3.2522 \(\int \frac{5-x}{(3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=147 \[ -\frac{2 (47 x+37)}{5 (2 x+3)^2 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{11808 \sqrt{3 x^2+5 x+2}}{125 (2 x+3)}+\frac{152 \sqrt{3 x^2+5 x+2}}{(2 x+3)^2}+\frac{4 (2112 x+1907)}{25 (2 x+3)^2 \sqrt{3 x^2+5 x+2}}+\frac{4884 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{125 \sqrt{5}} \]

[Out]

(-2*(37 + 47*x))/(5*(3 + 2*x)^2*(2 + 5*x + 3*x^2)^(3/2)) + (4*(1907 + 2112*x))/(
25*(3 + 2*x)^2*Sqrt[2 + 5*x + 3*x^2]) + (152*Sqrt[2 + 5*x + 3*x^2])/(3 + 2*x)^2
+ (11808*Sqrt[2 + 5*x + 3*x^2])/(125*(3 + 2*x)) + (4884*ArcTanh[(7 + 8*x)/(2*Sqr
t[5]*Sqrt[2 + 5*x + 3*x^2])])/(125*Sqrt[5])

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Rubi [A]  time = 0.301667, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ -\frac{2 (47 x+37)}{5 (2 x+3)^2 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{11808 \sqrt{3 x^2+5 x+2}}{125 (2 x+3)}+\frac{152 \sqrt{3 x^2+5 x+2}}{(2 x+3)^2}+\frac{4 (2112 x+1907)}{25 (2 x+3)^2 \sqrt{3 x^2+5 x+2}}+\frac{4884 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{125 \sqrt{5}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(37 + 47*x))/(5*(3 + 2*x)^2*(2 + 5*x + 3*x^2)^(3/2)) + (4*(1907 + 2112*x))/(
25*(3 + 2*x)^2*Sqrt[2 + 5*x + 3*x^2]) + (152*Sqrt[2 + 5*x + 3*x^2])/(3 + 2*x)^2
+ (11808*Sqrt[2 + 5*x + 3*x^2])/(125*(3 + 2*x)) + (4884*ArcTanh[(7 + 8*x)/(2*Sqr
t[5]*Sqrt[2 + 5*x + 3*x^2])])/(125*Sqrt[5])

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Rubi in Sympy [A]  time = 39.8382, size = 133, normalized size = 0.9 \[ - \frac{4884 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \left (- 8 x - 7\right )}{10 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{625} + \frac{11808 \sqrt{3 x^{2} + 5 x + 2}}{125 \left (2 x + 3\right )} - \frac{2 \left (141 x + 111\right )}{15 \left (2 x + 3\right )^{2} \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}} + \frac{4 \left (6336 x + 5721\right )}{75 \left (2 x + 3\right )^{2} \sqrt{3 x^{2} + 5 x + 2}} + \frac{152 \sqrt{3 x^{2} + 5 x + 2}}{\left (2 x + 3\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4884*sqrt(5)*atanh(sqrt(5)*(-8*x - 7)/(10*sqrt(3*x**2 + 5*x + 2)))/625 + 11808*
sqrt(3*x**2 + 5*x + 2)/(125*(2*x + 3)) - 2*(141*x + 111)/(15*(2*x + 3)**2*(3*x**
2 + 5*x + 2)**(3/2)) + 4*(6336*x + 5721)/(75*(2*x + 3)**2*sqrt(3*x**2 + 5*x + 2)
) + 152*sqrt(3*x**2 + 5*x + 2)/(2*x + 3)**2

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Mathematica [A]  time = 0.264379, size = 100, normalized size = 0.68 \[ \frac{1}{625} \left (-4884 \sqrt{5} \log \left (2 \sqrt{5} \sqrt{3 x^2+5 x+2}-8 x-7\right )+\frac{10 \left (106272 x^5+599148 x^4+1316616 x^3+1405814 x^2+727887 x+146063\right )}{(2 x+3)^2 \left (3 x^2+5 x+2\right )^{3/2}}+4884 \sqrt{5} \log (2 x+3)\right ) \]

Antiderivative was successfully verified.

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

[Out]

((10*(146063 + 727887*x + 1405814*x^2 + 1316616*x^3 + 599148*x^4 + 106272*x^5))/
((3 + 2*x)^2*(2 + 5*x + 3*x^2)^(3/2)) + 4884*Sqrt[5]*Log[3 + 2*x] - 4884*Sqrt[5]
*Log[-7 - 8*x + 2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2]])/625

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Maple [A]  time = 0.016, size = 148, normalized size = 1. \[ -{\frac{13}{40} \left ( x+{\frac{3}{2}} \right ) ^{-2} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}}-{\frac{177}{50} \left ( x+{\frac{3}{2}} \right ) ^{-1} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}}+{\frac{407}{50} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}}-{\frac{530+636\,x}{25} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}}+{\frac{14760+17712\,x}{125}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}}}+{\frac{2442}{125}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}}}-{\frac{4884\,\sqrt{5}}{625}{\it Artanh} \left ({\frac{2\,\sqrt{5}}{5} \left ( -{\frac{7}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-13/40/(x+3/2)^2/(3*(x+3/2)^2-4*x-19/4)^(3/2)-177/50/(x+3/2)/(3*(x+3/2)^2-4*x-19
/4)^(3/2)+407/50/(3*(x+3/2)^2-4*x-19/4)^(3/2)-106/25*(5+6*x)/(3*(x+3/2)^2-4*x-19
/4)^(3/2)+2952/125*(5+6*x)/(3*(x+3/2)^2-4*x-19/4)^(1/2)+2442/125/(3*(x+3/2)^2-4*
x-19/4)^(1/2)-4884/625*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 = 0.797834, size = 251, normalized size = 1.71 \[ -\frac{4884}{625} \, \sqrt{5} \log \left (\frac{\sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac{5}{2 \,{\left | 2 \, x + 3 \right |}} - 2\right ) + \frac{17712 \, x}{125 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} + \frac{17202}{125 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{636 \, x}{25 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} - \frac{13}{10 \,{\left (4 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x^{2} + 12 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + 9 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}\right )}} - \frac{177}{25 \,{\left (2 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + 3 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}\right )}} - \frac{653}{50 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} \]

