3.2309 \(\int \frac{1}{\sqrt{1+2 x} \left (2+3 x+5 x^2\right )^2} \, dx\)

Optimal. Leaf size=270 \[ \frac{\sqrt{2 x+1} (20 x+37)}{217 \left (5 x^2+3 x+2\right )}-\frac{1}{217} \sqrt{\frac{1}{434} \left (10325 \sqrt{35}-32678\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )+\frac{1}{217} \sqrt{\frac{1}{434} \left (10325 \sqrt{35}-32678\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )-\frac{1}{217} \sqrt{\frac{2}{217} \left (32678+10325 \sqrt{35}\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )+\frac{1}{217} \sqrt{\frac{2}{217} \left (32678+10325 \sqrt{35}\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right ) \]

[Out]

(Sqrt[1 + 2*x]*(37 + 20*x))/(217*(2 + 3*x + 5*x^2)) - (Sqrt[(2*(32678 + 10325*Sq
rt[35]))/217]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 +
Sqrt[35])]])/217 + (Sqrt[(2*(32678 + 10325*Sqrt[35]))/217]*ArcTan[(Sqrt[10*(2 +
Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/217 - (Sqrt[(-32678 +
10325*Sqrt[35])/434]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1
+ 2*x)])/217 + (Sqrt[(-32678 + 10325*Sqrt[35])/434]*Log[Sqrt[35] + Sqrt[10*(2 +
Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/217

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Rubi [A]  time = 1.09322, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ \frac{\sqrt{2 x+1} (20 x+37)}{217 \left (5 x^2+3 x+2\right )}-\frac{1}{217} \sqrt{\frac{1}{434} \left (10325 \sqrt{35}-32678\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )+\frac{1}{217} \sqrt{\frac{1}{434} \left (10325 \sqrt{35}-32678\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )-\frac{1}{217} \sqrt{\frac{2}{217} \left (32678+10325 \sqrt{35}\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )+\frac{1}{217} \sqrt{\frac{2}{217} \left (32678+10325 \sqrt{35}\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 + 2*x]*(37 + 20*x))/(217*(2 + 3*x + 5*x^2)) - (Sqrt[(2*(32678 + 10325*Sq
rt[35]))/217]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 +
Sqrt[35])]])/217 + (Sqrt[(2*(32678 + 10325*Sqrt[35]))/217]*ArcTan[(Sqrt[10*(2 +
Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/217 - (Sqrt[(-32678 +
10325*Sqrt[35])/434]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1
+ 2*x)])/217 + (Sqrt[(-32678 + 10325*Sqrt[35])/434]*Log[Sqrt[35] + Sqrt[10*(2 +
Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/217

