3.2307 \(\int \frac{(1+2 x)^{3/2}}{\left (2+3 x+5 x^2\right )^2} \, dx\)

Optimal. Leaf size=270 \[ -\frac{\sqrt{2 x+1} (5-4 x)}{31 \left (5 x^2+3 x+2\right )}-\frac{1}{31} \sqrt{\frac{1}{310} \left (47 \sqrt{35}-218\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )+\frac{1}{31} \sqrt{\frac{1}{310} \left (47 \sqrt{35}-218\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )-\frac{1}{31} \sqrt{\frac{2}{155} \left (218+47 \sqrt{35}\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )+\frac{1}{31} \sqrt{\frac{2}{155} \left (218+47 \sqrt{35}\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right ) \]

[Out]

-((5 - 4*x)*Sqrt[1 + 2*x])/(31*(2 + 3*x + 5*x^2)) - (Sqrt[(2*(218 + 47*Sqrt[35])
)/155]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35
])]])/31 + (Sqrt[(2*(218 + 47*Sqrt[35]))/155]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] +
10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/31 - (Sqrt[(-218 + 47*Sqrt[35])/310
]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/31 + (Sqr
t[(-218 + 47*Sqrt[35])/310]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x]
+ 5*(1 + 2*x)])/31

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Rubi [A]  time = 1.1621, 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 \[ -\frac{\sqrt{2 x+1} (5-4 x)}{31 \left (5 x^2+3 x+2\right )}-\frac{1}{31} \sqrt{\frac{1}{310} \left (47 \sqrt{35}-218\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )+\frac{1}{31} \sqrt{\frac{1}{310} \left (47 \sqrt{35}-218\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )-\frac{1}{31} \sqrt{\frac{2}{155} \left (218+47 \sqrt{35}\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )+\frac{1}{31} \sqrt{\frac{2}{155} \left (218+47 \sqrt{35}\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-((5 - 4*x)*Sqrt[1 + 2*x])/(31*(2 + 3*x + 5*x^2)) - (Sqrt[(2*(218 + 47*Sqrt[35])
)/155]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35
])]])/31 + (Sqrt[(2*(218 + 47*Sqrt[35]))/155]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] +
10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/31 - (Sqrt[(-218 + 47*Sqrt[35])/310
]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/31 + (Sqr
t[(-218 + 47*Sqrt[35])/310]*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 = 69.9454, size = 379, normalized size = 1.4 \[ - \frac{\left (- 4 x + 5\right ) \sqrt{2 x + 1}}{31 \left (5 x^{2} + 3 x + 2\right )} - \frac{\sqrt{14} \left (- \frac{2 \sqrt{35}}{5} + 7\right ) \log{\left (2 x - \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \sqrt{2 x + 1}}{5} + 1 + \frac{\sqrt{35}}{5} \right )}}{434 \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{14} \left (- \frac{2 \sqrt{35}}{5} + 7\right ) \log{\left (2 x + \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \sqrt{2 x + 1}}{5} + 1 + \frac{\sqrt{35}}{5} \right )}}{434 \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{35} \left (- \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \left (- \frac{4 \sqrt{35}}{5} + 14\right )}{10} + \frac{14 \sqrt{10} \sqrt{2 + \sqrt{35}}}{5}\right ) \operatorname{atan}{\left (\frac{\sqrt{10} \left (\sqrt{2 x + 1} - \frac{\sqrt{20 + 10 \sqrt{35}}}{10}\right )}{\sqrt{-2 + \sqrt{35}}} \right )}}{217 \sqrt{-2 + \sqrt{35}} \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{35} \left (- \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \left (- \frac{4 \sqrt{35}}{5} + 14\right )}{10} + \frac{14 \sqrt{10} \sqrt{2 + \sqrt{35}}}{5}\right ) \operatorname{atan}{\left (\frac{\sqrt{10} \left (\sqrt{2 x + 1} + \frac{\sqrt{20 + 10 \sqrt{35}}}{10}\right )}{\sqrt{-2 + \sqrt{35}}} \right )}}{217 \sqrt{-2 + \sqrt{35}} \sqrt{2 + \sqrt{35}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-4*x + 5)*sqrt(2*x + 1)/(31*(5*x**2 + 3*x + 2)) - sqrt(14)*(-2*sqrt(35)/5 + 7)
*log(2*x - sqrt(10)*sqrt(2 + sqrt(35))*sqrt(2*x + 1)/5 + 1 + sqrt(35)/5)/(434*sq
rt(2 + sqrt(35))) + sqrt(14)*(-2*sqrt(35)/5 + 7)*log(2*x + sqrt(10)*sqrt(2 + sqr
t(35))*sqrt(2*x + 1)/5 + 1 + sqrt(35)/5)/(434*sqrt(2 + sqrt(35))) + sqrt(35)*(-s
qrt(10)*sqrt(2 + sqrt(35))*(-4*sqrt(35)/5 + 14)/10 + 14*sqrt(10)*sqrt(2 + sqrt(3
5))/5)*atan(sqrt(10)*(sqrt(2*x + 1) - sqrt(20 + 10*sqrt(35))/10)/sqrt(-2 + sqrt(
35)))/(217*sqrt(-2 + sqrt(35))*sqrt(2 + sqrt(35))) + sqrt(35)*(-sqrt(10)*sqrt(2
+ sqrt(35))*(-4*sqrt(35)/5 + 14)/10 + 14*sqrt(10)*sqrt(2 + sqrt(35))/5)*atan(sqr
t(10)*(sqrt(2*x + 1) + sqrt(20 + 10*sqrt(35))/10)/sqrt(-2 + sqrt(35)))/(217*sqrt
(-2 + sqrt(35))*sqrt(2 + sqrt(35)))

