3.2411 \(\int \frac{(5-x) \sqrt{2+5 x+3 x^2}}{(3+2 x)^2} \, dx\)

Optimal. Leaf size=105 \[ -\frac{\sqrt{3 x^2+5 x+2} (x+8)}{2 (2 x+3)}+\frac{43 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{8 \sqrt{3}}-\frac{57 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{8 \sqrt{5}} \]

[Out]

-((8 + x)*Sqrt[2 + 5*x + 3*x^2])/(2*(3 + 2*x)) + (43*ArcTanh[(5 + 6*x)/(2*Sqrt[3
]*Sqrt[2 + 5*x + 3*x^2])])/(8*Sqrt[3]) - (57*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2
+ 5*x + 3*x^2])])/(8*Sqrt[5])

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Rubi [A]  time = 0.180983, antiderivative size = 105, 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 \[ -\frac{\sqrt{3 x^2+5 x+2} (x+8)}{2 (2 x+3)}+\frac{43 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{8 \sqrt{3}}-\frac{57 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{8 \sqrt{5}} \]

Antiderivative was successfully verified.

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

[Out]

-((8 + x)*Sqrt[2 + 5*x + 3*x^2])/(2*(3 + 2*x)) + (43*ArcTanh[(5 + 6*x)/(2*Sqrt[3
]*Sqrt[2 + 5*x + 3*x^2])])/(8*Sqrt[3]) - (57*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2
+ 5*x + 3*x^2])])/(8*Sqrt[5])

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Rubi in Sympy [A]  time = 25.3278, size = 95, normalized size = 0.9 \[ \frac{43 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{24} + \frac{57 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \left (- 8 x - 7\right )}{10 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{40} - \frac{\left (2 x + 16\right ) \sqrt{3 x^{2} + 5 x + 2}}{4 \left (2 x + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)*(3*x**2+5*x+2)**(1/2)/(3+2*x)**2,x)

[Out]

43*sqrt(3)*atanh(sqrt(3)*(6*x + 5)/(6*sqrt(3*x**2 + 5*x + 2)))/24 + 57*sqrt(5)*a
tanh(sqrt(5)*(-8*x - 7)/(10*sqrt(3*x**2 + 5*x + 2)))/40 - (2*x + 16)*sqrt(3*x**2
+ 5*x + 2)/(4*(2*x + 3))

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Mathematica [A]  time = 0.158848, size = 107, normalized size = 1.02 \[ \frac{1}{120} \left (-\frac{60 \sqrt{3 x^2+5 x+2} (x+8)}{2 x+3}+171 \sqrt{5} \log \left (2 \sqrt{5} \sqrt{3 x^2+5 x+2}-8 x-7\right )+215 \sqrt{3} \log \left (-2 \sqrt{9 x^2+15 x+6}-6 x-5\right )-171 \sqrt{5} \log (2 x+3)\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-60*(8 + x)*Sqrt[2 + 5*x + 3*x^2])/(3 + 2*x) - 171*Sqrt[5]*Log[3 + 2*x] + 171*
Sqrt[5]*Log[-7 - 8*x + 2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2]] + 215*Sqrt[3]*Log[-5 - 6
*x - 2*Sqrt[6 + 15*x + 9*x^2]])/120

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Maple [A]  time = 0.015, size = 121, normalized size = 1.2 \[ -{\frac{13}{10} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}-{\frac{57}{40}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}+{\frac{43\,\sqrt{3}}{24}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \right ) }+{\frac{57\,\sqrt{5}}{40}{\it Artanh} \left ({\frac{2\,\sqrt{5}}{5} \left ( -{\frac{7}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}}} \right ) }+{\frac{65+78\,x}{20}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)^2,x)

[Out]

-13/10/(x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(3/2)-57/40*(12*(x+3/2)^2-16*x-19)^(1/2)+4
3/24*ln(1/3*(5/2+3*x)*3^(1/2)+(3*(x+3/2)^2-4*x-19/4)^(1/2))*3^(1/2)+57/40*5^(1/2
)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))+13/20*(5+6*x)*(3*
(x+3/2)^2-4*x-19/4)^(1/2)

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Maxima [A]  time = 0.776571, size = 142, normalized size = 1.35 \[ \frac{43}{24} \, \sqrt{3} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac{5}{2}\right ) + \frac{57}{40} \, \sqrt{5} \log \left (\frac{\sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac{5}{2 \,{\left | 2 \, x + 3 \right |}} - 2\right ) - \frac{1}{4} \, \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{13 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{4 \,{\left (2 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

43/24*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2) + 57/40*sqrt(5)*log
(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 1/4*sqrt(3
*x^2 + 5*x + 2) - 13/4*sqrt(3*x^2 + 5*x + 2)/(2*x + 3)

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Fricas [A]  time = 0.28356, size = 192, normalized size = 1.83 \[ -\frac{\sqrt{5} \sqrt{3}{\left (8 \, \sqrt{5} \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (x + 8\right )} - 43 \, \sqrt{5}{\left (2 \, x + 3\right )} \log \left (\sqrt{3}{\left (72 \, x^{2} + 120 \, x + 49\right )} + 12 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )}\right ) - 57 \, \sqrt{3}{\left (2 \, x + 3\right )} \log \left (\frac{\sqrt{5}{\left (124 \, x^{2} + 212 \, x + 89\right )} - 20 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{240 \,{\left (2 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/240*sqrt(5)*sqrt(3)*(8*sqrt(5)*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(x + 8) - 43*sqr
t(5)*(2*x + 3)*log(sqrt(3)*(72*x^2 + 120*x + 49) + 12*sqrt(3*x^2 + 5*x + 2)*(6*x
+ 5)) - 57*sqrt(3)*(2*x + 3)*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)))/(2*x + 3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{5 \sqrt{3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \frac{x \sqrt{3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5-x)*(3*x**2+5*x+2)**(1/2)/(3+2*x)**2,x)

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError