}}{92}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

119/21296*ln(5*x^2+3*x+2)+247/330088*arctan(1/31*(3+10*x)*31^(1/2))*31^(1/2)-1/2
662*(8206/529*x^3-43978/529*x^2+81191/1058*x-77902/529)/(2*x^2-x+3)^2-119/21296*
ln(8*x^2-4*x+12)+53403/129554216*23^(1/2)*arctan(1/92*(16*x-4)*23^(1/2))

_______________________________________________________________________________________

Maxima [A]  time = 0.767025, size = 132, normalized size = 1.15 \[ \frac{247}{330088} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + \frac{53403}{129554216} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{1492 \, x^{3} - 7996 \, x^{2} + 7381 \, x - 14164}{256036 \,{\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} + \frac{119}{21296} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) - \frac{119}{21296} \, \log \left (2 \, x^{2} - x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

247/330088*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 53403/129554216*sqrt(23)*
arctan(1/23*sqrt(23)*(4*x - 1)) - 1/256036*(1492*x^3 - 7996*x^2 + 7381*x - 14164
)/(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9) + 119/21296*log(5*x^2 + 3*x + 2) - 119/2129
6*log(2*x^2 - x + 3)

_______________________________________________________________________________________

Fricas [A]  time = 0.271669, size = 275, normalized size = 2.39 \[ \frac{\sqrt{31} \sqrt{23}{\left (62951 \, \sqrt{31} \sqrt{23}{\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \log \left (5 \, x^{2} + 3 \, x + 2\right ) - 62951 \, \sqrt{31} \sqrt{23}{\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \log \left (2 \, x^{2} - x + 3\right ) + 261326 \, \sqrt{23}{\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + 106806 \, \sqrt{31}{\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - 44 \, \sqrt{31} \sqrt{23}{\left (1492 \, x^{3} - 7996 \, x^{2} + 7381 \, x - 14164\right )}\right )}}{8032361392 \,{\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8032361392*sqrt(31)*sqrt(23)*(62951*sqrt(31)*sqrt(23)*(4*x^4 - 4*x^3 + 13*x^2
- 6*x + 9)*log(5*x^2 + 3*x + 2) - 62951*sqrt(31)*sqrt(23)*(4*x^4 - 4*x^3 + 13*x^
2 - 6*x + 9)*log(2*x^2 - x + 3) + 261326*sqrt(23)*(4*x^4 - 4*x^3 + 13*x^2 - 6*x
+ 9)*arctan(1/31*sqrt(31)*(10*x + 3)) + 106806*sqrt(31)*(4*x^4 - 4*x^3 + 13*x^2
- 6*x + 9)*arctan(1/23*sqrt(23)*(4*x - 1)) - 44*sqrt(31)*sqrt(23)*(1492*x^3 - 79
96*x^2 + 7381*x - 14164))/(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)

_______________________________________________________________________________________

Sympy [A]  time = 0.486023, size = 122, normalized size = 1.06 \[ - \frac{1492 x^{3} - 7996 x^{2} + 7381 x - 14164}{1024144 x^{4} - 1024144 x^{3} + 3328468 x^{2} - 1536216 x + 2304324} - \frac{119 \log{\left (x^{2} - \frac{x}{2} + \frac{3}{2} \right )}}{21296} + \frac{119 \log{\left (x^{2} + \frac{3 x}{5} + \frac{2}{5} \right )}}{21296} + \frac{53403 \sqrt{23} \operatorname{atan}{\left (\frac{4 \sqrt{23} x}{23} - \frac{\sqrt{23}}{23} \right )}}{129554216} + \frac{247 \sqrt{31} \operatorname{atan}{\left (\frac{10 \sqrt{31} x}{31} + \frac{3 \sqrt{31}}{31} \right )}}{330088} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(1492*x**3 - 7996*x**2 + 7381*x - 14164)/(1024144*x**4 - 1024144*x**3 + 3328468
*x**2 - 1536216*x + 2304324) - 119*log(x**2 - x/2 + 3/2)/21296 + 119*log(x**2 +
3*x/5 + 2/5)/21296 + 53403*sqrt(23)*atan(4*sqrt(23)*x/23 - sqrt(23)/23)/12955421
6 + 247*sqrt(31)*atan(10*sqrt(31)*x/31 + 3*sqrt(31)/31)/330088

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.265309, size = 119, normalized size = 1.03 \[ \frac{247}{330088} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + \frac{53403}{129554216} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{1492 \, x^{3} - 7996 \, x^{2} + 7381 \, x - 14164}{256036 \,{\left (2 \, x^{2} - x + 3\right )}^{2}} + \frac{119}{21296} \,{\rm ln}\left (5 \, x^{2} + 3 \, x + 2\right ) - \frac{119}{21296} \,{\rm ln}\left (2 \, x^{2} - x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

247/330088*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 53403/129554216*sqrt(23)*
arctan(1/23*sqrt(23)*(4*x - 1)) - 1/256036*(1492*x^3 - 7996*x^2 + 7381*x - 14164
)/(2*x^2 - x + 3)^2 + 119/21296*ln(5*x^2 + 3*x + 2) - 119/21296*ln(2*x^2 - x + 3
)