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Mathematica [C]  time = 0.83775, size = 145, normalized size = 0.54 \[ \frac{\sqrt{2 x+1} (4 x-5)}{31 \left (5 x^2+3 x+2\right )}+\frac{2 \left (62-39 i \sqrt{31}\right ) \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2-i \sqrt{31}}}\right )}{961 \sqrt{-10-5 i \sqrt{31}}}+\frac{2 \left (62+39 i \sqrt{31}\right ) \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2+i \sqrt{31}}}\right )}{961 \sqrt{5 i \left (\sqrt{31}+2 i\right )}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 + 2*x]*(-5 + 4*x))/(31*(2 + 3*x + 5*x^2)) + (2*(62 - (39*I)*Sqrt[31])*Ar
cTan[Sqrt[5 + 10*x]/Sqrt[-2 - I*Sqrt[31]]])/(961*Sqrt[-10 - (5*I)*Sqrt[31]]) + (
2*(62 + (39*I)*Sqrt[31])*ArcTan[Sqrt[5 + 10*x]/Sqrt[-2 + I*Sqrt[31]]])/(961*Sqrt
[(5*I)*(2*I + Sqrt[31])])

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Maple [B]  time = 0.034, size = 485, normalized size = 1.8 \[ 16\,{\frac{1}{ \left ( 1+2\,x \right ) ^{2}-8/5\,x+3/5} \left ({\frac{ \left ( 1+2\,x \right ) ^{3/2}}{310}}-{\frac{7\,\sqrt{1+2\,x}}{620}} \right ) }+{\frac{2\,\sqrt{20+10\,\sqrt{35}}\sqrt{35}}{4805}\ln \left ( 5+10\,x+\sqrt{35}-\sqrt{1+2\,x}\sqrt{20+10\,\sqrt{35}} \right ) }-{\frac{39\,\sqrt{20+10\,\sqrt{35}}}{9610}\ln \left ( 5+10\,x+\sqrt{35}-\sqrt{1+2\,x}\sqrt{20+10\,\sqrt{35}} \right ) }+{\frac{ \left ( 80+40\,\sqrt{35} \right ) \sqrt{35}}{4805\,\sqrt{-20+10\,\sqrt{35}}}\arctan \left ({\frac{1}{\sqrt{-20+10\,\sqrt{35}}} \left ( 10\,\sqrt{1+2\,x}-\sqrt{20+10\,\sqrt{35}} \right ) } \right ) }-{\frac{780+390\,\sqrt{35}}{4805\,\sqrt{-20+10\,\sqrt{35}}}\arctan \left ({\frac{1}{\sqrt{-20+10\,\sqrt{35}}} \left ( 10\,\sqrt{1+2\,x}-\sqrt{20+10\,\sqrt{35}} \right ) } \right ) }+{\frac{4\,\sqrt{35}}{31\,\sqrt{-20+10\,\sqrt{35}}}\arctan \left ({\frac{1}{\sqrt{-20+10\,\sqrt{35}}} \left ( 10\,\sqrt{1+2\,x}-\sqrt{20+10\,\sqrt{35}} \right ) } \right ) }-{\frac{2\,\sqrt{20+10\,\sqrt{35}}\sqrt{35}}{4805}\ln \left ( 5+10\,x+\sqrt{35}+\sqrt{1+2\,x}\sqrt{20+10\,\sqrt{35}} \right ) }+{\frac{39\,\sqrt{20+10\,\sqrt{35}}}{9610}\ln \left ( 5+10\,x+\sqrt{35}+\sqrt{1+2\,x}\sqrt{20+10\,\sqrt{35}} \right ) }+{\frac{ \left ( 80+40\,\sqrt{35} \right ) \sqrt{35}}{4805\,\sqrt{-20+10\,\sqrt{35}}}\arctan \left ({\frac{1}{\sqrt{-20+10\,\sqrt{35}}} \left ( 10\,\sqrt{1+2\,x}+\sqrt{20+10\,\sqrt{35}} \right ) } \right ) }-{\frac{780+390\,\sqrt{35}}{4805\,\sqrt{-20+10\,\sqrt{35}}}\arctan \left ({\frac{1}{\sqrt{-20+10\,\sqrt{35}}} \left ( 10\,\sqrt{1+2\,x}+\sqrt{20+10\,\sqrt{35}} \right ) } \right ) }+{\frac{4\,\sqrt{35}}{31\,\sqrt{-20+10\,\sqrt{35}}}\arctan \left ({\frac{1}{\sqrt{-20+10\,\sqrt{35}}} \left ( 10\,\sqrt{1+2\,x}+\sqrt{20+10\,\sqrt{35}} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

