3.2521 \(\int \frac{5-x}{(3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=124 \[ -\frac{2 (47 x+37)}{5 (2 x+3) \left (3 x^2+5 x+2\right )^{3/2}}+\frac{4416 \sqrt{3 x^2+5 x+2}}{25 (2 x+3)}+\frac{4 (462 x+401)}{5 (2 x+3) \sqrt{3 x^2+5 x+2}}+\frac{408 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{25 \sqrt{5}} \]

[Out]

(-2*(37 + 47*x))/(5*(3 + 2*x)*(2 + 5*x + 3*x^2)^(3/2)) + (4*(401 + 462*x))/(5*(3
+ 2*x)*Sqrt[2 + 5*x + 3*x^2]) + (4416*Sqrt[2 + 5*x + 3*x^2])/(25*(3 + 2*x)) + (
408*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(25*Sqrt[5])

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Rubi [A]  time = 0.234905, antiderivative size = 124, 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 \[ -\frac{2 (47 x+37)}{5 (2 x+3) \left (3 x^2+5 x+2\right )^{3/2}}+\frac{4416 \sqrt{3 x^2+5 x+2}}{25 (2 x+3)}+\frac{4 (462 x+401)}{5 (2 x+3) \sqrt{3 x^2+5 x+2}}+\frac{408 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{25 \sqrt{5}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(37 + 47*x))/(5*(3 + 2*x)*(2 + 5*x + 3*x^2)^(3/2)) + (4*(401 + 462*x))/(5*(3
+ 2*x)*Sqrt[2 + 5*x + 3*x^2]) + (4416*Sqrt[2 + 5*x + 3*x^2])/(25*(3 + 2*x)) + (
408*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(25*Sqrt[5])

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Rubi in Sympy [A]  time = 31.076, size = 107, normalized size = 0.86 \[ - \frac{408 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \left (- 8 x - 7\right )}{10 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{125} - \frac{2 \left (141 x + 111\right )}{15 \left (2 x + 3\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}} + \frac{4 \left (6930 x + 6015\right )}{75 \left (2 x + 3\right ) \sqrt{3 x^{2} + 5 x + 2}} + \frac{4416 \sqrt{3 x^{2} + 5 x + 2}}{25 \left (2 x + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-408*sqrt(5)*atanh(sqrt(5)*(-8*x - 7)/(10*sqrt(3*x**2 + 5*x + 2)))/125 - 2*(141*
x + 111)/(15*(2*x + 3)*(3*x**2 + 5*x + 2)**(3/2)) + 4*(6930*x + 6015)/(75*(2*x +
3)*sqrt(3*x**2 + 5*x + 2)) + 4416*sqrt(3*x**2 + 5*x + 2)/(25*(2*x + 3))

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Mathematica [A]  time = 0.180198, size = 95, normalized size = 0.77 \[ \frac{2}{125} \left (-204 \sqrt{5} \log \left (2 \sqrt{5} \sqrt{3 x^2+5 x+2}-8 x-7\right )+\frac{5 \left (19872 x^4+80100 x^3+116826 x^2+73215 x+16667\right )}{(2 x+3) \left (3 x^2+5 x+2\right )^{3/2}}+204 \sqrt{5} \log (2 x+3)\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*((5*(16667 + 73215*x + 116826*x^2 + 80100*x^3 + 19872*x^4))/((3 + 2*x)*(2 + 5
*x + 3*x^2)^(3/2)) + 204*Sqrt[5]*Log[3 + 2*x] - 204*Sqrt[5]*Log[-7 - 8*x + 2*Sqr
t[5]*Sqrt[2 + 5*x + 3*x^2]]))/125

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Maple [A]  time = 0.016, size = 127, normalized size = 1. \[ -{\frac{13}{10} \left ( x+{\frac{3}{2}} \right ) ^{-1} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}}+{\frac{17}{5} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}}-{\frac{80+96\,x}{5} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}}+{\frac{5520+6624\,x}{25}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}}}+{\frac{204}{25}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}}}-{\frac{408\,\sqrt{5}}{125}{\it Artanh} \left ({\frac{2\,\sqrt{5}}{5} \left ( -{\frac{7}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-13/10/(x+3/2)/(3*(x+3/2)^2-4*x-19/4)^(3/2)+17/5/(3*(x+3/2)^2-4*x-19/4)^(3/2)-16
/5*(5+6*x)/(3*(x+3/2)^2-4*x-19/4)^(3/2)+1104/25*(5+6*x)/(3*(x+3/2)^2-4*x-19/4)^(
1/2)+204/25/(3*(x+3/2)^2-4*x-19/4)^(1/2)-408/125*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 = 0.792977, size = 182, normalized size = 1.47 \[ -\frac{408}{125} \, \sqrt{5} \log \left (\frac{\sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac{5}{2 \,{\left | 2 \, x + 3 \right |}} - 2\right ) + \frac{6624 \, x}{25 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} + \frac{5724}{25 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{96 \, x}{5 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} - \frac{13}{5 \,{\left (2 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + 3 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}\right )}} - \frac{63}{5 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} \]

