3.2412 \(\int \frac{(5-x) \sqrt{2+5 x+3 x^2}}{(3+2 x)^3} \, dx\)

Optimal. Leaf size=107 \[ \frac{\sqrt{3 x^2+5 x+2} (124 x+121)}{40 (2 x+3)^2}-\frac{1}{8} \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )+\frac{27 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{80 \sqrt{5}} \]

[Out]

((121 + 124*x)*Sqrt[2 + 5*x + 3*x^2])/(40*(3 + 2*x)^2) - (Sqrt[3]*ArcTanh[(5 + 6
*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/8 + (27*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqr
t[2 + 5*x + 3*x^2])])/(80*Sqrt[5])

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Rubi [A]  time = 0.172303, antiderivative size = 107, 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 \[ \frac{\sqrt{3 x^2+5 x+2} (124 x+121)}{40 (2 x+3)^2}-\frac{1}{8} \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )+\frac{27 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{80 \sqrt{5}} \]

Antiderivative was successfully verified.

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

[Out]

((121 + 124*x)*Sqrt[2 + 5*x + 3*x^2])/(40*(3 + 2*x)^2) - (Sqrt[3]*ArcTanh[(5 + 6
*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/8 + (27*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqr
t[2 + 5*x + 3*x^2])])/(80*Sqrt[5])

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Rubi in Sympy [A]  time = 25.1327, size = 97, normalized size = 0.91 \[ - \frac{\sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{8} - \frac{27 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \left (- 8 x - 7\right )}{10 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{400} + \frac{\left (124 x + 121\right ) \sqrt{3 x^{2} + 5 x + 2}}{40 \left (2 x + 3\right )^{2}} \]

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)**3,x)

[Out]

-sqrt(3)*atanh(sqrt(3)*(6*x + 5)/(6*sqrt(3*x**2 + 5*x + 2)))/8 - 27*sqrt(5)*atan
h(sqrt(5)*(-8*x - 7)/(10*sqrt(3*x**2 + 5*x + 2)))/400 + (124*x + 121)*sqrt(3*x**
2 + 5*x + 2)/(40*(2*x + 3)**2)

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Mathematica [A]  time = 0.164839, size = 109, normalized size = 1.02 \[ \frac{1}{400} \left (\frac{10 \sqrt{3 x^2+5 x+2} (124 x+121)}{(2 x+3)^2}-27 \sqrt{5} \log \left (2 \sqrt{5} \sqrt{3 x^2+5 x+2}-8 x-7\right )-50 \sqrt{3} \log \left (-2 \sqrt{9 x^2+15 x+6}-6 x-5\right )+27 \sqrt{5} \log (2 x+3)\right ) \]

Antiderivative was successfully verified.

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

[Out]

((10*(121 + 124*x)*Sqrt[2 + 5*x + 3*x^2])/(3 + 2*x)^2 + 27*Sqrt[5]*Log[3 + 2*x]
- 27*Sqrt[5]*Log[-7 - 8*x + 2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2]] - 50*Sqrt[3]*Log[-5
- 6*x - 2*Sqrt[6 + 15*x + 9*x^2]])/400

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Maple [A]  time = 0.016, size = 142, normalized size = 1.3 \[ -{\frac{13}{40} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{21}{50} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}+{\frac{27}{400}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}-{\frac{27\,\sqrt{5}}{400}{\it Artanh} \left ({\frac{2\,\sqrt{5}}{5} \left ( -{\frac{7}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}}} \right ) }+{\frac{105+126\,x}{100}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}-{\frac{\sqrt{3}}{8}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \right ) } \]

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)^3,x)

[Out]

-13/40/(x+3/2)^2*(3*(x+3/2)^2-4*x-19/4)^(3/2)-21/50/(x+3/2)*(3*(x+3/2)^2-4*x-19/
4)^(3/2)+27/400*(12*(x+3/2)^2-16*x-19)^(1/2)-27/400*5^(1/2)*arctanh(2/5*(-7/2-4*
x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))+21/100*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(
1/2)-1/8*ln(1/3*(5/2+3*x)*3^(1/2)+(3*(x+3/2)^2-4*x-19/4)^(1/2))*3^(1/2)

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Maxima [A]  time = 0.775225, size = 177, normalized size = 1.65 \[ -\frac{1}{8} \, \sqrt{3} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac{5}{2}\right ) - \frac{27}{400} \, \sqrt{5} \log \left (\frac{\sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac{5}{2 \,{\left | 2 \, x + 3 \right |}} - 2\right ) + \frac{39}{40} \, \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{13 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}}{10 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{21 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{20 \,{\left (2 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2) - 27/400*sqrt(5)*log
(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) + 39/40*sqrt
(3*x^2 + 5*x + 2) - 13/10*(3*x^2 + 5*x + 2)^(3/2)/(4*x^2 + 12*x + 9) - 21/20*sqr
t(3*x^2 + 5*x + 2)/(2*x + 3)

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Fricas [A]  time = 0.286288, size = 204, normalized size = 1.91 \[ \frac{\sqrt{5}{\left (10 \, \sqrt{5} \sqrt{3}{\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (-4 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + 4 \, \sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (124 \, x + 121\right )} + 27 \,{\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (\frac{\sqrt{5}{\left (124 \, x^{2} + 212 \, x + 89\right )} + 20 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{800 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/800*sqrt(5)*(10*sqrt(5)*sqrt(3)*(4*x^2 + 12*x + 9)*log(-4*sqrt(3)*sqrt(3*x^2 +
5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49) + 4*sqrt(5)*sqrt(3*x^2 + 5*x + 2)*(12
4*x + 121) + 27*(4*x^2 + 12*x + 9)*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)))/(4*x^2 + 12*x + 9)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{5 \sqrt{3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \frac{x \sqrt{3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\, dx \]

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)**3,x)

[Out]

-Integral(-5*sqrt(3*x**2 + 5*x + 2)/(8*x**3 + 36*x**2 + 54*x + 27), x) - Integra
l(x*sqrt(3*x**2 + 5*x + 2)/(8*x**3 + 36*x**2 + 54*x + 27), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \mathit{undef} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

undef