3.2219 \(\int \frac{1}{(1+2 x)^2 \left (2+3 x+5 x^2\right )^3} \, dx\)

Optimal. Leaf size=114 \[ \frac{20 x+37}{434 (2 x+1) \left (5 x^2+3 x+2\right )^2}+\frac{5820 x+6427}{47089 (2 x+1) \left (5 x^2+3 x+2\right )}-\frac{192 \log \left (5 x^2+3 x+2\right )}{2401}-\frac{51516}{329623 (2 x+1)}+\frac{384 \log (2 x+1)}{2401}-\frac{1065012 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{2307361 \sqrt{31}} \]

[Out]

-51516/(329623*(1 + 2*x)) + (37 + 20*x)/(434*(1 + 2*x)*(2 + 3*x + 5*x^2)^2) + (6
427 + 5820*x)/(47089*(1 + 2*x)*(2 + 3*x + 5*x^2)) - (1065012*ArcTan[(3 + 10*x)/S
qrt[31]])/(2307361*Sqrt[31]) + (384*Log[1 + 2*x])/2401 - (192*Log[2 + 3*x + 5*x^
2])/2401

_______________________________________________________________________________________

Rubi [A]  time = 0.195953, antiderivative size = 114, 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 \[ \frac{20 x+37}{434 (2 x+1) \left (5 x^2+3 x+2\right )^2}+\frac{5820 x+6427}{47089 (2 x+1) \left (5 x^2+3 x+2\right )}-\frac{192 \log \left (5 x^2+3 x+2\right )}{2401}-\frac{51516}{329623 (2 x+1)}+\frac{384 \log (2 x+1)}{2401}-\frac{1065012 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{2307361 \sqrt{31}} \]

Antiderivative was successfully verified.

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

[Out]

-51516/(329623*(1 + 2*x)) + (37 + 20*x)/(434*(1 + 2*x)*(2 + 3*x + 5*x^2)^2) + (6
427 + 5820*x)/(47089*(1 + 2*x)*(2 + 3*x + 5*x^2)) - (1065012*ArcTan[(3 + 10*x)/S
qrt[31]])/(2307361*Sqrt[31]) + (384*Log[1 + 2*x])/2401 - (192*Log[2 + 3*x + 5*x^
2])/2401

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 29.8789, size = 100, normalized size = 0.88 \[ \frac{384 \log{\left (2 x + 1 \right )}}{2401} - \frac{192 \log{\left (5 x^{2} + 3 x + 2 \right )}}{2401} - \frac{1065012 \sqrt{31} \operatorname{atan}{\left (\sqrt{31} \left (\frac{10 x}{31} + \frac{3}{31}\right ) \right )}}{71528191} + \frac{20 x + 37}{434 \left (2 x + 1\right ) \left (5 x^{2} + 3 x + 2\right )^{2}} + \frac{11640 x + 12854}{94178 \left (2 x + 1\right ) \left (5 x^{2} + 3 x + 2\right )} - \frac{51516}{329623 \left (2 x + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

384*log(2*x + 1)/2401 - 192*log(5*x**2 + 3*x + 2)/2401 - 1065012*sqrt(31)*atan(s
qrt(31)*(10*x/31 + 3/31))/71528191 + (20*x + 37)/(434*(2*x + 1)*(5*x**2 + 3*x +
2)**2) + (11640*x + 12854)/(94178*(2*x + 1)*(5*x**2 + 3*x + 2)) - 51516/(329623*
(2*x + 1))

_______________________________________________________________________________________

Mathematica [A]  time = 0.134821, size = 98, normalized size = 0.86 \[ \frac{4 \left (-\frac{47089 (270 x-43)}{8 \left (5 x^2+3 x+2\right )^2}-\frac{217 (51910 x-15179)}{4 \left (5 x^2+3 x+2\right )}-1429968 \log \left (4 \left (5 x^2+3 x+2\right )\right )-\frac{1668296}{2 x+1}+2859936 \log (2 x+1)-266253 \sqrt{31} \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )\right )}{71528191} \]

Antiderivative was successfully verified.

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

[Out]

(4*(-1668296/(1 + 2*x) - (47089*(-43 + 270*x))/(8*(2 + 3*x + 5*x^2)^2) - (217*(-
15179 + 51910*x))/(4*(2 + 3*x + 5*x^2)) - 266253*Sqrt[31]*ArcTan[(3 + 10*x)/Sqrt
[31]] + 2859936*Log[1 + 2*x] - 1429968*Log[4*(2 + 3*x + 5*x^2)]))/71528191

_______________________________________________________________________________________

Maple [A]  time = 0.017, size = 77, normalized size = 0.7 \[ -{\frac{32}{343+686\,x}}+{\frac{384\,\ln \left ( 1+2\,x \right ) }{2401}}-{\frac{25}{2401\, \left ( 5\,{x}^{2}+3\,x+2 \right ) ^{2}} \left ({\frac{72674\,{x}^{3}}{961}}+{\frac{111769\,{x}^{2}}{4805}}+{\frac{613046\,x}{24025}}-{\frac{490329}{48050}} \right ) }-{\frac{192\,\ln \left ( 125\,{x}^{2}+75\,x+50 \right ) }{2401}}-{\frac{1065012\,\sqrt{31}}{71528191}\arctan \left ({\frac{ \left ( 250\,x+75 \right ) \sqrt{31}}{775}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-32/343/(1+2*x)+384/2401*ln(1+2*x)-25/2401*(72674/961*x^3+111769/4805*x^2+613046
/24025*x-490329/48050)/(5*x^2+3*x+2)^2-192/2401*ln(125*x^2+75*x+50)-1065012/7152
8191*31^(1/2)*arctan(1/775*(250*x+75)*31^(1/2))

