3.2310 \(\int \frac{1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^2} \, dx\)

Optimal. Leaf size=283 \[ \frac{20 x+37}{217 \sqrt{2 x+1} \left (5 x^2+3 x+2\right )}-\frac{604}{1519 \sqrt{2 x+1}}-\frac{\sqrt{\frac{1}{434} \left (5682718+968975 \sqrt{35}\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{1519}+\frac{\sqrt{\frac{1}{434} \left (5682718+968975 \sqrt{35}\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{1519}+\frac{\sqrt{\frac{2}{217} \left (968975 \sqrt{35}-5682718\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )}{1519}-\frac{\sqrt{\frac{2}{217} \left (968975 \sqrt{35}-5682718\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )}{1519} \]

[Out]

-604/(1519*Sqrt[1 + 2*x]) + (37 + 20*x)/(217*Sqrt[1 + 2*x]*(2 + 3*x + 5*x^2)) +
(Sqrt[(2*(-5682718 + 968975*Sqrt[35]))/217]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10
*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/1519 - (Sqrt[(2*(-5682718 + 968975*Sq
rt[35]))/217]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 +
Sqrt[35])]])/1519 - (Sqrt[(5682718 + 968975*Sqrt[35])/434]*Log[Sqrt[35] - Sqrt[1
0*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/1519 + (Sqrt[(5682718 + 968975*S
qrt[35])/434]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)
])/1519

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Rubi [A]  time = 1.18793, antiderivative size = 283, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ \frac{20 x+37}{217 \sqrt{2 x+1} \left (5 x^2+3 x+2\right )}-\frac{604}{1519 \sqrt{2 x+1}}-\frac{\sqrt{\frac{1}{434} \left (5682718+968975 \sqrt{35}\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{1519}+\frac{\sqrt{\frac{1}{434} \left (5682718+968975 \sqrt{35}\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{1519}+\frac{\sqrt{\frac{2}{217} \left (968975 \sqrt{35}-5682718\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )}{1519}-\frac{\sqrt{\frac{2}{217} \left (968975 \sqrt{35}-5682718\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )}{1519} \]

Antiderivative was successfully verified.

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

[Out]

-604/(1519*Sqrt[1 + 2*x]) + (37 + 20*x)/(217*Sqrt[1 + 2*x]*(2 + 3*x + 5*x^2)) +
(Sqrt[(2*(-5682718 + 968975*Sqrt[35]))/217]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10
*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/1519 - (Sqrt[(2*(-5682718 + 968975*Sq
rt[35]))/217]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 +
Sqrt[35])]])/1519 - (Sqrt[(5682718 + 968975*Sqrt[35])/434]*Log[Sqrt[35] - Sqrt[1
0*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/1519 + (Sqrt[(5682718 + 968975*S
qrt[35])/434]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)
])/1519