16*(1/310*(1+2*x)^(3/2)-7/620*(1+2*x)^(1/2))/((1+2*x)^2-8/5*x+3/5)+2/4805*ln(5+1
0*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)-39/9610*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)+4/4805/(-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)-39/4805/(-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))+4/31/(-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)-2/4805*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)+39/9610
*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)
+4/4805/(-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)-39/4805/(-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))*(2
0+10*35^(1/2))+4/31/(-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)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.272005, size = 1297, normalized size = 4.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/434054870*4805^(3/4)*sqrt(94)*sqrt(31)*(4805^(1/4)*sqrt(94)*sqrt(31)*(235*sqrt
(7)*(4*x - 5) - 218*sqrt(5)*(4*x - 5))*sqrt(2*x + 1)*sqrt((218*sqrt(7)*sqrt(5) -
1645)/(20492*sqrt(7)*sqrt(5) - 124839)) + 3844*15463^(1/4)*sqrt(5)*(5*x^2 + 3*x
+ 2)*arctan(7285*15463^(1/4)*sqrt(31)*(20*sqrt(7) - 39*sqrt(5))/(4805^(1/4)*sqr
t(1457/7)*sqrt(94)*sqrt(31)*(235*sqrt(7) - 218*sqrt(5))*sqrt(sqrt(5)*(15463^(1/4
)*4805^(1/4)*sqrt(94)*(3570960498393839969371901698*sqrt(7)*sqrt(5) - 2112608722
4934542119570823165)*sqrt(2*x + 1)*sqrt((218*sqrt(7)*sqrt(5) - 1645)/(20492*sqrt
(7)*sqrt(5) - 124839)) + 1645*sqrt(5)*(76250177874576326902188860*sqrt(7)*sqrt(5
)*(2*x + 1) - 902204272783668527795291598*x - 451102136391834263897645799) + 164
5*sqrt(7)*(76250177874576326902188860*sqrt(7)*sqrt(5) - 451102136391834263897645
799))/(76250177874576326902188860*sqrt(7)*sqrt(5) - 451102136391834263897645799)
)*sqrt((218*sqrt(7)*sqrt(5) - 1645)/(20492*sqrt(7)*sqrt(5) - 124839)) + 7285*480
5^(1/4)*sqrt(94)*sqrt(2*x + 1)*(235*sqrt(7) - 218*sqrt(5))*sqrt((218*sqrt(7)*sqr
t(5) - 1645)/(20492*sqrt(7)*sqrt(5) - 124839)) + 225835*15463^(1/4)*(5*sqrt(7) -
2*sqrt(5)))) + 3844*15463^(1/4)*sqrt(5)*(5*x^2 + 3*x + 2)*arctan(7285*15463^(1/
4)*sqrt(31)*(20*sqrt(7) - 39*sqrt(5))/(4805^(1/4)*sqrt(1457/7)*sqrt(94)*sqrt(31)