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Rubi in Sympy [A]  time = 68.9657, size = 372, normalized size = 1.38 \[ \frac{\sqrt{2 x + 1} \left (20 x + 37\right )}{217 \left (5 x^{2} + 3 x + 2\right )} - \frac{\sqrt{14} \left (- 2 \sqrt{35} + 97\right ) \log{\left (2 x - \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \sqrt{2 x + 1}}{5} + 1 + \frac{\sqrt{35}}{5} \right )}}{3038 \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{14} \left (- 2 \sqrt{35} + 97\right ) \log{\left (2 x + \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \sqrt{2 x + 1}}{5} + 1 + \frac{\sqrt{35}}{5} \right )}}{3038 \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{35} \left (- \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \left (- 4 \sqrt{35} + 194\right )}{10} + \frac{194 \sqrt{10} \sqrt{2 + \sqrt{35}}}{5}\right ) \operatorname{atan}{\left (\frac{\sqrt{10} \left (\sqrt{2 x + 1} - \frac{\sqrt{20 + 10 \sqrt{35}}}{10}\right )}{\sqrt{-2 + \sqrt{35}}} \right )}}{1519 \sqrt{-2 + \sqrt{35}} \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{35} \left (- \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \left (- 4 \sqrt{35} + 194\right )}{10} + \frac{194 \sqrt{10} \sqrt{2 + \sqrt{35}}}{5}\right ) \operatorname{atan}{\left (\frac{\sqrt{10} \left (\sqrt{2 x + 1} + \frac{\sqrt{20 + 10 \sqrt{35}}}{10}\right )}{\sqrt{-2 + \sqrt{35}}} \right )}}{1519 \sqrt{-2 + \sqrt{35}} \sqrt{2 + \sqrt{35}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2*x + 1)*(20*x + 37)/(217*(5*x**2 + 3*x + 2)) - sqrt(14)*(-2*sqrt(35) + 97)
*log(2*x - sqrt(10)*sqrt(2 + sqrt(35))*sqrt(2*x + 1)/5 + 1 + sqrt(35)/5)/(3038*s
qrt(2 + sqrt(35))) + sqrt(14)*(-2*sqrt(35) + 97)*log(2*x + sqrt(10)*sqrt(2 + sqr
t(35))*sqrt(2*x + 1)/5 + 1 + sqrt(35)/5)/(3038*sqrt(2 + sqrt(35))) + sqrt(35)*(-
sqrt(10)*sqrt(2 + sqrt(35))*(-4*sqrt(35) + 194)/10 + 194*sqrt(10)*sqrt(2 + sqrt(
35))/5)*atan(sqrt(10)*(sqrt(2*x + 1) - sqrt(20 + 10*sqrt(35))/10)/sqrt(-2 + sqrt
(35)))/(1519*sqrt(-2 + sqrt(35))*sqrt(2 + sqrt(35))) + sqrt(35)*(-sqrt(10)*sqrt(
2 + sqrt(35))*(-4*sqrt(35) + 194)/10 + 194*sqrt(10)*sqrt(2 + sqrt(35))/5)*atan(s
qrt(10)*(sqrt(2*x + 1) + sqrt(20 + 10*sqrt(35))/10)/sqrt(-2 + sqrt(35)))/(1519*s
qrt(-2 + sqrt(35))*sqrt(2 + sqrt(35)))

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Mathematica [C]  time = 0.745805, size = 147, normalized size = 0.54 \[ \frac{2 \left (\frac{31 \sqrt{2 x+1} (20 x+37)}{10 x^2+6 x+4}+\frac{\left (62-101 i \sqrt{31}\right ) \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2-i \sqrt{31}}}\right )}{\sqrt{-\frac{1}{5} i \left (\sqrt{31}-2 i\right )}}+\frac{\left (62+101 i \sqrt{31}\right ) \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2+i \sqrt{31}}}\right )}{\sqrt{\frac{1}{5} i \left (\sqrt{31}+2 i\right )}}\right )}{6727} \]

Antiderivative was successfully verified.

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

[Out]

(2*((31*Sqrt[1 + 2*x]*(37 + 20*x))/(4 + 6*x + 10*x^2) + ((62 - (101*I)*Sqrt[31])
*ArcTan[Sqrt[5 + 10*x]/Sqrt[-2 - I*Sqrt[31]]])/Sqrt[(-I/5)*(-2*I + Sqrt[31])] +
((62 + (101*I)*Sqrt[31])*ArcTan[Sqrt[5 + 10*x]/Sqrt[-2 + I*Sqrt[31]]])/Sqrt[(I/5
)*(2*I + Sqrt[31])]))/6727