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Rubi in Sympy [A]  time = 77.6577, size = 386, normalized size = 1.36 \[ - \frac{\sqrt{14} \left (814 + 151 \sqrt{35}\right ) \log{\left (2 x - \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \sqrt{2 x + 1}}{5} + 1 + \frac{\sqrt{35}}{5} \right )}}{21266 \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{14} \left (814 + 151 \sqrt{35}\right ) \log{\left (2 x + \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \sqrt{2 x + 1}}{5} + 1 + \frac{\sqrt{35}}{5} \right )}}{21266 \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{35} \left (- \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \left (1628 + 302 \sqrt{35}\right )}{10} + \frac{1628 \sqrt{10} \sqrt{2 + \sqrt{35}}}{5}\right ) \operatorname{atan}{\left (\frac{\sqrt{10} \left (\sqrt{2 x + 1} - \frac{\sqrt{20 + 10 \sqrt{35}}}{10}\right )}{\sqrt{-2 + \sqrt{35}}} \right )}}{10633 \sqrt{-2 + \sqrt{35}} \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{35} \left (- \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \left (1628 + 302 \sqrt{35}\right )}{10} + \frac{1628 \sqrt{10} \sqrt{2 + \sqrt{35}}}{5}\right ) \operatorname{atan}{\left (\frac{\sqrt{10} \left (\sqrt{2 x + 1} + \frac{\sqrt{20 + 10 \sqrt{35}}}{10}\right )}{\sqrt{-2 + \sqrt{35}}} \right )}}{10633 \sqrt{-2 + \sqrt{35}} \sqrt{2 + \sqrt{35}}} + \frac{20 x + 37}{217 \sqrt{2 x + 1} \left (5 x^{2} + 3 x + 2\right )} - \frac{604}{1519 \sqrt{2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(14)*(814 + 151*sqrt(35))*log(2*x - sqrt(10)*sqrt(2 + sqrt(35))*sqrt(2*x +
1)/5 + 1 + sqrt(35)/5)/(21266*sqrt(2 + sqrt(35))) + sqrt(14)*(814 + 151*sqrt(35)
)*log(2*x + sqrt(10)*sqrt(2 + sqrt(35))*sqrt(2*x + 1)/5 + 1 + sqrt(35)/5)/(21266
*sqrt(2 + sqrt(35))) + sqrt(35)*(-sqrt(10)*sqrt(2 + sqrt(35))*(1628 + 302*sqrt(3
5))/10 + 1628*sqrt(10)*sqrt(2 + sqrt(35))/5)*atan(sqrt(10)*(sqrt(2*x + 1) - sqrt
(20 + 10*sqrt(35))/10)/sqrt(-2 + sqrt(35)))/(10633*sqrt(-2 + sqrt(35))*sqrt(2 +
sqrt(35))) + sqrt(35)*(-sqrt(10)*sqrt(2 + sqrt(35))*(1628 + 302*sqrt(35))/10 + 1
628*sqrt(10)*sqrt(2 + sqrt(35))/5)*atan(sqrt(10)*(sqrt(2*x + 1) + sqrt(20 + 10*s
qrt(35))/10)/sqrt(-2 + sqrt(35)))/(10633*sqrt(-2 + sqrt(35))*sqrt(2 + sqrt(35)))
+ (20*x + 37)/(217*sqrt(2*x + 1)*(5*x**2 + 3*x + 2)) - 604/(1519*sqrt(2*x + 1))

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Mathematica [C]  time = 1.06107, size = 160, normalized size = 0.57 \[ \frac{2 \left (-\frac{31 \left (3020 x^2+1672 x+949\right )}{2 \sqrt{2 x+1} \left (5 x^2+3 x+2\right )}-\frac{i \left (512 \sqrt{31}-4681 i\right ) \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2-i \sqrt{31}}}\right )}{\sqrt{-\frac{1}{5} i \left (\sqrt{31}-2 i\right )}}+\frac{i \left (512 \sqrt{31}+4681 i\right ) \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2+i \sqrt{31}}}\right )}{\sqrt{\frac{1}{5} i \left (\sqrt{31}+2 i\right )}}\right )}{47089} \]

Antiderivative was successfully verified.

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

[Out]

(2*((-31*(949 + 1672*x + 3020*x^2))/(2*Sqrt[1 + 2*x]*(2 + 3*x + 5*x^2)) - (I*(-4
681*I + 512*Sqrt[31])*ArcTan[Sqrt[5 + 10*x]/Sqrt[-2 - I*Sqrt[31]]])/Sqrt[(-I/5)*
(-2*I + Sqrt[31])] + (I*(4681*I + 512*Sqrt[31])*ArcTan[Sqrt[5 + 10*x]/Sqrt[-2 +
I*Sqrt[31]]])/Sqrt[(I/5)*(2*I + Sqrt[31])]))/47089

