3.2308 \(\int \frac{\sqrt{1+2 x}}{\left (2+3 x+5 x^2\right )^2} \, dx\)

Optimal. Leaf size=270 \[ \frac{\sqrt{2 x+1} (10 x+3)}{31 \left (5 x^2+3 x+2\right )}+\frac{1}{31} \sqrt{\frac{1}{434} \left (47 \sqrt{35}-218\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )-\frac{1}{31} \sqrt{\frac{1}{434} \left (47 \sqrt{35}-218\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )-\frac{1}{31} \sqrt{\frac{2}{217} \left (218+47 \sqrt{35}\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )+\frac{1}{31} \sqrt{\frac{2}{217} \left (218+47 \sqrt{35}\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right ) \]

[Out]

(Sqrt[1 + 2*x]*(3 + 10*x))/(31*(2 + 3*x + 5*x^2)) - (Sqrt[(2*(218 + 47*Sqrt[35])
)/217]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35
])]])/31 + (Sqrt[(2*(218 + 47*Sqrt[35]))/217]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] +
10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/31 + (Sqrt[(-218 + 47*Sqrt[35])/434
]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/31 - (Sqr
t[(-218 + 47*Sqrt[35])/434]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x]
+ 5*(1 + 2*x)])/31

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Rubi [A]  time = 1.06701, 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 \[ \frac{\sqrt{2 x+1} (10 x+3)}{31 \left (5 x^2+3 x+2\right )}+\frac{1}{31} \sqrt{\frac{1}{434} \left (47 \sqrt{35}-218\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )-\frac{1}{31} \sqrt{\frac{1}{434} \left (47 \sqrt{35}-218\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )-\frac{1}{31} \sqrt{\frac{2}{217} \left (218+47 \sqrt{35}\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )+\frac{1}{31} \sqrt{\frac{2}{217} \left (218+47 \sqrt{35}\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 + 2*x]*(3 + 10*x))/(31*(2 + 3*x + 5*x^2)) - (Sqrt[(2*(218 + 47*Sqrt[35])
)/217]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35
])]])/31 + (Sqrt[(2*(218 + 47*Sqrt[35]))/217]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] +
10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/31 + (Sqrt[(-218 + 47*Sqrt[35])/434
]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/31 - (Sqr
t[(-218 + 47*Sqrt[35])/434]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x]
+ 5*(1 + 2*x)])/31

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Rubi in Sympy [A]  time = 69.966, size = 369, normalized size = 1.37 \[ \frac{\sqrt{2 x + 1} \left (10 x + 3\right )}{31 \left (5 x^{2} + 3 x + 2\right )} - \frac{\sqrt{14} \left (- \sqrt{35} + 2\right ) \log{\left (2 x - \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \sqrt{2 x + 1}}{5} + 1 + \frac{\sqrt{35}}{5} \right )}}{434 \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{14} \left (- \sqrt{35} + 2\right ) \log{\left (2 x + \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \sqrt{2 x + 1}}{5} + 1 + \frac{\sqrt{35}}{5} \right )}}{434 \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{35} \left (- \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \left (- 2 \sqrt{35} + 4\right )}{10} + \frac{4 \sqrt{10} \sqrt{2 + \sqrt{35}}}{5}\right ) \operatorname{atan}{\left (\frac{\sqrt{10} \left (\sqrt{2 x + 1} - \frac{\sqrt{20 + 10 \sqrt{35}}}{10}\right )}{\sqrt{-2 + \sqrt{35}}} \right )}}{217 \sqrt{-2 + \sqrt{35}} \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{35} \left (- \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \left (- 2 \sqrt{35} + 4\right )}{10} + \frac{4 \sqrt{10} \sqrt{2 + \sqrt{35}}}{5}\right ) \operatorname{atan}{\left (\frac{\sqrt{10} \left (\sqrt{2 x + 1} + \frac{\sqrt{20 + 10 \sqrt{35}}}{10}\right )}{\sqrt{-2 + \sqrt{35}}} \right )}}{217 \sqrt{-2 + \sqrt{35}} \sqrt{2 + \sqrt{35}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2*x + 1)*(10*x + 3)/(31*(5*x**2 + 3*x + 2)) - sqrt(14)*(-sqrt(35) + 2)*log(
2*x - sqrt(10)*sqrt(2 + sqrt(35))*sqrt(2*x + 1)/5 + 1 + sqrt(35)/5)/(434*sqrt(2
+ sqrt(35))) + sqrt(14)*(-sqrt(35) + 2)*log(2*x + sqrt(10)*sqrt(2 + sqrt(35))*sq
rt(2*x + 1)/5 + 1 + sqrt(35)/5)/(434*sqrt(2 + sqrt(35))) + sqrt(35)*(-sqrt(10)*s
qrt(2 + sqrt(35))*(-2*sqrt(35) + 4)/10 + 4*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)))/(217*sq
rt(-2 + sqrt(35))*sqrt(2 + sqrt(35))) + sqrt(35)*(-sqrt(10)*sqrt(2 + sqrt(35))*(
-2*sqrt(35) + 4)/10 + 4*sqrt(10)*sqrt(2 + sqrt(35))/5)*atan(sqrt(10)*(sqrt(2*x +
1) + sqrt(20 + 10*sqrt(35))/10)/sqrt(-2 + sqrt(35)))/(217*sqrt(-2 + sqrt(35))*s
qrt(2 + sqrt(35)))