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Maple [B]  time = 0.227, size = 1288, normalized size = 4.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/47089*(-2/37375*(-3244150*5^(1/2)*7^(1/2)+6488300)/(2*5^(1/2)-5*7^(1/2))*5^(1
/2)*(1+2*x)^(1/2)+1/7475/(2*5^(1/2)-5*7^(1/2))*(2*5^(1/2)*7^(1/2)+4)^(1/2)*(-194
6490*5^(1/2)*7^(1/2)[phone]))/(1/5*5^(1/2)*7^(1/2)+2*x+1-1/5*(2*5^(1/2)*7^(1/2
)+4)^(1/2)*5^(1/2)*(1+2*x)^(1/2))-101575/94178/(-50*5^(1/2)+125*7^(1/2))*ln(5*(2
*5^(1/2)-5*7^(1/2))*((2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(1+2*x)^(1/2)-5^(1/2)*7
^(1/2)-10*x-5))*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*7^(1/2)+29125/6727/(-50*5^(1
/2)+125*7^(1/2))*ln(5*(2*5^(1/2)-5*7^(1/2))*((2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)
*(1+2*x)^(1/2)-5^(1/2)*7^(1/2)-10*x-5))*(2*5^(1/2)*7^(1/2)+4)^(1/2)+388/217/(94*
5^(1/2)*7^(1/2)-436)^(1/2)*arctan(1/25*(2*(-50*5^(1/2)+125*7^(1/2))*(1+2*x)^(1/2
)+5*(2*5^(1/2)-5*7^(1/2))*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2))/(94*5^(1/2)*7^(1/
2)-436)^(1/2))*5^(1/2)-776/1519/(94*5^(1/2)*7^(1/2)-436)^(1/2)*arctan(1/25*(2*(-
50*5^(1/2)+125*7^(1/2))*(1+2*x)^(1/2)+5*(2*5^(1/2)-5*7^(1/2))*(2*5^(1/2)*7^(1/2)
+4)^(1/2)*5^(1/2))/(94*5^(1/2)*7^(1/2)-436)^(1/2))*7^(1/2)+610900/47089/(94*5^(1
/2)*7^(1/2)-436)^(1/2)*arctan(1/25*(2*(-50*5^(1/2)+125*7^(1/2))*(1+2*x)^(1/2)+5*
(2*5^(1/2)-5*7^(1/2))*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2))/(94*5^(1/2)*7^(1/2)-4
36)^(1/2))*(2*5^(1/2)*7^(1/2)+4)*5^(1/2)/(-50*5^(1/2)+125*7^(1/2))*7^(1/2)-62437
5/6727/(94*5^(1/2)*7^(1/2)-436)^(1/2)*arctan(1/25*(2*(-50*5^(1/2)+125*7^(1/2))*(
1+2*x)^(1/2)+5*(2*5^(1/2)-5*7^(1/2))*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2))/(94*5^
(1/2)*7^(1/2)-436)^(1/2))*(2*5^(1/2)*7^(1/2)+4)/(-50*5^(1/2)+125*7^(1/2))+5/4708
9*(2/37375*(-3244150*5^(1/2)*7^(1/2)+6488300)/(2*5^(1/2)-5*7^(1/2))*5^(1/2)*(1+2
*x)^(1/2)+1/7475/(2*5^(1/2)-5*7^(1/2))*(2*5^(1/2)*7^(1/2)+4)^(1/2)*(-1946490*5^(
1/2)*7^(1/2)[phone]))/(1/5*5^(1/2)*7^(1/2)+2*x+1+1/5*(2*5^(1/2)*7^(1/2)+4)^(1/
2)*5^(1/2)*(1+2*x)^(1/2))+101575/94178/(-50*5^(1/2)+125*7^(1/2))*ln(-5*(2*5^(1/2
)-5*7^(1/2))*(5^(1/2)*7^(1/2)+10*x+5+(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(1+2*x)
^(1/2)))*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*7^(1/2)-29125/6727/(-50*5^(1/2)+125
*7^(1/2))*ln(-5*(2*5^(1/2)-5*7^(1/2))*(5^(1/2)*7^(1/2)+10*x+5+(2*5^(1/2)*7^(1/2)
+4)^(1/2)*5^(1/2)*(1+2*x)^(1/2)))*(2*5^(1/2)*7^(1/2)+4)^(1/2)+388/217/(94*5^(1/2
)*7^(1/2)-436)^(1/2)*arctan(1/25*(2*(-50*5^(1/2)+125*7^(1/2))*(1+2*x)^(1/2)-5*(2
*5^(1/2)-5*7^(1/2))*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2))/(94*5^(1/2)*7^(1/2)-436
)^(1/2))*5^(1/2)-776/1519/(94*5^(1/2)*7^(1/2)-436)^(1/2)*arctan(1/25*(2*(-50*5^(
1/2)+125*7^(1/2))*(1+2*x)^(1/2)-5*(2*5^(1/2)-5*7^(1/2))*(2*5^(1/2)*7^(1/2)+4)^(1
/2)*5^(1/2))/(94*5^(1/2)*7^(1/2)-436)^(1/2))*7^(1/2)+610900/47089/(94*5^(1/2)*7^
(1/2)-436)^(1/2)*arctan(1/25*(2*(-50*5^(1/2)+125*7^(1/2))*(1+2*x)^(1/2)-5*(2*5^(
1/2)-5*7^(1/2))*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2))/(94*5^(1/2)*7^(1/2)-436)^(1
/2))*(2*5^(1/2)*7^(1/2)+4)*5^(1/2)/(-50*5^(1/2)+125*7^(1/2))*7^(1/2)-624375/6727
/(94*5^(1/2)*7^(1/2)-436)^(1/2)*arctan(1/25*(2*(-50*5^(1/2)+125*7^(1/2))*(1+2*x)
^(1/2)-5*(2*5^(1/2)-5*7^(1/2))*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2))/(94*5^(1/2)*
7^(1/2)-436)^(1/2))*(2*5^(1/2)*7^(1/2)+4)/(-50*5^(1/2)+125*7^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{2} \sqrt{2 \, x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((5*x^2 + 3*x + 2)^2*sqrt(2*x + 1)), x)