*(235*sqrt(7) - 218*sqrt(5))*sqrt(-sqrt(5)*(15463^(1/4)*4805^(1/4)*sqrt(94)*(357
0960498393839969371901698*sqrt(7)*sqrt(5) - 21126087224934542119570823165)*sqrt(
2*x + 1)*sqrt((218*sqrt(7)*sqrt(5) - 1645)/(20492*sqrt(7)*sqrt(5) - 124839)) - 1
645*sqrt(5)*(76250177874576326902188860*sqrt(7)*sqrt(5)*(2*x + 1) - 902204272783
668527795291598*x - 451102136391834263897645799) - 1645*sqrt(7)*(762501778745763
26902188860*sqrt(7)*sqrt(5) - 451102136391834263897645799))/(7625017787457632690
2188860*sqrt(7)*sqrt(5) - 451102136391834263897645799))*sqrt((218*sqrt(7)*sqrt(5
) - 1645)/(20492*sqrt(7)*sqrt(5) - 124839)) + 7285*4805^(1/4)*sqrt(94)*sqrt(2*x
+ 1)*(235*sqrt(7) - 218*sqrt(5))*sqrt((218*sqrt(7)*sqrt(5) - 1645)/(20492*sqrt(7
)*sqrt(5) - 124839)) - 225835*15463^(1/4)*(5*sqrt(7) - 2*sqrt(5)))) + 15463^(1/4
)*sqrt(31)*(235*sqrt(7)*(5*x^2 + 3*x + 2) - 218*sqrt(5)*(5*x^2 + 3*x + 2))*log(5
828/175*sqrt(5)*(15463^(1/4)*4805^(1/4)*sqrt(94)*(3570960498393839969371901698*s
qrt(7)*sqrt(5) - 21126087224934542119570823165)*sqrt(2*x + 1)*sqrt((218*sqrt(7)*
sqrt(5) - 1645)/(20492*sqrt(7)*sqrt(5) - 124839)) + 1645*sqrt(5)*(76250177874576
326902188860*sqrt(7)*sqrt(5)*(2*x + 1) - 902204272783668527795291598*x - 4511021
36391834263897645799) + 1645*sqrt(7)*(76250177874576326902188860*sqrt(7)*sqrt(5)
- 451102136391834263897645799))/(76250177874576326902188860*sqrt(7)*sqrt(5) - 4
51102136391834263897645799)) - 15463^(1/4)*sqrt(31)*(235*sqrt(7)*(5*x^2 + 3*x +
2) - 218*sqrt(5)*(5*x^2 + 3*x + 2))*log(-5828/175*sqrt(5)*(15463^(1/4)*4805^(1/4
)*sqrt(94)*(3570960498393839969371901698*sqrt(7)*sqrt(5) - 211260872249345421195
70823165)*sqrt(2*x + 1)*sqrt((218*sqrt(7)*sqrt(5) - 1645)/(20492*sqrt(7)*sqrt(5)
- 124839)) - 1645*sqrt(5)*(76250177874576326902188860*sqrt(7)*sqrt(5)*(2*x + 1)
- 902204272783668527795291598*x - 451102136391834263897645799) - 1645*sqrt(7)*(
76250177874576326902188860*sqrt(7)*sqrt(5) - 451102136391834263897645799))/(7625
0177874576326902188860*sqrt(7)*sqrt(5) - 451102136391834263897645799)))/((235*sq
rt(7)*(5*x^2 + 3*x + 2) - 218*sqrt(5)*(5*x^2 + 3*x + 2))*sqrt((218*sqrt(7)*sqrt(
5) - 1645)/(20492*sqrt(7)*sqrt(5) - 124839)))