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Mathematica [C]  time = 0.878634, size = 143, normalized size = 0.53 \[ \frac{2}{961} \left (\frac{31 \sqrt{2 x+1} (10 x+3)}{10 x^2+6 x+4}+\frac{\left (\sqrt{31}-4 i\right ) \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2-i \sqrt{31}}}\right )}{\sqrt{-\frac{1}{155} i \left (\sqrt{31}-2 i\right )}}+\frac{\left (\sqrt{31}+4 i\right ) \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2+i \sqrt{31}}}\right )}{\sqrt{\frac{1}{155} i \left (\sqrt{31}+2 i\right )}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*((31*Sqrt[1 + 2*x]*(3 + 10*x))/(4 + 6*x + 10*x^2) + ((-4*I + Sqrt[31])*ArcTan
[Sqrt[5 + 10*x]/Sqrt[-2 - I*Sqrt[31]]])/Sqrt[(-I/155)*(-2*I + Sqrt[31])] + ((4*I
+ Sqrt[31])*ArcTan[Sqrt[5 + 10*x]/Sqrt[-2 + I*Sqrt[31]]])/Sqrt[(I/155)*(2*I + S
qrt[31])]))/961

_______________________________________________________________________________________

Maple [B]  time = 0.232, size = 1286, normalized size = 4.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/6727*(-2/25*(-5425*7^(1/2)+2170*5^(1/2))/(2*5^(1/2)-5*7^(1/2))*(1+2*x)^(1/2)+
1/25*7^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)*(-1085*5^(1/2)+310*7^(1/2))/(2*5^(1/2)-
5*7^(1/2)))/(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))-545/6727/(-10*5^(1/2)+25*7^(1/2))*ln((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)+1175/1922/(-10*5^(1/2)+25*7^(1/2))*ln((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)+8/31/(94*5^(1/2)*7^(1/2)-436)^(1/2)*arctan(
1/5*(2*(-10*5^(1/2)+25*7^(1/2))*(1+2*x)^(1/2)+(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)-16/217/(94*5^(1
/2)*7^(1/2)-436)^(1/2)*arctan(1/5*(2*(-10*5^(1/2)+25*7^(1/2))*(1+2*x)^(1/2)+(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)+10405/6727/(94*5^(1/2)*7^(1/2)-436)^(1/2)*arctan(1/5*(2*(-10*5^(1
/2)+25*7^(1/2))*(1+2*x)^(1/2)+(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)/(-10*5^(1
/2)+25*7^(1/2))*7^(1/2)-7800/961/(94*5^(1/2)*7^(1/2)-436)^(1/2)*arctan(1/5*(2*(-
10*5^(1/2)+25*7^(1/2))*(1+2*x)^(1/2)+(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)/(-10*5^(1/
2)+25*7^(1/2))+5/6727*(2/25*(-5425*7^(1/2)+2170*5^(1/2))/(2*5^(1/2)-5*7^(1/2))*(
1+2*x)^(1/2)+1/25*7^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)*(-1085*5^(1/2)+310*7^(1/2)
)/(2*5^(1/2)-5*7^(1/2)))/(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))+545/6727/(-10*5^(1/2)+25*7^(1/2))*ln(-(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)-1175/1922/(-10*5^(1/2)+25*7^(1/2)
)*ln(-(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)+8/31/(94*5^(1/2)*7^(1/2)-436
