3.2440 \(\int \frac{(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^5} \, dx\)

Optimal. Leaf size=167 \[ -\frac{(4 x+19) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^4}-\frac{(2898 x+3727) \left (3 x^2+5 x+2\right )^{3/2}}{384 (2 x+3)^3}+\frac{(5718 x+12265) \sqrt{3 x^2+5 x+2}}{512 (2 x+3)}-\frac{1875}{256} \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )+\frac{29047 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{1024 \sqrt{5}} \]

[Out]

((12265 + 5718*x)*Sqrt[2 + 5*x + 3*x^2])/(512*(3 + 2*x)) - ((3727 + 2898*x)*(2 +
5*x + 3*x^2)^(3/2))/(384*(3 + 2*x)^3) - ((19 + 4*x)*(2 + 5*x + 3*x^2)^(5/2))/(1
6*(3 + 2*x)^4) - (1875*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2
])])/256 + (29047*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(1024*Sq
rt[5])

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Rubi [A]  time = 0.320556, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{(4 x+19) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^4}-\frac{(2898 x+3727) \left (3 x^2+5 x+2\right )^{3/2}}{384 (2 x+3)^3}+\frac{(5718 x+12265) \sqrt{3 x^2+5 x+2}}{512 (2 x+3)}-\frac{1875}{256} \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )+\frac{29047 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{1024 \sqrt{5}} \]

Antiderivative was successfully verified.

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

[Out]

((12265 + 5718*x)*Sqrt[2 + 5*x + 3*x^2])/(512*(3 + 2*x)) - ((3727 + 2898*x)*(2 +
5*x + 3*x^2)^(3/2))/(384*(3 + 2*x)^3) - ((19 + 4*x)*(2 + 5*x + 3*x^2)^(5/2))/(1
6*(3 + 2*x)^4) - (1875*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2
])])/256 + (29047*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(1024*Sq
rt[5])

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Rubi in Sympy [A]  time = 41.1445, size = 150, normalized size = 0.9 \[ - \frac{1875 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{256} - \frac{29047 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \left (- 8 x - 7\right )}{10 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{5120} + \frac{\left (45744 x + 98120\right ) \sqrt{3 x^{2} + 5 x + 2}}{4096 \left (2 x + 3\right )} - \frac{\left (11592 x + 14908\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{1536 \left (2 x + 3\right )^{3}} - \frac{\left (8 x + 38\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{32 \left (2 x + 3\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1875*sqrt(3)*atanh(sqrt(3)*(6*x + 5)/(6*sqrt(3*x**2 + 5*x + 2)))/256 - 29047*sq
rt(5)*atanh(sqrt(5)*(-8*x - 7)/(10*sqrt(3*x**2 + 5*x + 2)))/5120 + (45744*x + 98
120)*sqrt(3*x**2 + 5*x + 2)/(4096*(2*x + 3)) - (11592*x + 14908)*(3*x**2 + 5*x +
2)**(3/2)/(1536*(2*x + 3)**3) - (8*x + 38)*(3*x**2 + 5*x + 2)**(5/2)/(32*(2*x +
3)**4)

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Mathematica [A]  time = 0.170916, size = 131, normalized size = 0.78 \[ \frac{-29047 \sqrt{5} \log \left (2 \sqrt{5} \sqrt{3 x^2+5 x+2}-8 x-7\right )-37500 \sqrt{3} \log \left (-2 \sqrt{9 x^2+15 x+6}-6 x-5\right )-\frac{10 \sqrt{3 x^2+5 x+2} \left (3456 x^5-39744 x^4-533280 x^3-1672268 x^2-2059268 x-896721\right )}{3 (2 x+3)^4}+29047 \sqrt{5} \log (2 x+3)}{5120} \]

Antiderivative was successfully verified.

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

[Out]

((-10*Sqrt[2 + 5*x + 3*x^2]*(-896721 - 2059268*x - 1672268*x^2 - 533280*x^3 - 39
744*x^4 + 3456*x^5))/(3*(3 + 2*x)^4) + 29047*Sqrt[5]*Log[3 + 2*x] - 29047*Sqrt[5
]*Log[-7 - 8*x + 2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2]] - 37500*Sqrt[3]*Log[-5 - 6*x -
2*Sqrt[6 + 15*x + 9*x^2]])/5120