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Maple [B]  time = 0.043, size = 494, normalized size = 1.8 \[ -{\frac{16}{49}{\frac{1}{\sqrt{1+2\,x}}}}-{\frac{16}{49} \left ({\frac{27}{124} \left ( 1+2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{89}{310}\sqrt{1+2\,x}} \right ) \left ( \left ( 1+2\,x \right ) ^{2}-{\frac{8\,x}{5}}+{\frac{3}{5}} \right ) ^{-1}}-{\frac{3657\,\sqrt{20+10\,\sqrt{35}}\sqrt{35}}{3296230}\ln \left ( 5+10\,x+\sqrt{35}-\sqrt{1+2\,x}\sqrt{20+10\,\sqrt{35}} \right ) }-{\frac{256\,\sqrt{20+10\,\sqrt{35}}}{47089}\ln \left ( 5+10\,x+\sqrt{35}-\sqrt{1+2\,x}\sqrt{20+10\,\sqrt{35}} \right ) }-{\frac{ \left ( 73140+36570\,\sqrt{35} \right ) \sqrt{35}}{1648115\,\sqrt{-20+10\,\sqrt{35}}}\arctan \left ({\frac{1}{\sqrt{-20+10\,\sqrt{35}}} \left ( 10\,\sqrt{1+2\,x}-\sqrt{20+10\,\sqrt{35}} \right ) } \right ) }-{\frac{10240+5120\,\sqrt{35}}{47089\,\sqrt{-20+10\,\sqrt{35}}}\arctan \left ({\frac{1}{\sqrt{-20+10\,\sqrt{35}}} \left ( 10\,\sqrt{1+2\,x}-\sqrt{20+10\,\sqrt{35}} \right ) } \right ) }+{\frac{3256\,\sqrt{35}}{10633\,\sqrt{-20+10\,\sqrt{35}}}\arctan \left ({\frac{1}{\sqrt{-20+10\,\sqrt{35}}} \left ( 10\,\sqrt{1+2\,x}-\sqrt{20+10\,\sqrt{35}} \right ) } \right ) }+{\frac{3657\,\sqrt{20+10\,\sqrt{35}}\sqrt{35}}{3296230}\ln \left ( 5+10\,x+\sqrt{35}+\sqrt{1+2\,x}\sqrt{20+10\,\sqrt{35}} \right ) }+{\frac{256\,\sqrt{20+10\,\sqrt{35}}}{47089}\ln \left ( 5+10\,x+\sqrt{35}+\sqrt{1+2\,x}\sqrt{20+10\,\sqrt{35}} \right ) }-{\frac{ \left ( 73140+36570\,\sqrt{35} \right ) \sqrt{35}}{1648115\,\sqrt{-20+10\,\sqrt{35}}}\arctan \left ({\frac{1}{\sqrt{-20+10\,\sqrt{35}}} \left ( 10\,\sqrt{1+2\,x}+\sqrt{20+10\,\sqrt{35}} \right ) } \right ) }-{\frac{10240+5120\,\sqrt{35}}{47089\,\sqrt{-20+10\,\sqrt{35}}}\arctan \left ({\frac{1}{\sqrt{-20+10\,\sqrt{35}}} \left ( 10\,\sqrt{1+2\,x}+\sqrt{20+10\,\sqrt{35}} \right ) } \right ) }+{\frac{3256\,\sqrt{35}}{10633\,\sqrt{-20+10\,\sqrt{35}}}\arctan \left ({\frac{1}{\sqrt{-20+10\,\sqrt{35}}} \left ( 10\,\sqrt{1+2\,x}+\sqrt{20+10\,\sqrt{35}} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-16/49/(1+2*x)^(1/2)-16/49*(27/124*(1+2*x)^(3/2)-89/310*(1+2*x)^(1/2))/((1+2*x)^
2-8/5*x+3/5)-3657/3296230*ln(5+10*x+35^(1/2)-(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2
))*(20+10*35^(1/2))^(1/2)*35^(1/2)-256/47089*ln(5+10*x+35^(1/2)-(1+2*x)^(1/2)*(2
0+10*35^(1/2))^(1/2))*(20+10*35^(1/2))^(1/2)-3657/1648115/(-20+10*35^(1/2))^(1/2
)*arctan((10*(1+2*x)^(1/2)-(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(20+
10*35^(1/2))*35^(1/2)-512/47089/(-20+10*35^(1/2))^(1/2)*arctan((10*(1+2*x)^(1/2)
-(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(20+10*35^(1/2))+3256/10633/(-
20+10*35^(1/2))^(1/2)*arctan((10*(1+2*x)^(1/2)-(20+10*35^(1/2))^(1/2))/(-20+10*3
5^(1/2))^(1/2))*35^(1/2)+3657/3296230*ln(5+10*x+35^(1/2)+(1+2*x)^(1/2)*(20+10*35
^(1/2))^(1/2))*(20+10*35^(1/2))^(1/2)*35^(1/2)+256/47089*ln(5+10*x+35^(1/2)+(1+2
*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(20+10*35^(1/2))^(1/2)-3657/1648115/(-20+10*35
^(1/2))^(1/2)*arctan((10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))