)^(1/2)*arctan(1/5*(2*(-10*5^(1/2)+25*7^(1/2))*(1+2*x)^(1/2)-(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)-
16/217/(94*5^(1/2)*7^(1/2)-436)^(1/2)*arctan(1/5*(2*(-10*5^(1/2)+25*7^(1/2))*(1+
2*x)^(1/2)-(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)+10405/6727/(94*5^(1/2)*7^(1/2)-436)^(1/2)*arctan(1
/5*(2*(-10*5^(1/2)+25*7^(1/2))*(1+2*x)^(1/2)-(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)/(-10*5^(1/2)+25*7^(1/2))*7^(1/2)-7800/961/(94*5^(1/2)*7^(1/2)-436)^(1/2)*a
rctan(1/5*(2*(-10*5^(1/2)+25*7^(1/2))*(1+2*x)^(1/2)-(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)/(-10*5^(1/2)+25*7^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.276663, size = 1297, normalized size = 4.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/607676818*6727^(3/4)*sqrt(94)*sqrt(31)*(6727^(1/4)*sqrt(94)*sqrt(31)*(218*sqrt
(7)*(10*x + 3) - 329*sqrt(5)*(10*x + 3))*sqrt(2*x + 1)*sqrt((218*sqrt(7)*sqrt(5)
- 1645)/(20492*sqrt(7)*sqrt(5) - 124839)) + 3844*11045^(1/4)*sqrt(7)*(5*x^2 + 3
*x + 2)*arctan(10199*11045^(1/4)*sqrt(31)*(20*sqrt(7) - 39*sqrt(5))/(6727^(1/4)*
sqrt(7285/7)*sqrt(94)*sqrt(31)*(218*sqrt(7) - 329*sqrt(5))*sqrt(sqrt(7)*(11045^(
1/4)*6727^(1/4)*sqrt(94)*(603602492140986917702023519*sqrt(7)*sqrt(5) - 35709604
98393839969371901698)*sqrt(2*x + 1)*sqrt((218*sqrt(7)*sqrt(5) - 1645)/(20492*sqr
t(7)*sqrt(5) - 124839)) + 235*sqrt(7)*(76250177874576326902188860*sqrt(7)*sqrt(5
)*(2*x + 1) - 902204272783668527795291598*x - 451102136391834263897645799) + 329
*sqrt(5)*(76250177874576326902188860*sqrt(7)*sqrt(5) - 4511021363918342638976457
99))/(76250177874576326902188860*sqrt(7)*sqrt(5) - 451102136391834263897645799))
*sqrt((218*sqrt(7)*sqrt(5) - 1645)/(20492*sqrt(7)*sqrt(5) - 124839)) + 7285*6727
^(1/4)*sqrt(94)*sqrt(2*x + 1)*(218*sqrt(7) - 329*sqrt(5))*sqrt((218*sqrt(7)*sqrt
(5) - 1645)/(20492*sqrt(7)*sqrt(5) - 124839)) + 316169*11045^(1/4)*(5*sqrt(7) -
2*sqrt(5)))) + 3844*11045^(1/4)*sqrt(7)*(5*x^2 + 3*x + 2)*arctan(10199*11045^(1/
4)*sqrt(31)*(20*sqrt(7) - 39*sqrt(5))/(6727^(1/4)*sqrt(7285/7)*sqrt(94)*sqrt(31)
*(218*sqrt(7) - 329*sqrt(5))*sqrt(-sqrt(7)*(11045^(1/4)*6727^(1/4)*sqrt(94)*(603
602492140986917702023519*sqrt(7)*sqrt(5) - 3570960498393839969371901698)*sqrt(2*
x + 1)*sqrt((218*sqrt(7)*sqrt(5) - 1645)/(20492*sqrt(7)*sqrt(5) - 124839)) - 235
*sqrt(7)*(76250177874576326902188860*sqrt(7)*sqrt(5)*(2*x + 1) - 902204272783668
527795291598*x - 451102136391834263897645799) - 329*sqrt(5)*(7625017787457632690
2188860*sqrt(7)*sqrt(5) - 451102136391834263897645799))/(76250177874576326902188