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Maple [A]  time = 0.017, size = 258, normalized size = 1.5 \[ -{\frac{13}{320} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-4}}-{\frac{1}{75} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{1627}{12000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}+{\frac{1307}{2500} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}+{\frac{29047}{20000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{6935+8322\,x}{2400} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{2305+2766\,x}{320}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}-{\frac{1875\,\sqrt{3}}{256}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \right ) }+{\frac{29047}{9600} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{29047}{5120}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}-{\frac{29047\,\sqrt{5}}{5120}{\it Artanh} \left ({\frac{2\,\sqrt{5}}{5} \left ( -{\frac{7}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}}} \right ) }-{\frac{6535+7842\,x}{5000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-13/320/(x+3/2)^4*(3*(x+3/2)^2-4*x-19/4)^(7/2)-1/75/(x+3/2)^3*(3*(x+3/2)^2-4*x-1
9/4)^(7/2)-1627/12000/(x+3/2)^2*(3*(x+3/2)^2-4*x-19/4)^(7/2)+1307/2500/(x+3/2)*(
3*(x+3/2)^2-4*x-19/4)^(7/2)+29047/20000*(3*(x+3/2)^2-4*x-19/4)^(5/2)-1387/2400*(
5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(3/2)-461/320*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(1/2)
-1875/256*ln(1/3*(5/2+3*x)*3^(1/2)+(3*(x+3/2)^2-4*x-19/4)^(1/2))*3^(1/2)+29047/9
600*(3*(x+3/2)^2-4*x-19/4)^(3/2)+29047/5120*(12*(x+3/2)^2-16*x-19)^(1/2)-29047/5
120*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))-1307/50
00*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(5/2)

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Maxima [A]  time = 0.810615, size = 346, normalized size = 2.07 \[ \frac{1627}{4000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} - \frac{13 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{20 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{8 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{75 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{1627 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{3000 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{1387}{400} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + \frac{1307}{9600} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} + \frac{1307 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{1000 \,{\left (2 \, x + 3\right )}} - \frac{1383}{160} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x - \frac{1875}{256} \, \sqrt{3} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac{5}{2}\right ) - \frac{29047}{5120} \, \sqrt{5} \log \left (\frac{\sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac{5}{2 \,{\left | 2 \, x + 3 \right |}} - 2\right ) + \frac{10607}{2560} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1627/4000*(3*x^2 + 5*x + 2)^(5/2) - 13/20*(3*x^2 + 5*x + 2)^(7/2)/(16*x^4 + 96*x
^3 + 216*x^2 + 216*x + 81) - 8/75*(3*x^2 + 5*x + 2)^(7/2)/(8*x^3 + 36*x^2 + 54*x
+ 27) - 1627/3000*(3*x^2 + 5*x + 2)^(7/2)/(4*x^2 + 12*x + 9) - 1387/400*(3*x^2
+ 5*x + 2)^(3/2)*x + 1307/9600*(3*x^2 + 5*x + 2)^(3/2) + 1307/1000*(3*x^2 + 5*x
+ 2)^(5/2)/(2*x + 3) - 1383/160*sqrt(3*x^2 + 5*x + 2)*x - 1875/256*sqrt(3)*log(s
qrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2) - 29047/5120*sqrt(5)*log(sqrt(5)*sqrt(
3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) + 10607/2560*sqrt(3*x^2 +
5*x + 2)

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Fricas [A]  time = 0.292608, size = 271, normalized size = 1.62 \[ \frac{\sqrt{5}{\left (22500 \, \sqrt{5} \sqrt{3}{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (-4 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) - 4 \, \sqrt{5}{\left (3456 \, x^{5} - 39744 \, x^{4} - 533280 \, x^{3} - 1672268 \, x^{2} - 2059268 \, x - 896721\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + 87141 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (\frac{\sqrt{5}{\left (124 \, x^{2} + 212 \, x + 89\right )} + 20 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{30720 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/30720*sqrt(5)*(22500*sqrt(5)*sqrt(3)*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*
log(-4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49) - 4*sqrt(5
)*(3456*x^5 - 39744*x^4 - 533280*x^3 - 1672268*x^2 - 2059268*x - 896721)*sqrt(3*
x^2 + 5*x + 2) + 87141*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log((sqrt(5)*(12
4*x^2 + 212*x + 89) + 20*sqrt(3*x^2 + 5*x + 2)*(8*x + 7))/(4*x^2 + 12*x + 9)))/(
16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{20 \sqrt{3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \left (- \frac{96 x \sqrt{3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \left (- \frac{165 x^{2} \sqrt{3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \left (- \frac{113 x^{3} \sqrt{3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \left (- \frac{15 x^{4} \sqrt{3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \frac{9 x^{5} \sqrt{3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(-20*sqrt(3*x**2 + 5*x + 2)/(32*x**5 + 240*x**4 + 720*x**3 + 1080*x**2
+ 810*x + 243), x) - Integral(-96*x*sqrt(3*x**2 + 5*x + 2)/(32*x**5 + 240*x**4 +
720*x**3 + 1080*x**2 + 810*x + 243), x) - Integral(-165*x**2*sqrt(3*x**2 + 5*x
+ 2)/(32*x**5 + 240*x**4 + 720*x**3 + 1080*x**2 + 810*x + 243), x) - Integral(-1
13*x**3*sqrt(3*x**2 + 5*x + 2)/(32*x**5 + 240*x**4 + 720*x**3 + 1080*x**2 + 810*
x + 243), x) - Integral(-15*x**4*sqrt(3*x**2 + 5*x + 2)/(32*x**5 + 240*x**4 + 72
0*x**3 + 1080*x**2 + 810*x + 243), x) - Integral(9*x**5*sqrt(3*x**2 + 5*x + 2)/(
32*x**5 + 240*x**4 + 720*x**3 + 1080*x**2 + 810*x + 243), x)