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Sympy [A]  time = 77.1158, size = 211, normalized size = 0.78 \[ \frac{64 \left (2 x + 1\right )^{\frac{3}{2}}}{- 992 x + 620 \left (2 x + 1\right )^{2} + 372} - \frac{224 \left (2 x + 1\right )^{\frac{3}{2}}}{- 6944 x + 4340 \left (2 x + 1\right )^{2} + 2604} - \frac{128 \sqrt{2 x + 1}}{5 \left (- 992 x + 620 \left (2 x + 1\right )^{2} + 372\right )} - \frac{3024 \sqrt{2 x + 1}}{5 \left (- 6944 x + 4340 \left (2 x + 1\right )^{2} + 2604\right )} + \frac{64 \operatorname{RootSum}{\left (407144088666112 t^{4} + 3325152256 t^{2} + 11045, \left ( t \mapsto t \log{\left (\frac{33312534528 t^{3}}{235} + \frac{166784 t}{235} + \sqrt{2 x + 1} \right )} \right )\right )}}{5} - \frac{112 \operatorname{RootSum}{\left (19950060344639488 t^{4} + 498437272576 t^{2} + 10878125, \left ( t \mapsto t \log{\left (- \frac{11049511452672 t^{3}}{2205125} + \frac{307918256 t}{2205125} + \sqrt{2 x + 1} \right )} \right )\right )}}{5} + \frac{16 \operatorname{RootSum}{\left (1722112 t^{4} + 1984 t^{2} + 5, \left ( t \mapsto t \log{\left (- \frac{27776 t^{3}}{5} + \frac{108 t}{5} + \sqrt{2 x + 1} \right )} \right )\right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

64*(2*x + 1)**(3/2)/(-992*x + 620*(2*x + 1)**2 + 372) - 224*(2*x + 1)**(3/2)/(-6
944*x + 4340*(2*x + 1)**2 + 2604) - 128*sqrt(2*x + 1)/(5*(-992*x + 620*(2*x + 1)
**2 + 372)) - 3024*sqrt(2*x + 1)/(5*(-6944*x + 4340*(2*x + 1)**2 + 2604)) + 64*R
ootSum(407144088666112*_t**4 + 3325152256*_t**2 + 11045, Lambda(_t, _t*log(33312
534528*_t**3/235 + 166784*_t/235 + sqrt(2*x + 1))))/5 - 112*RootSum(199500603446
39488*_t**4 + 498437272576*_t**2 + 10878125, Lambda(_t, _t*log(-11049511452672*_
t**3/2205125 + 307918256*_t/2205125 + sqrt(2*x + 1))))/5 + 16*RootSum(1722112*_t
**4 + 1984*_t**2 + 5, Lambda(_t, _t*log(-27776*_t**3/5 + 108*_t/5 + sqrt(2*x + 1
))))/5

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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