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)^2),x, algorithm="maxima")

[Out]

-408/125*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x +
3) - 2) + 6624/25*x/sqrt(3*x^2 + 5*x + 2) + 5724/25/sqrt(3*x^2 + 5*x + 2) - 96/5
*x/(3*x^2 + 5*x + 2)^(3/2) - 13/5/(2*(3*x^2 + 5*x + 2)^(3/2)*x + 3*(3*x^2 + 5*x
+ 2)^(3/2)) - 63/5/(3*x^2 + 5*x + 2)^(3/2)

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Fricas [A]  time = 0.286449, size = 194, normalized size = 1.56 \[ \frac{2 \, \sqrt{5}{\left (\sqrt{5}{\left (19872 \, x^{4} + 80100 \, x^{3} + 116826 \, x^{2} + 73215 \, x + 16667\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + 102 \,{\left (18 \, x^{5} + 87 \, x^{4} + 164 \, x^{3} + 151 \, x^{2} + 68 \, x + 12\right )} \log \left (\frac{\sqrt{5}{\left (124 \, x^{2} + 212 \, x + 89\right )} + 20 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{125 \,{\left (18 \, x^{5} + 87 \, x^{4} + 164 \, x^{3} + 151 \, x^{2} + 68 \, x + 12\right )}} \]

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)^2),x, algorithm="fricas")

[Out]

2/125*sqrt(5)*(sqrt(5)*(19872*x^4 + 80100*x^3 + 116826*x^2 + 73215*x + 16667)*sq
rt(3*x^2 + 5*x + 2) + 102*(18*x^5 + 87*x^4 + 164*x^3 + 151*x^2 + 68*x + 12)*log(
(sqrt(5)*(124*x^2 + 212*x + 89) + 20*sqrt(3*x^2 + 5*x + 2)*(8*x + 7))/(4*x^2 + 1
2*x + 9)))/(18*x^5 + 87*x^4 + 164*x^3 + 151*x^2 + 68*x + 12)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{x}{36 x^{6} \sqrt{3 x^{2} + 5 x + 2} + 228 x^{5} \sqrt{3 x^{2} + 5 x + 2} + 589 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 794 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 589 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 228 x \sqrt{3 x^{2} + 5 x + 2} + 36 \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac{5}{36 x^{6} \sqrt{3 x^{2} + 5 x + 2} + 228 x^{5} \sqrt{3 x^{2} + 5 x + 2} + 589 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 794 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 589 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 228 x \sqrt{3 x^{2} + 5 x + 2} + 36 \sqrt{3 x^{2} + 5 x + 2}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(x/(36*x**6*sqrt(3*x**2 + 5*x + 2) + 228*x**5*sqrt(3*x**2 + 5*x + 2) +
589*x**4*sqrt(3*x**2 + 5*x + 2) + 794*x**3*sqrt(3*x**2 + 5*x + 2) + 589*x**2*sqr
t(3*x**2 + 5*x + 2) + 228*x*sqrt(3*x**2 + 5*x + 2) + 36*sqrt(3*x**2 + 5*x + 2)),
x) - Integral(-5/(36*x**6*sqrt(3*x**2 + 5*x + 2) + 228*x**5*sqrt(3*x**2 + 5*x +
2) + 589*x**4*sqrt(3*x**2 + 5*x + 2) + 794*x**3*sqrt(3*x**2 + 5*x + 2) + 589*x*
*2*sqrt(3*x**2 + 5*x + 2) + 228*x*sqrt(3*x**2 + 5*x + 2) + 36*sqrt(3*x**2 + 5*x
+ 2)), x)

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

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)^2),x, algorithm="giac")

[Out]

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