3.2511 \(\int \frac{5-x}{(3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=94 \[ -\frac{6 (47 x+37)}{5 (2 x+3) \sqrt{3 x^2+5 x+2}}-\frac{856 \sqrt{3 x^2+5 x+2}}{25 (2 x+3)}+\frac{302 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{25 \sqrt{5}} \]

[Out]

(-6*(37 + 47*x))/(5*(3 + 2*x)*Sqrt[2 + 5*x + 3*x^2]) - (856*Sqrt[2 + 5*x + 3*x^2
])/(25*(3 + 2*x)) + (302*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(
25*Sqrt[5])

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Rubi [A]  time = 0.160557, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ -\frac{6 (47 x+37)}{5 (2 x+3) \sqrt{3 x^2+5 x+2}}-\frac{856 \sqrt{3 x^2+5 x+2}}{25 (2 x+3)}+\frac{302 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{25 \sqrt{5}} \]

Antiderivative was successfully verified.

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

[Out]

(-6*(37 + 47*x))/(5*(3 + 2*x)*Sqrt[2 + 5*x + 3*x^2]) - (856*Sqrt[2 + 5*x + 3*x^2
])/(25*(3 + 2*x)) + (302*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 = 22.8878, size = 83, normalized size = 0.88 \[ - \frac{302 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \left (- 8 x - 7\right )}{10 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{125} - \frac{2 \left (141 x + 111\right )}{5 \left (2 x + 3\right ) \sqrt{3 x^{2} + 5 x + 2}} - \frac{856 \sqrt{3 x^{2} + 5 x + 2}}{25 \left (2 x + 3\right )} \]

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)**(3/2),x)

[Out]

-302*sqrt(5)*atanh(sqrt(5)*(-8*x - 7)/(10*sqrt(3*x**2 + 5*x + 2)))/125 - 2*(141*
x + 111)/(5*(2*x + 3)*sqrt(3*x**2 + 5*x + 2)) - 856*sqrt(3*x**2 + 5*x + 2)/(25*(
2*x + 3))

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Mathematica [A]  time = 0.179428, size = 85, normalized size = 0.9 \[ \frac{2}{125} \left (-\frac{5 \left (1284 x^2+2845 x+1411\right )}{(2 x+3) \sqrt{3 x^2+5 x+2}}-151 \sqrt{5} \log \left (2 \sqrt{5} \sqrt{3 x^2+5 x+2}-8 x-7\right )+151 \sqrt{5} \log (2 x+3)\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*((-5*(1411 + 2845*x + 1284*x^2))/((3 + 2*x)*Sqrt[2 + 5*x + 3*x^2]) + 151*Sqrt
[5]*Log[3 + 2*x] - 151*Sqrt[5]*Log[-7 - 8*x + 2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2]]))
/125

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Maple [A]  time = 0.016, size = 90, normalized size = 1. \[ -{\frac{13}{10} \left ( x+{\frac{3}{2}} \right ) ^{-1}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}}}+{\frac{151}{25}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}}}-{\frac{1070+1284\,x}{25}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}}}-{\frac{302\,\sqrt{5}}{125}{\it Artanh} \left ({\frac{2\,\sqrt{5}}{5} \left ( -{\frac{7}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-13/10/(x+3/2)/(3*(x+3/2)^2-4*x-19/4)^(1/2)+151/25/(3*(x+3/2)^2-4*x-19/4)^(1/2)-
214/25*(5+6*x)/(3*(x+3/2)^2-4*x-19/4)^(1/2)-302/125*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 = 0.796656, size = 143, normalized size = 1.52 \[ -\frac{302}{125} \, \sqrt{5} \log \left (\frac{\sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac{5}{2 \,{\left | 2 \, x + 3 \right |}} - 2\right ) - \frac{1284 \, x}{25 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{919}{25 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{13}{5 \,{\left (2 \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + 3 \, \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-302/125*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x +
3) - 2) - 1284/25*x/sqrt(3*x^2 + 5*x + 2) - 919/25/sqrt(3*x^2 + 5*x + 2) - 13/5/
(2*sqrt(3*x^2 + 5*x + 2)*x + 3*sqrt(3*x^2 + 5*x + 2))

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Fricas [A]  time = 0.282123, size = 155, normalized size = 1.65 \[ -\frac{\sqrt{5}{\left (2 \, \sqrt{5}{\left (1284 \, x^{2} + 2845 \, x + 1411\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - 151 \,{\left (6 \, x^{3} + 19 \, x^{2} + 19 \, x + 6\right )} \log \left (\frac{\sqrt{5}{\left (124 \, x^{2} + 212 \, x + 89\right )} + 20 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{125 \,{\left (6 \, x^{3} + 19 \, x^{2} + 19 \, x + 6\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/125*sqrt(5)*(2*sqrt(5)*(1284*x^2 + 2845*x + 1411)*sqrt(3*x^2 + 5*x + 2) - 151
*(6*x^3 + 19*x^2 + 19*x + 6)*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)))/(6*x^3 + 19*x^2 + 19*x + 6)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{x}{12 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 56 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 95 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 69 x \sqrt{3 x^{2} + 5 x + 2} + 18 \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac{5}{12 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 56 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 95 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 69 x \sqrt{3 x^{2} + 5 x + 2} + 18 \sqrt{3 x^{2} + 5 x + 2}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(x/(12*x**4*sqrt(3*x**2 + 5*x + 2) + 56*x**3*sqrt(3*x**2 + 5*x + 2) + 9
5*x**2*sqrt(3*x**2 + 5*x + 2) + 69*x*sqrt(3*x**2 + 5*x + 2) + 18*sqrt(3*x**2 + 5
*x + 2)), x) - Integral(-5/(12*x**4*sqrt(3*x**2 + 5*x + 2) + 56*x**3*sqrt(3*x**2
+ 5*x + 2) + 95*x**2*sqrt(3*x**2 + 5*x + 2) + 69*x*sqrt(3*x**2 + 5*x + 2) + 18*
sqrt(3*x**2 + 5*x + 2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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