^(1/2))*(20+10*35^(1/2))*35^(1/2)-512/47089/(-20+10*35^(1/2))^(1/2)*arctan((10*(
1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(20+10*35^(1/2))+3
256/10633/(-20+10*35^(1/2))^(1/2)*arctan((10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2
))/(-20+10*35^(1/2))^(1/2))*35^(1/2)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.28272, size = 1351, normalized size = 4.77 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24555198801154*329623^(3/4)*sqrt(1582)*sqrt(31)*(117561052*63845^(1/4)*sqrt(7)
*(5*x^2 + 3*x + 2)*sqrt(2*x + 1)*arctan(1201529*63845^(1/4)*sqrt(31)*(2560*sqrt(
7) + 3657*sqrt(5))/(329623^(1/4)*sqrt(17515)*sqrt(1582)*sqrt(31)*(5682718*sqrt(7
) + 6782825*sqrt(5))*sqrt(sqrt(7)*(329623^(1/4)*63845^(1/4)*sqrt(1582)*(54213539
77109919349375231176591971670210740279768440185458887323725713849*sqrt(7)*sqrt(5
) + 32073162661006692460155647633513671868488934691072016679539595765063499686)*
sqrt(2*x + 1)*sqrt((5682718*sqrt(7)*sqrt(5) + 33914125)/(11012823348100*sqrt(7)*
sqrt(5) + 65155223139399)) + 3955*sqrt(7)*(3175344062231234807493149055402797662
344572468937043301020715443940500*sqrt(7)*sqrt(5)*(2*x + 1) + 375711776219032346
50010699940319130769036533643095191237457284137193998*x + 1878558881095161732500
5349970159565384518266821547595618728642068596999) + 5537*sqrt(5)*(3175344062231
234807493149055402797662344572468937043301020715443940500*sqrt(7)*sqrt(5) + 1878
5588810951617325005349970159565384518266821547595618728642068596999))/(317534406
2231234807493149055402797662344572468937043301020715443940500*sqrt(7)*sqrt(5) +
18785588810951617325005349970159565384518266821547595618728642068596999))*sqrt((
5682718*sqrt(7)*sqrt(5) + 33914125)/(11012823348100*sqrt(7)*sqrt(5) + 6515522313
9399)) + 122605*329623^(1/4)*sqrt(1582)*sqrt(2*x + 1)*(5682718*sqrt(7) + 6782825
*sqrt(5))*sqrt((5682718*sqrt(7)*sqrt(5) + 33914125)/(11012823348100*sqrt(7)*sqrt
(5) + 65155223139399)) + 37247399*63845^(1/4)*(755*sqrt(7) + 814*sqrt(5)))) + 11
7561052*63845^(1/4)*sqrt(7)*(5*x^2 + 3*x + 2)*sqrt(2*x + 1)*arctan(1201529*63845
^(1/4)*sqrt(31)*(2560*sqrt(7) + 3657*sqrt(5))/(329623^(1/4)*sqrt(17515)*sqrt(158
2)*sqrt(31)*(5682718*sqrt(7) + 6782825*sqrt(5))*sqrt(-sqrt(7)*(329623^(1/4)*6384
5^(1/4)*sqrt(1582)*(542135397710991934937523117659197167021074027976844018545888
7323725713849*sqrt(7)*sqrt(5) + 320731626610066924601556476335136718684889346910
72016679539595765063499686)*sqrt(2*x + 1)*sqrt((5682718*sqrt(7)*sqrt(5) + 339141
25)/(11012823348100*sqrt(7)*sqrt(5) + 65155223139399)) - 3955*sqrt(7)*(317534406
2231234807493149055402797662344572468937043301020715443940500*sqrt(7)*sqrt(5)*(2
*x + 1) + 3757117762190323465001069994031913076903653364309519123745728413719399
8*x + 18785588810951617325005349970159565384518266821547595618728642068596999) -
5537*sqrt(5)*(31753440622312348074931490554027976623445724689370433010207154439
40500*sqrt(7)*sqrt(5) + 18785588810951617325005349970159565384518266821547595618
728642068596999))/(3175344062231234807493149055402797662344572468937043301020715
443940500*sqrt(7)*sqrt(5) + 1878558881095161732500534997015956538451826682154759
5618728642068596999))*sqrt((5682718*sqrt(7)*sqrt(5) + 33914125)/(11012823348100*
sqrt(7)*sqrt(5) + 65155223139399)) + 122605*329623^(1/4)*sqrt(1582)*sqrt(2*x + 1
)*(5682718*sqrt(7) + 6782825*sqrt(5))*sqrt((5682718*sqrt(7)*sqrt(5) + 33914125)/
(11012823348100*sqrt(7)*sqrt(5) + 65155223139399)) - 37247399*63845^(1/4)*(755*s
qrt(7) + 814*sqrt(5)))) + 7*63845^(1/4)*sqrt(31)*(5682718*sqrt(7)*(5*x^2 + 3*x +
2) + 6782825*sqrt(5)*(5*x^2 + 3*x + 2))*sqrt(2*x + 1)*log(43787500*sqrt(7)*(329
623^(1/4)*63845^(1/4)*sqrt(1582)*(5421353977109919349375231176591971670210740279
768440185458887323725713849*sqrt(7)*sqrt(5) + 3207316266100669246015564763351367
1868488934691072016679539595765063499686)*sqrt(2*x + 1)*sqrt((5682718*sqrt(7)*sq
rt(5) + 33914125)/(11012823348100*sqrt(7)*sqrt(5) + 65155223139399)) + 3955*sqrt
(7)*(3175344062231234807493149055402797662344572468937043301020715443940500*sqrt
(7)*sqrt(5)*(2*x + 1) + 37571177621903234650010699940319130769036533643095191237
457284137193998*x + 187855888109516173250053499701595653845182668215475956187286
42068596999) + 5537*sqrt(5)*(317534406223123480749314905540279766234457246893704
3301020715443940500*sqrt(7)*sqrt(5) + 187855888109516173250053499701595653845182
66821547595618728642068596999))/(31753440622312348074931490554027976623445724689
37043301020715443940500*sqrt(7)*sqrt(5) + 18785588810951617325005349970159565384
518266821547595618728642068596999)) - 7*63845^(1/4)*sqrt(31)*(5682718*sqrt(7)*(5
*x^2 + 3*x + 2) + 6782825*sqrt(5)*(5*x^2 + 3*x + 2))*sqrt(2*x + 1)*log(-43787500
*sqrt(7)*(329623^(1/4)*63845^(1/4)*sqrt(1582)*(542135397710991934937523117659197
1670210740279768440185458887323725713849*sqrt(7)*sqrt(5) + 320731626610066924601
55647633513671868488934691072016679539595765063499686)*sqrt(2*x + 1)*sqrt((56827
18*sqrt(7)*sqrt(5) + 33914125)/(11012823348100*sqrt(7)*sqrt(5) + 65155223139399)
) - 3955*sqrt(7)*(31753440622312348074931490554027976623445724689370433010207154
43940500*sqrt(7)*sqrt(5)*(2*x + 1) + 3757117762190323465001069994031913076903653
3643095191237457284137193998*x + 18785588810951617325005349970159565384518266821
547595618728642068596999) - 5537*sqrt(5)*(31753440622312348074931490554027976623
44572468937043301020715443940500*sqrt(7)*sqrt(5) + 18785588810951617325005349970
159565384518266821547595618728642068596999))/(3175344062231234807493149055402797
662344572468937043301020715443940500*sqrt(7)*sqrt(5) + 1878558881095161732500534
9970159565384518266821547595618728642068596999)) - 329623^(1/4)*sqrt(1582)*sqrt(
31)*(5682718*sqrt(7)*(3020*x^2 + 1672*x + 949) + 6782825*sqrt(5)*(3020*x^2 + 167
2*x + 949))*sqrt((5682718*sqrt(7)*sqrt(5) + 33914125)/(11012823348100*sqrt(7)*sq
rt(5) + 65155223139399)))/((5682718*sqrt(7)*(5*x^2 + 3*x + 2) + 6782825*sqrt(5)*
(5*x^2 + 3*x + 2))*sqrt(2*x + 1)*sqrt((5682718*sqrt(7)*sqrt(5) + 33914125)/(1101
2823348100*sqrt(7)*sqrt(5) + 65155223139399)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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