3.2523 \(\int \frac{5-x}{(3+2 x)^4 \left (2+5 x+3 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=174 \[ -\frac{2 (47 x+37)}{5 (2 x+3)^3 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{9696 \sqrt{3 x^2+5 x+2}}{625 (2 x+3)}+\frac{1048 \sqrt{3 x^2+5 x+2}}{15 (2 x+3)^2}+\frac{47552 \sqrt{3 x^2+5 x+2}}{375 (2 x+3)^3}+\frac{12 (638 x+603)}{25 (2 x+3)^3 \sqrt{3 x^2+5 x+2}}+\frac{46108 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{625 \sqrt{5}} \]

[Out]

(-2*(37 + 47*x))/(5*(3 + 2*x)^3*(2 + 5*x + 3*x^2)^(3/2)) + (12*(603 + 638*x))/(2
5*(3 + 2*x)^3*Sqrt[2 + 5*x + 3*x^2]) + (47552*Sqrt[2 + 5*x + 3*x^2])/(375*(3 + 2
*x)^3) + (1048*Sqrt[2 + 5*x + 3*x^2])/(15*(3 + 2*x)^2) + (9696*Sqrt[2 + 5*x + 3*
x^2])/(625*(3 + 2*x)) + (46108*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2
])])/(625*Sqrt[5])

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Rubi [A]  time = 0.374681, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ -\frac{2 (47 x+37)}{5 (2 x+3)^3 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{9696 \sqrt{3 x^2+5 x+2}}{625 (2 x+3)}+\frac{1048 \sqrt{3 x^2+5 x+2}}{15 (2 x+3)^2}+\frac{47552 \sqrt{3 x^2+5 x+2}}{375 (2 x+3)^3}+\frac{12 (638 x+603)}{25 (2 x+3)^3 \sqrt{3 x^2+5 x+2}}+\frac{46108 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{625 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]  Int[(5 - x)/((3 + 2*x)^4*(2 + 5*x + 3*x^2)^(5/2)),x]

[Out]

(-2*(37 + 47*x))/(5*(3 + 2*x)^3*(2 + 5*x + 3*x^2)^(3/2)) + (12*(603 + 638*x))/(2
5*(3 + 2*x)^3*Sqrt[2 + 5*x + 3*x^2]) + (47552*Sqrt[2 + 5*x + 3*x^2])/(375*(3 + 2
*x)^3) + (1048*Sqrt[2 + 5*x + 3*x^2])/(15*(3 + 2*x)^2) + (9696*Sqrt[2 + 5*x + 3*
x^2])/(625*(3 + 2*x)) + (46108*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2
])])/(625*Sqrt[5])

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Rubi in Sympy [A]  time = 47.6103, size = 158, normalized size = 0.91 \[ - \frac{46108 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \left (- 8 x - 7\right )}{10 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{3125} + \frac{9696 \sqrt{3 x^{2} + 5 x + 2}}{625 \left (2 x + 3\right )} + \frac{1048 \sqrt{3 x^{2} + 5 x + 2}}{15 \left (2 x + 3\right )^{2}} - \frac{2 \left (141 x + 111\right )}{15 \left (2 x + 3\right )^{3} \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}} + \frac{4 \left (5742 x + 5427\right )}{75 \left (2 x + 3\right )^{3} \sqrt{3 x^{2} + 5 x + 2}} + \frac{47552 \sqrt{3 x^{2} + 5 x + 2}}{375 \left (2 x + 3\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)/(3+2*x)**4/(3*x**2+5*x+2)**(5/2),x)

[Out]

-46108*sqrt(5)*atanh(sqrt(5)*(-8*x - 7)/(10*sqrt(3*x**2 + 5*x + 2)))/3125 + 9696
*sqrt(3*x**2 + 5*x + 2)/(625*(2*x + 3)) + 1048*sqrt(3*x**2 + 5*x + 2)/(15*(2*x +
3)**2) - 2*(141*x + 111)/(15*(2*x + 3)**3*(3*x**2 + 5*x + 2)**(3/2)) + 4*(5742*
x + 5427)/(75*(2*x + 3)**3*sqrt(3*x**2 + 5*x + 2)) + 47552*sqrt(3*x**2 + 5*x + 2
)/(375*(2*x + 3)**3)

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Mathematica [A]  time = 0.264654, size = 105, normalized size = 0.6 \[ \frac{2 \left (-69162 \sqrt{5} \log \left (2 \sqrt{5} \sqrt{3 x^2+5 x+2}-8 x-7\right )+\frac{5 \left (523584 x^6+4495032 x^5+15334836 x^4+26717636 x^3+25105026 x^2+12060957 x+2313929\right )}{(2 x+3)^3 \left (3 x^2+5 x+2\right )^{3/2}}+69162 \sqrt{5} \log (2 x+3)\right )}{9375} \]

Antiderivative was successfully verified.