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Fricas [A]  time = 0.275888, size = 1299, normalized size = 4.81 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1831550858746*329623^(3/4)*sqrt(826)*sqrt(31)*(329623^(1/4)*sqrt(826)*sqrt(31)
*(32678*sqrt(7)*(20*x + 37) - 72275*sqrt(5)*(20*x + 37))*sqrt(2*x + 1)*sqrt((326
78*sqrt(7)*sqrt(5) - 361375)/(674800700*sqrt(7)*sqrt(5) - 4799048559)) - 8045492
*17405^(1/4)*sqrt(7)*(5*x^2 + 3*x + 2)*arctan(627347*17405^(1/4)*sqrt(31)*(505*s
qrt(7) - 264*sqrt(5))/(329623^(1/4)*sqrt(9145)*sqrt(826)*sqrt(31)*(32678*sqrt(7)
- 72275*sqrt(5))*sqrt(sqrt(7)*(329623^(1/4)*17405^(1/4)*sqrt(826)*(482989380583
726445493049940619221886265474170993098*sqrt(7)*sqrt(5) - 2857432837452652734362
918220094532883072047512224503)*sqrt(2*x + 1)*sqrt((32678*sqrt(7)*sqrt(5) - 3613
75)/(674800700*sqrt(7)*sqrt(5) - 4799048559)) + 2065*sqrt(7)*(443792256356847122
0638688941619965174410065763500*sqrt(7)*sqrt(5)*(2*x + 1) - 52510891917584737091
097113282085264347697791933598*x - 262554459587923685455485566410426321738488959
66799) + 2891*sqrt(5)*(4437922563568471220638688941619965174410065763500*sqrt(7)
*sqrt(5) - 26255445958792368545548556641042632173848895966799))/(443792256356847
1220638688941619965174410065763500*sqrt(7)*sqrt(5) - 262554459587923685455485566
41042632173848895966799))*sqrt((32678*sqrt(7)*sqrt(5) - 361375)/(674800700*sqrt(
7)*sqrt(5) - 4799048559)) + 64015*329623^(1/4)*sqrt(826)*sqrt(2*x + 1)*(32678*sq
rt(7) - 72275*sqrt(5))*sqrt((32678*sqrt(7)*sqrt(5) - 361375)/(674800700*sqrt(7)*
sqrt(5) - 4799048559)) + 19447757*17405^(1/4)*(10*sqrt(7) - 97*sqrt(5)))) - 8045
492*17405^(1/4)*sqrt(7)*(5*x^2 + 3*x + 2)*arctan(627347*17405^(1/4)*sqrt(31)*(50
5*sqrt(7) - 264*sqrt(5))/(329623^(1/4)*sqrt(9145)*sqrt(826)*sqrt(31)*(32678*sqrt
(7) - 72275*sqrt(5))*sqrt(-sqrt(7)*(329623^(1/4)*17405^(1/4)*sqrt(826)*(48298938
0583726445493049940619221886265474170993098*sqrt(7)*sqrt(5) - 285743283745265273
4362918220094532883072047512224503)*sqrt(2*x + 1)*sqrt((32678*sqrt(7)*sqrt(5) -
361375)/(674800700*sqrt(7)*sqrt(5) - 4799048559)) - 2065*sqrt(7)*(44379225635684
71220638688941619965174410065763500*sqrt(7)*sqrt(5)*(2*x + 1) - 5251089191758473
7091097113282085264347697791933598*x - 26255445958792368545548556641042632173848
895966799) - 2891*sqrt(5)*(4437922563568471220638688941619965174410065763500*sqr
t(7)*sqrt(5) - 26255445958792368545548556641042632173848895966799))/(44379225635
68471220638688941619965174410065763500*sqrt(7)*sqrt(5) - 26255445958792368545548