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)^3),x, algorithm="maxima")

[Out]

-4884/625*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x +
3) - 2) + 17712/125*x/sqrt(3*x^2 + 5*x + 2) + 17202/125/sqrt(3*x^2 + 5*x + 2) -
636/25*x/(3*x^2 + 5*x + 2)^(3/2) - 13/10/(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)) - 177/25/(2*(3*x^2 + 5*x +
2)^(3/2)*x + 3*(3*x^2 + 5*x + 2)^(3/2)) - 653/50/(3*x^2 + 5*x + 2)^(3/2)

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Fricas [A]  time = 0.283243, size = 215, normalized size = 1.46 \[ \frac{2 \, \sqrt{5}{\left (\sqrt{5}{\left (106272 \, x^{5} + 599148 \, x^{4} + 1316616 \, x^{3} + 1405814 \, x^{2} + 727887 \, x + 146063\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + 1221 \,{\left (36 \, x^{6} + 228 \, x^{5} + 589 \, x^{4} + 794 \, x^{3} + 589 \, x^{2} + 228 \, x + 36\right )} \log \left (\frac{\sqrt{5}{\left (124 \, x^{2} + 212 \, x + 89\right )} + 20 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{625 \,{\left (36 \, x^{6} + 228 \, x^{5} + 589 \, x^{4} + 794 \, x^{3} + 589 \, x^{2} + 228 \, x + 36\right )}} \]

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)^3),x, algorithm="fricas")

[Out]

2/625*sqrt(5)*(sqrt(5)*(106272*x^5 + 599148*x^4 + 1316616*x^3 + 1405814*x^2 + 72
7887*x + 146063)*sqrt(3*x^2 + 5*x + 2) + 1221*(36*x^6 + 228*x^5 + 589*x^4 + 794*
x^3 + 589*x^2 + 228*x + 36)*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)))/(36*x^6 + 228*x^5 + 589*x^4 + 794*x^3
+ 589*x^2 + 228*x + 36)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{x}{72 x^{7} \sqrt{3 x^{2} + 5 x + 2} + 564 x^{6} \sqrt{3 x^{2} + 5 x + 2} + 1862 x^{5} \sqrt{3 x^{2} + 5 x + 2} + 3355 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 3560 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 2223 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 756 x \sqrt{3 x^{2} + 5 x + 2} + 108 \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac{5}{72 x^{7} \sqrt{3 x^{2} + 5 x + 2} + 564 x^{6} \sqrt{3 x^{2} + 5 x + 2} + 1862 x^{5} \sqrt{3 x^{2} + 5 x + 2} + 3355 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 3560 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 2223 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 756 x \sqrt{3 x^{2} + 5 x + 2} + 108 \sqrt{3 x^{2} + 5 x + 2}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(x/(72*x**7*sqrt(3*x**2 + 5*x + 2) + 564*x**6*sqrt(3*x**2 + 5*x + 2) +
1862*x**5*sqrt(3*x**2 + 5*x + 2) + 3355*x**4*sqrt(3*x**2 + 5*x + 2) + 3560*x**3*
sqrt(3*x**2 + 5*x + 2) + 2223*x**2*sqrt(3*x**2 + 5*x + 2) + 756*x*sqrt(3*x**2 +
5*x + 2) + 108*sqrt(3*x**2 + 5*x + 2)), x) - Integral(-5/(72*x**7*sqrt(3*x**2 +
5*x + 2) + 564*x**6*sqrt(3*x**2 + 5*x + 2) + 1862*x**5*sqrt(3*x**2 + 5*x + 2) +
3355*x**4*sqrt(3*x**2 + 5*x + 2) + 3560*x**3*sqrt(3*x**2 + 5*x + 2) + 2223*x**2*
sqrt(3*x**2 + 5*x + 2) + 756*x*sqrt(3*x**2 + 5*x + 2) + 108*sqrt(3*x**2 + 5*x +
2)), x)

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GIAC/XCAS [A]  time = 0.307478, size = 316, normalized size = 2.15 \[ \frac{4884}{625} \, \sqrt{5}{\rm ln}\left (\frac{{\left | -4 \, \sqrt{3} x - 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt{3} x + 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}\right ) + \frac{2 \,{\left ({\left (6 \,{\left (23826 \, x + 61591\right )} x + 309599\right )} x + 84259\right )}}{625 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} - \frac{8 \,{\left (4106 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 16447 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 57729 \, \sqrt{3} x + 20987 \, \sqrt{3} - 57729 \, \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}}{625 \,{\left (2 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{2}} \]

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)^3),x, algorithm="giac")

[Out]

4884/625*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
/625*((6*(23826*x + 61591)*x + 309599)*x + 84259)/(3*x^2 + 5*x + 2)^(3/2) - 8/62
5*(4106*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 + 16447*sqrt(3)*(sqrt(3)*x - sqrt(
3*x^2 + 5*x + 2))^2 + 57729*sqrt(3)*x + 20987*sqrt(3) - 57729*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)^2