[In]  Integrate[(5 - x)/((3 + 2*x)^4*(2 + 5*x + 3*x^2)^(5/2)),x]

[Out]

(2*((5*(2313929 + 12060957*x + 25105026*x^2 + 26717636*x^3 + 15334836*x^4 + 4495
032*x^5 + 523584*x^6))/((3 + 2*x)^3*(2 + 5*x + 3*x^2)^(3/2)) + 69162*Sqrt[5]*Log
[3 + 2*x] - 69162*Sqrt[5]*Log[-7 - 8*x + 2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2]]))/9375

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Maple [A]  time = 0.018, size = 169, normalized size = 1. \[ -{\frac{13}{120} \left ( x+{\frac{3}{2}} \right ) ^{-3} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}}-{\frac{151}{200} \left ( x+{\frac{3}{2}} \right ) ^{-2} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}}-{\frac{862}{125} \left ( x+{\frac{3}{2}} \right ) ^{-1} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}}+{\frac{11527}{750} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}}-{\frac{11830+14196\,x}{375} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}}+{\frac{12120+14544\,x}{625}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}}}+{\frac{23054}{625}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}}}-{\frac{46108\,\sqrt{5}}{3125}{\it Artanh} \left ({\frac{2\,\sqrt{5}}{5} \left ( -{\frac{7}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)/(3+2*x)^4/(3*x^2+5*x+2)^(5/2),x)

[Out]

-13/120/(x+3/2)^3/(3*(x+3/2)^2-4*x-19/4)^(3/2)-151/200/(x+3/2)^2/(3*(x+3/2)^2-4*
x-19/4)^(3/2)-862/125/(x+3/2)/(3*(x+3/2)^2-4*x-19/4)^(3/2)+11527/750/(3*(x+3/2)^
2-4*x-19/4)^(3/2)-2366/375*(5+6*x)/(3*(x+3/2)^2-4*x-19/4)^(3/2)+2424/625*(5+6*x)
/(3*(x+3/2)^2-4*x-19/4)^(1/2)+23054/625/(3*(x+3/2)^2-4*x-19/4)^(1/2)-46108/3125*
5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))

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Maxima [A]  time = 0.801597, size = 343, normalized size = 1.97 \[ -\frac{46108}{3125} \, \sqrt{5} \log \left (\frac{\sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac{5}{2 \,{\left | 2 \, x + 3 \right |}} - 2\right ) + \frac{14544 \, x}{625 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} + \frac{35174}{625 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{4732 \, x}{125 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} - \frac{13}{15 \,{\left (8 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x^{3} + 36 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x^{2} + 54 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + 27 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}\right )}} - \frac{151}{50 \,{\left (4 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x^{2} + 12 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + 9 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}\right )}} - \frac{1724}{125 \,{\left (2 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + 3 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}\right )}} - \frac{12133}{750 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x - 5)/((3*x^2 + 5*x + 2)^(5/2)*(2*x + 3)^4),x, algorithm="maxima")

[Out]

-46108/3125*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x
+ 3) - 2) + 14544/625*x/sqrt(3*x^2 + 5*x + 2) + 35174/625/sqrt(3*x^2 + 5*x + 2)
- 4732/125*x/(3*x^2 + 5*x + 2)^(3/2) - 13/15/(8*(3*x^2 + 5*x + 2)^(3/2)*x^3 + 3
6*(3*x^2 + 5*x + 2)^(3/2)*x^2 + 54*(3*x^2 + 5*x + 2)^(3/2)*x + 27*(3*x^2 + 5*x +
2)^(3/2)) - 151/50/(4*(3*x^2 + 5*x + 2)^(3/2)*x^2 + 12*(3*x^2 + 5*x + 2)^(3/2)*
x + 9*(3*x^2 + 5*x + 2)^(3/2)) - 1724/125/(2*(3*x^2 + 5*x + 2)^(3/2)*x + 3*(3*x^
2 + 5*x + 2)^(3/2)) - 12133/750/(3*x^2 + 5*x + 2)^(3/2)

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Fricas [A]  time = 0.288845, size = 235, normalized size = 1.35 \[ \frac{2 \, \sqrt{5}{\left (\sqrt{5}{\left (523584 \, x^{6} + 4495032 \, x^{5} + 15334836 \, x^{4} + 26717636 \, x^{3} + 25105026 \, x^{2} + 12060957 \, x + 2313929\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + 34581 \,{\left (72 \, x^{7} + 564 \, x^{6} + 1862 \, x^{5} + 3355 \, x^{4} + 3560 \, x^{3} + 2223 \, x^{2} + 756 \, x + 108\right )} \log \left (\frac{\sqrt{5}{\left (124 \, x^{2} + 212 \, x + 89\right )} + 20 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{9375 \,{\left (72 \, x^{7} + 564 \, x^{6} + 1862 \, x^{5} + 3355 \, x^{4} + 3560 \, x^{3} + 2223 \, x^{2} + 756 \, x + 108\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x - 5)/((3*x^2 + 5*x + 2)^(5/2)*(2*x + 3)^4),x, algorithm="fricas")