_______________________________________________________________________________________

Maxima [A]  time = 0.90807, size = 117, normalized size = 1.03 \[ -\frac{1065012}{71528191} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) - \frac{2575800 \, x^{4} + 2683560 \, x^{3} + 2293598 \, x^{2} + 773110 \, x + 175969}{659246 \,{\left (50 \, x^{5} + 85 \, x^{4} + 88 \, x^{3} + 53 \, x^{2} + 20 \, x + 4\right )}} - \frac{192}{2401} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) + \frac{384}{2401} \, \log \left (2 \, x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1065012/71528191*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) - 1/659246*(2575800*
x^4 + 2683560*x^3 + 2293598*x^2 + 773110*x + 175969)/(50*x^5 + 85*x^4 + 88*x^3 +
53*x^2 + 20*x + 4) - 192/2401*log(5*x^2 + 3*x + 2) + 384/2401*log(2*x + 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.229563, size = 234, normalized size = 2.05 \[ -\frac{\sqrt{31}{\left (369024 \, \sqrt{31}{\left (50 \, x^{5} + 85 \, x^{4} + 88 \, x^{3} + 53 \, x^{2} + 20 \, x + 4\right )} \log \left (5 \, x^{2} + 3 \, x + 2\right ) - 738048 \, \sqrt{31}{\left (50 \, x^{5} + 85 \, x^{4} + 88 \, x^{3} + 53 \, x^{2} + 20 \, x + 4\right )} \log \left (2 \, x + 1\right ) + 2130024 \,{\left (50 \, x^{5} + 85 \, x^{4} + 88 \, x^{3} + 53 \, x^{2} + 20 \, x + 4\right )} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + 7 \, \sqrt{31}{\left (2575800 \, x^{4} + 2683560 \, x^{3} + 2293598 \, x^{2} + 773110 \, x + 175969\right )}\right )}}{143056382 \,{\left (50 \, x^{5} + 85 \, x^{4} + 88 \, x^{3} + 53 \, x^{2} + 20 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/143056382*sqrt(31)*(369024*sqrt(31)*(50*x^5 + 85*x^4 + 88*x^3 + 53*x^2 + 20*x
+ 4)*log(5*x^2 + 3*x + 2) - 738048*sqrt(31)*(50*x^5 + 85*x^4 + 88*x^3 + 53*x^2
+ 20*x + 4)*log(2*x + 1) + 2130024*(50*x^5 + 85*x^4 + 88*x^3 + 53*x^2 + 20*x + 4
)*arctan(1/31*sqrt(31)*(10*x + 3)) + 7*sqrt(31)*(2575800*x^4 + 2683560*x^3 + 229
3598*x^2 + 773110*x + 175969))/(50*x^5 + 85*x^4 + 88*x^3 + 53*x^2 + 20*x + 4)

_______________________________________________________________________________________

Sympy [A]  time = 0.791288, size = 100, normalized size = 0.88 \[ - \frac{2575800 x^{4} + 2683560 x^{3} + 2293598 x^{2} + 773110 x + 175969}{32962300 x^{5} + 56035910 x^{4} + 58013648 x^{3} + 34940038 x^{2} + 13184920 x + 2636984} + \frac{384 \log{\left (x + \frac{1}{2} \right )}}{2401} - \frac{192 \log{\left (x^{2} + \frac{3 x}{5} + \frac{2}{5} \right )}}{2401} - \frac{1065012 \sqrt{31} \operatorname{atan}{\left (\frac{10 \sqrt{31} x}{31} + \frac{3 \sqrt{31}}{31} \right )}}{71528191} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2575800*x**4 + 2683560*x**3 + 2293598*x**2 + 773110*x + 175969)/(32962300*x**5
+ 56035910*x**4 + 58013648*x**3 + 34940038*x**2 + 13184920*x + 2636984) + 384*l
og(x + 1/2)/2401 - 192*log(x**2 + 3*x/5 + 2/5)/2401 - 1065012*sqrt(31)*atan(10*s
qrt(31)*x/31 + 3*sqrt(31)/31)/71528191

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.207171, size = 146, normalized size = 1.28 \[ -\frac{1065012}{71528191} \, \sqrt{31} \arctan \left (-\frac{1}{31} \, \sqrt{31}{\left (\frac{7}{2 \, x + 1} - 2\right )}\right ) - \frac{32}{343 \,{\left (2 \, x + 1\right )}} + \frac{4 \,{\left (\frac{1178375}{2 \, x + 1} - \frac{2320190}{{\left (2 \, x + 1\right )}^{2}} + \frac{87843}{{\left (2 \, x + 1\right )}^{3}} - 1304250\right )}}{2307361 \,{\left (\frac{4}{2 \, x + 1} - \frac{7}{{\left (2 \, x + 1\right )}^{2}} - 5\right )}^{2}} - \frac{192}{2401} \,{\rm ln}\left (-\frac{4}{2 \, x + 1} + \frac{7}{{\left (2 \, x + 1\right )}^{2}} + 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1065012/71528191*sqrt(31)*arctan(-1/31*sqrt(31)*(7/(2*x + 1) - 2)) - 32/343/(2*
x + 1) + 4/2307361*(1178375/(2*x + 1) - 2320190/(2*x + 1)^2 + 87843/(2*x + 1)^3
- 1304250)/(4/(2*x + 1) - 7/(2*x + 1)^2 - 5)^2 - 192/2401*ln(-4/(2*x + 1) + 7/(2
*x + 1)^2 + 5)