556641042632173848895966799))*sqrt((32678*sqrt(7)*sqrt(5) - 361375)/(674800700*s
qrt(7)*sqrt(5) - 4799048559)) + 64015*329623^(1/4)*sqrt(826)*sqrt(2*x + 1)*(3267
8*sqrt(7) - 72275*sqrt(5))*sqrt((32678*sqrt(7)*sqrt(5) - 361375)/(674800700*sqrt
(7)*sqrt(5) - 4799048559)) - 19447757*17405^(1/4)*(10*sqrt(7) - 97*sqrt(5)))) +
7*17405^(1/4)*sqrt(31)*(32678*sqrt(7)*(5*x^2 + 3*x + 2) - 72275*sqrt(5)*(5*x^2 +
3*x + 2))*log(22862500/49*sqrt(7)*(329623^(1/4)*17405^(1/4)*sqrt(826)*(48298938
0583726445493049940619221886265474170993098*sqrt(7)*sqrt(5) - 285743283745265273
4362918220094532883072047512224503)*sqrt(2*x + 1)*sqrt((32678*sqrt(7)*sqrt(5) -
361375)/(674800700*sqrt(7)*sqrt(5) - 4799048559)) + 2065*sqrt(7)*(44379225635684
71220638688941619965174410065763500*sqrt(7)*sqrt(5)*(2*x + 1) - 5251089191758473
7091097113282085264347697791933598*x - 26255445958792368545548556641042632173848
895966799) + 2891*sqrt(5)*(4437922563568471220638688941619965174410065763500*sqr
t(7)*sqrt(5) - 26255445958792368545548556641042632173848895966799))/(44379225635
68471220638688941619965174410065763500*sqrt(7)*sqrt(5) - 26255445958792368545548
556641042632173848895966799)) - 7*17405^(1/4)*sqrt(31)*(32678*sqrt(7)*(5*x^2 + 3
*x + 2) - 72275*sqrt(5)*(5*x^2 + 3*x + 2))*log(-22862500/49*sqrt(7)*(329623^(1/4
)*17405^(1/4)*sqrt(826)*(482989380583726445493049940619221886265474170993098*sqr
t(7)*sqrt(5) - 2857432837452652734362918220094532883072047512224503)*sqrt(2*x +
1)*sqrt((32678*sqrt(7)*sqrt(5) - 361375)/(674800700*sqrt(7)*sqrt(5) - 4799048559
)) - 2065*sqrt(7)*(4437922563568471220638688941619965174410065763500*sqrt(7)*sqr
t(5)*(2*x + 1) - 52510891917584737091097113282085264347697791933598*x - 26255445
958792368545548556641042632173848895966799) - 2891*sqrt(5)*(44379225635684712206
38688941619965174410065763500*sqrt(7)*sqrt(5) - 26255445958792368545548556641042
632173848895966799))/(4437922563568471220638688941619965174410065763500*sqrt(7)*
sqrt(5) - 26255445958792368545548556641042632173848895966799)))/((32678*sqrt(7)*
(5*x^2 + 3*x + 2) - 72275*sqrt(5)*(5*x^2 + 3*x + 2))*sqrt((32678*sqrt(7)*sqrt(5)
- 361375)/(674800700*sqrt(7)*sqrt(5) - 4799048559)))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{2 x + 1} \left (5 x^{2} + 3 x + 2\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(sqrt(2*x + 1)*(5*x**2 + 3*x + 2)**2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{2} \sqrt{2 \, x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((5*x^2 + 3*x + 2)^2*sqrt(2*x + 1)), x)