[Out]

2/9375*sqrt(5)*(sqrt(5)*(523584*x^6 + 4495032*x^5 + 15334836*x^4 + 26717636*x^3
+ 25105026*x^2 + 12060957*x + 2313929)*sqrt(3*x^2 + 5*x + 2) + 34581*(72*x^7 + 5
64*x^6 + 1862*x^5 + 3355*x^4 + 3560*x^3 + 2223*x^2 + 756*x + 108)*log((sqrt(5)*(
124*x^2 + 212*x + 89) + 20*sqrt(3*x^2 + 5*x + 2)*(8*x + 7))/(4*x^2 + 12*x + 9)))
/(72*x^7 + 564*x^6 + 1862*x^5 + 3355*x^4 + 3560*x^3 + 2223*x^2 + 756*x + 108)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{x}{144 x^{8} \sqrt{3 x^{2} + 5 x + 2} + 1344 x^{7} \sqrt{3 x^{2} + 5 x + 2} + 5416 x^{6} \sqrt{3 x^{2} + 5 x + 2} + 12296 x^{5} \sqrt{3 x^{2} + 5 x + 2} + 17185 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 15126 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 8181 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 2484 x \sqrt{3 x^{2} + 5 x + 2} + 324 \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac{5}{144 x^{8} \sqrt{3 x^{2} + 5 x + 2} + 1344 x^{7} \sqrt{3 x^{2} + 5 x + 2} + 5416 x^{6} \sqrt{3 x^{2} + 5 x + 2} + 12296 x^{5} \sqrt{3 x^{2} + 5 x + 2} + 17185 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 15126 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 8181 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 2484 x \sqrt{3 x^{2} + 5 x + 2} + 324 \sqrt{3 x^{2} + 5 x + 2}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5-x)/(3+2*x)**4/(3*x**2+5*x+2)**(5/2),x)

[Out]

-Integral(x/(144*x**8*sqrt(3*x**2 + 5*x + 2) + 1344*x**7*sqrt(3*x**2 + 5*x + 2)
+ 5416*x**6*sqrt(3*x**2 + 5*x + 2) + 12296*x**5*sqrt(3*x**2 + 5*x + 2) + 17185*x
**4*sqrt(3*x**2 + 5*x + 2) + 15126*x**3*sqrt(3*x**2 + 5*x + 2) + 8181*x**2*sqrt(
3*x**2 + 5*x + 2) + 2484*x*sqrt(3*x**2 + 5*x + 2) + 324*sqrt(3*x**2 + 5*x + 2)),
x) - Integral(-5/(144*x**8*sqrt(3*x**2 + 5*x + 2) + 1344*x**7*sqrt(3*x**2 + 5*x
+ 2) + 5416*x**6*sqrt(3*x**2 + 5*x + 2) + 12296*x**5*sqrt(3*x**2 + 5*x + 2) + 1
7185*x**4*sqrt(3*x**2 + 5*x + 2) + 15126*x**3*sqrt(3*x**2 + 5*x + 2) + 8181*x**2
*sqrt(3*x**2 + 5*x + 2) + 2484*x*sqrt(3*x**2 + 5*x + 2) + 324*sqrt(3*x**2 + 5*x
+ 2)), x)

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GIAC/XCAS [A]  time = 0.310575, size = 385, normalized size = 2.21 \[ \frac{46108}{3125} \, \sqrt{5}{\rm ln}\left (\frac{{\left | -4 \, \sqrt{3} x - 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt{3} x + 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}\right ) + \frac{2 \,{\left ({\left (12 \,{\left (19992 \, x + 58207\right )} x + 636631\right )} x + 184301\right )}}{3125 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} - \frac{8 \,{\left (296724 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 2103870 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 16891990 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 21246975 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 38063715 \, \sqrt{3} x + 8723544 \, \sqrt{3} - 38063715 \, \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}}{9375 \,{\left (2 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x - 5)/((3*x^2 + 5*x + 2)^(5/2)*(2*x + 3)^4),x, algorithm="giac")

[Out]

46108/3125*sqrt(5)*ln(abs(-4*sqrt(3)*x - 2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 +
5*x + 2))/abs(-4*sqrt(3)*x + 2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))) +
2/3125*((12*(19992*x + 58207)*x + 636631)*x + 184301)/(3*x^2 + 5*x + 2)^(3/2) -
8/9375*(296724*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 + 2103870*sqrt(3)*(sqrt(3)
*x - sqrt(3*x^2 + 5*x + 2))^4 + 16891990*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 +
21246975*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 38063715*sqrt(3)*x + 8
723544*sqrt(3) - 38063715*sqrt(3*x^2 + 5*x + 2))/(2*(sqrt(3)*x - sqrt(3*x^2 + 5*
x + 2))^2 + 6*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) + 11)^3