3.2520 \(\int \frac{5-x}{(3+2 x) \left (2+5 x+3 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=85 \[ -\frac{2 (47 x+37)}{5 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{12 (836 x+701)}{25 \sqrt{3 x^2+5 x+2}}+\frac{104 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{25 \sqrt{5}} \]

[Out]

(-2*(37 + 47*x))/(5*(2 + 5*x + 3*x^2)^(3/2)) + (12*(701 + 836*x))/(25*Sqrt[2 + 5
*x + 3*x^2]) + (104*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(25*Sq
rt[5])

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Rubi [A]  time = 0.170323, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ -\frac{2 (47 x+37)}{5 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{12 (836 x+701)}{25 \sqrt{3 x^2+5 x+2}}+\frac{104 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{25 \sqrt{5}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(37 + 47*x))/(5*(2 + 5*x + 3*x^2)^(3/2)) + (12*(701 + 836*x))/(25*Sqrt[2 + 5
*x + 3*x^2]) + (104*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(25*Sq
rt[5])

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Rubi in Sympy [A]  time = 25.087, size = 76, normalized size = 0.89 \[ - \frac{2 \left (141 x + 111\right )}{15 \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}} + \frac{4 \left (7524 x + 6309\right )}{75 \sqrt{3 x^{2} + 5 x + 2}} - \frac{104 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \left (- 8 x - 7\right )}{10 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(141*x + 111)/(15*(3*x**2 + 5*x + 2)**(3/2)) + 4*(7524*x + 6309)/(75*sqrt(3*x
**2 + 5*x + 2)) - 104*sqrt(5)*atanh(sqrt(5)*(-8*x - 7)/(10*sqrt(3*x**2 + 5*x + 2
)))/125

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Mathematica [A]  time = 0.152814, size = 83, normalized size = 0.98 \[ \frac{2}{125} \left (-52 \sqrt{5} \log \left (2 \sqrt{5} \sqrt{3 x^2+5 x+2}-8 x-7\right )+\frac{5 \left (15048 x^3+37698 x^2+30827 x+8227\right )}{\left (3 x^2+5 x+2\right )^{3/2}}+52 \sqrt{5} \log (2 x+3)\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*((5*(8227 + 30827*x + 37698*x^2 + 15048*x^3))/(2 + 5*x + 3*x^2)^(3/2) + 52*Sq
rt[5]*Log[3 + 2*x] - 52*Sqrt[5]*Log[-7 - 8*x + 2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2]])
)/125

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Maple [B]  time = 0.012, size = 144, normalized size = 1.7 \[{\frac{5+6\,x}{3} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{-{\frac{3}{2}}}}-8\,{\frac{5+6\,x}{\sqrt{3\,{x}^{2}+5\,x+2}}}+{\frac{13}{15} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}}-{\frac{260+312\,x}{15} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}}+{\frac{9360+11232\,x}{25}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}}}+{\frac{52}{25}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}}}-{\frac{104\,\sqrt{5}}{125}{\it Artanh} \left ({\frac{2\,\sqrt{5}}{5} \left ( -{\frac{7}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(5+6*x)/(3*x^2+5*x+2)^(3/2)-8*(5+6*x)/(3*x^2+5*x+2)^(1/2)+13/15/(3*(x+3/2)^2
-4*x-19/4)^(3/2)-52/15*(5+6*x)/(3*(x+3/2)^2-4*x-19/4)^(3/2)+1872/25*(5+6*x)/(3*(
x+3/2)^2-4*x-19/4)^(1/2)+52/25/(3*(x+3/2)^2-4*x-19/4)^(1/2)-104/125*5^(1/2)*arct
anh(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 = 0.788857, size = 136, normalized size = 1.6 \[ -\frac{104}{125} \, \sqrt{5} \log \left (\frac{\sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac{5}{2 \,{\left | 2 \, x + 3 \right |}} - 2\right ) + \frac{10032 \, x}{25 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} + \frac{8412}{25 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{94 \, x}{5 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} - \frac{74}{5 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} \]

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)),x, algorithm="maxima")

[Out]

-104/125*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x +
3) - 2) + 10032/25*x/sqrt(3*x^2 + 5*x + 2) + 8412/25/sqrt(3*x^2 + 5*x + 2) - 94/
5*x/(3*x^2 + 5*x + 2)^(3/2) - 74/5/(3*x^2 + 5*x + 2)^(3/2)

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Fricas [A]  time = 0.282452, size = 174, normalized size = 2.05 \[ \frac{2 \, \sqrt{5}{\left (\sqrt{5}{\left (15048 \, x^{3} + 37698 \, x^{2} + 30827 \, x + 8227\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + 26 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (\frac{\sqrt{5}{\left (124 \, x^{2} + 212 \, x + 89\right )} + 20 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{125 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \]

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)),x, algorithm="fricas")

[Out]

2/125*sqrt(5)*(sqrt(5)*(15048*x^3 + 37698*x^2 + 30827*x + 8227)*sqrt(3*x^2 + 5*x
+ 2) + 26*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*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)))/(9*x^4 + 30*x^3 +
37*x^2 + 20*x + 4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{x}{18 x^{5} \sqrt{3 x^{2} + 5 x + 2} + 87 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 164 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 151 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 68 x \sqrt{3 x^{2} + 5 x + 2} + 12 \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac{5}{18 x^{5} \sqrt{3 x^{2} + 5 x + 2} + 87 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 164 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 151 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 68 x \sqrt{3 x^{2} + 5 x + 2} + 12 \sqrt{3 x^{2} + 5 x + 2}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(x/(18*x**5*sqrt(3*x**2 + 5*x + 2) + 87*x**4*sqrt(3*x**2 + 5*x + 2) + 1
64*x**3*sqrt(3*x**2 + 5*x + 2) + 151*x**2*sqrt(3*x**2 + 5*x + 2) + 68*x*sqrt(3*x
**2 + 5*x + 2) + 12*sqrt(3*x**2 + 5*x + 2)), x) - Integral(-5/(18*x**5*sqrt(3*x*
*2 + 5*x + 2) + 87*x**4*sqrt(3*x**2 + 5*x + 2) + 164*x**3*sqrt(3*x**2 + 5*x + 2)
+ 151*x**2*sqrt(3*x**2 + 5*x + 2) + 68*x*sqrt(3*x**2 + 5*x + 2) + 12*sqrt(3*x**
2 + 5*x + 2)), x)

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GIAC/XCAS [A]  time = 0.311614, size = 138, normalized size = 1.62 \[ \frac{104}{125} \, \sqrt{5}{\rm ln}\left (\frac{{\left | -4 \, \sqrt{3} x - 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt{3} x + 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}\right ) + \frac{2 \,{\left ({\left (6 \,{\left (2508 \, x + 6283\right )} x + 30827\right )} x + 8227\right )}}{25 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} \]

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)),x, algorithm="giac")

[Out]

104/125*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/
25*((6*(2508*x + 6283)*x + 30827)*x + 8227)/(3*x^2 + 5*x + 2)^(3/2)