860*sqrt(7)*sqrt(5) - 451102136391834263897645799))*sqrt((218*sqrt(7)*sqrt(5) -
1645)/(20492*sqrt(7)*sqrt(5) - 124839)) + 7285*6727^(1/4)*sqrt(94)*sqrt(2*x + 1)
*(218*sqrt(7) - 329*sqrt(5))*sqrt((218*sqrt(7)*sqrt(5) - 1645)/(20492*sqrt(7)*sq
rt(5) - 124839)) - 316169*11045^(1/4)*(5*sqrt(7) - 2*sqrt(5)))) + 11045^(1/4)*sq
rt(31)*(218*sqrt(7)*(5*x^2 + 3*x + 2) - 329*sqrt(5)*(5*x^2 + 3*x + 2))*log(29140
/7*sqrt(7)*(11045^(1/4)*6727^(1/4)*sqrt(94)*(603602492140986917702023519*sqrt(7)
*sqrt(5) - 3570960498393839969371901698)*sqrt(2*x + 1)*sqrt((218*sqrt(7)*sqrt(5)
- 1645)/(20492*sqrt(7)*sqrt(5) - 124839)) + 235*sqrt(7)*(7625017787457632690218
8860*sqrt(7)*sqrt(5)*(2*x + 1) - 902204272783668527795291598*x - 451102136391834
263897645799) + 329*sqrt(5)*(76250177874576326902188860*sqrt(7)*sqrt(5) - 451102
136391834263897645799))/(76250177874576326902188860*sqrt(7)*sqrt(5) - 4511021363
91834263897645799)) - 11045^(1/4)*sqrt(31)*(218*sqrt(7)*(5*x^2 + 3*x + 2) - 329*
sqrt(5)*(5*x^2 + 3*x + 2))*log(-29140/7*sqrt(7)*(11045^(1/4)*6727^(1/4)*sqrt(94)
*(603602492140986917702023519*sqrt(7)*sqrt(5) - 3570960498393839969371901698)*sq
rt(2*x + 1)*sqrt((218*sqrt(7)*sqrt(5) - 1645)/(20492*sqrt(7)*sqrt(5) - 124839))
- 235*sqrt(7)*(76250177874576326902188860*sqrt(7)*sqrt(5)*(2*x + 1) - 9022042727
83668527795291598*x - 451102136391834263897645799) - 329*sqrt(5)*(76250177874576
326902188860*sqrt(7)*sqrt(5) - 451102136391834263897645799))/(762501778745763269
02188860*sqrt(7)*sqrt(5) - 451102136391834263897645799)))/((218*sqrt(7)*(5*x^2 +
3*x + 2) - 329*sqrt(5)*(5*x^2 + 3*x + 2))*sqrt((218*sqrt(7)*sqrt(5) - 1645)/(20
492*sqrt(7)*sqrt(5) - 124839)))

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Sympy [A]  time = 11.6101, size = 83, normalized size = 0.31 \[ \frac{80 \left (2 x + 1\right )^{\frac{3}{2}}}{- 992 x + 620 \left (2 x + 1\right )^{2} + 372} - \frac{32 \sqrt{2 x + 1}}{- 992 x + 620 \left (2 x + 1\right )^{2} + 372} + 16 \operatorname{RootSum}{\left (407144088666112 t^{4} + 3325152256 t^{2} + 11045, \left ( t \mapsto t \log{\left (\frac{33312534528 t^{3}}{235} + \frac{166784 t}{235} + \sqrt{2 x + 1} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

80*(2*x + 1)**(3/2)/(-992*x + 620*(2*x + 1)**2 + 372) - 32*sqrt(2*x + 1)/(-992*x
+ 620*(2*x + 1)**2 + 372) + 16*RootSum(407144088666112*_t**4 + 3325152256*_t**2
+ 11045, Lambda(_t, _t*log(33312534528*_t**3/235 + 166784*_t/235 + sqrt(2*x + 1
))))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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