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GIAC/XCAS [A]  time = 0.538695, size = 601, normalized size = 3.6 \[ \frac{1875}{256} \, \sqrt{3}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{3} + 2 \, \sqrt{-\frac{8}{2 \, x + 3} + \frac{5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{2 \, \sqrt{5}}{2 \, x + 3} \right |}}{{\left | 2 \, \sqrt{3} + 2 \, \sqrt{-\frac{8}{2 \, x + 3} + \frac{5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{2 \, \sqrt{5}}{2 \, x + 3} \right |}}\right ){\rm sign}\left (\frac{1}{2 \, x + 3}\right ) - \frac{29047}{5120} \, \sqrt{5}{\rm ln}\left ({\left | \sqrt{5}{\left (\sqrt{-\frac{8}{2 \, x + 3} + \frac{5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{\sqrt{5}}{2 \, x + 3}\right )} - 4 \right |}\right ){\rm sign}\left (\frac{1}{2 \, x + 3}\right ) - \frac{1}{3072} \,{\left (\frac{\frac{10 \,{\left (\frac{195 \,{\rm sign}\left (\frac{1}{2 \, x + 3}\right )}{2 \, x + 3} - 904 \,{\rm sign}\left (\frac{1}{2 \, x + 3}\right )\right )}}{2 \, x + 3} + 18577 \,{\rm sign}\left (\frac{1}{2 \, x + 3}\right )}{2 \, x + 3} - 27132 \,{\rm sign}\left (\frac{1}{2 \, x + 3}\right )\right )} \sqrt{-\frac{8}{2 \, x + 3} + \frac{5}{{\left (2 \, x + 3\right )}^{2}} + 3} - \frac{9 \,{\left (157 \,{\left (\sqrt{-\frac{8}{2 \, x + 3} + \frac{5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{\sqrt{5}}{2 \, x + 3}\right )}^{3}{\rm sign}\left (\frac{1}{2 \, x + 3}\right ) - 126 \, \sqrt{5}{\left (\sqrt{-\frac{8}{2 \, x + 3} + \frac{5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{\sqrt{5}}{2 \, x + 3}\right )}^{2}{\rm sign}\left (\frac{1}{2 \, x + 3}\right ) - 409 \,{\left (\sqrt{-\frac{8}{2 \, x + 3} + \frac{5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{\sqrt{5}}{2 \, x + 3}\right )}{\rm sign}\left (\frac{1}{2 \, x + 3}\right ) + 330 \, \sqrt{5}{\rm sign}\left (\frac{1}{2 \, x + 3}\right )\right )}}{128 \,{\left ({\left (\sqrt{-\frac{8}{2 \, x + 3} + \frac{5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{\sqrt{5}}{2 \, x + 3}\right )}^{2} - 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1875/256*sqrt(3)*ln(abs(-2*sqrt(3) + 2*sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) +
2*sqrt(5)/(2*x + 3))/abs(2*sqrt(3) + 2*sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) +
2*sqrt(5)/(2*x + 3)))*sign(1/(2*x + 3)) - 29047/5120*sqrt(5)*ln(abs(sqrt(5)*(sqr
t(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3)) - 4))*sign(1/(2*x + 3))
- 1/3072*((10*(195*sign(1/(2*x + 3))/(2*x + 3) - 904*sign(1/(2*x + 3)))/(2*x +
3) + 18577*sign(1/(2*x + 3)))/(2*x + 3) - 27132*sign(1/(2*x + 3)))*sqrt(-8/(2*x
+ 3) + 5/(2*x + 3)^2 + 3) - 9/128*(157*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) +
sqrt(5)/(2*x + 3))^3*sign(1/(2*x + 3)) - 126*sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*
x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^2*sign(1/(2*x + 3)) - 409*(sqrt(-8/(2*x + 3)
+ 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))*sign(1/(2*x + 3)) + 330*sqrt(5)*sign(1
/(2*x + 3)))/((sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^2 - 3
)^2