3.2516 \(\int \frac{(5-x) (3+2 x)^3}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=92 \[ -\frac{2 (139 x+121) (2 x+3)^2}{9 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{4 (2834 x+2481)}{9 \sqrt{3 x^2+5 x+2}}-\frac{8 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{9 \sqrt{3}} \]

[Out]

(-2*(3 + 2*x)^2*(121 + 139*x))/(9*(2 + 5*x + 3*x^2)^(3/2)) + (4*(2481 + 2834*x))
/(9*Sqrt[2 + 5*x + 3*x^2]) - (8*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^
2])])/(9*Sqrt[3])

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Rubi [A]  time = 0.135184, antiderivative size = 92, 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 \[ -\frac{2 (139 x+121) (2 x+3)^2}{9 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{4 (2834 x+2481)}{9 \sqrt{3 x^2+5 x+2}}-\frac{8 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{9 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(3 + 2*x)^2*(121 + 139*x))/(9*(2 + 5*x + 3*x^2)^(3/2)) + (4*(2481 + 2834*x))
/(9*Sqrt[2 + 5*x + 3*x^2]) - (8*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^
2])])/(9*Sqrt[3])

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Rubi in Sympy [A]  time = 17.1685, size = 83, normalized size = 0.9 \[ - \frac{2 \left (2 x + 3\right )^{2} \left (139 x + 121\right )}{9 \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}} + \frac{4 \left (8502 x + 7443\right )}{27 \sqrt{3 x^{2} + 5 x + 2}} - \frac{8 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{27} \]

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]

-2*(2*x + 3)**2*(139*x + 121)/(9*(3*x**2 + 5*x + 2)**(3/2)) + 4*(8502*x + 7443)/
(27*sqrt(3*x**2 + 5*x + 2)) - 8*sqrt(3)*atanh(sqrt(3)*(6*x + 5)/(6*sqrt(3*x**2 +
5*x + 2)))/27

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Mathematica [A]  time = 0.0905008, size = 65, normalized size = 0.71 \[ \frac{2 \left (16448 x^3+41074 x^2+33443 x+8835\right )}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac{8 \log \left (-2 \sqrt{9 x^2+15 x+6}-6 x-5\right )}{9 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(8835 + 33443*x + 41074*x^2 + 16448*x^3))/(9*(2 + 5*x + 3*x^2)^(3/2)) - (8*Lo
g[-5 - 6*x - 2*Sqrt[6 + 15*x + 9*x^2]])/(9*Sqrt[3])

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Maple [B]  time = 0.008, size = 161, normalized size = 1.8 \[ -{\frac{20165+24198\,x}{486} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{-{\frac{3}{2}}}}+{\frac{82160+98592\,x}{81}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}}-{\frac{10855}{486} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{607\,x}{27} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{32\,{x}^{2}}{9} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{-{\frac{3}{2}}}}+{\frac{8\,{x}^{3}}{9} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{-{\frac{3}{2}}}}+{\frac{8\,x}{9}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}}-{\frac{20}{27}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}}-{\frac{8\,\sqrt{3}}{27}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) } \]

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]

-4033/486*(5+6*x)/(3*x^2+5*x+2)^(3/2)+16432/81*(5+6*x)/(3*x^2+5*x+2)^(1/2)-10855
/486/(3*x^2+5*x+2)^(3/2)-607/27*x/(3*x^2+5*x+2)^(3/2)-32/9*x^2/(3*x^2+5*x+2)^(3/
2)+8/9*x^3/(3*x^2+5*x+2)^(3/2)+8/9*x/(3*x^2+5*x+2)^(1/2)-20/27/(3*x^2+5*x+2)^(1/
2)-8/27*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)

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Maxima [A]  time = 0.795694, size = 266, normalized size = 2.89 \[ \frac{8}{27} \, x{\left (\frac{1410 \, x}{\sqrt{3 \, x^{2} + 5 \, x + 2}} + \frac{9 \, x^{2}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} + \frac{1175}{\sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{55 \, x}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} - \frac{46}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}}\right )} - \frac{8}{27} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac{3760}{27} \, \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{42272 \, x}{27 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{4 \, x^{2}}{3 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} + \frac{11680}{9 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{2318 \, x}{27 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} - \frac{2030}{27 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8/27*x*(1410*x/sqrt(3*x^2 + 5*x + 2) + 9*x^2/(3*x^2 + 5*x + 2)^(3/2) + 1175/sqrt
(3*x^2 + 5*x + 2) - 55*x/(3*x^2 + 5*x + 2)^(3/2) - 46/(3*x^2 + 5*x + 2)^(3/2)) -
8/27*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) - 3760/27*sqrt(3*x^
2 + 5*x + 2) + 42272/27*x/sqrt(3*x^2 + 5*x + 2) - 4/3*x^2/(3*x^2 + 5*x + 2)^(3/2
) + 11680/9/sqrt(3*x^2 + 5*x + 2) - 2318/27*x/(3*x^2 + 5*x + 2)^(3/2) - 2030/27/
(3*x^2 + 5*x + 2)^(3/2)

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Fricas [A]  time = 0.283183, size = 157, normalized size = 1.71 \[ \frac{2 \, \sqrt{3}{\left (\sqrt{3}{\left (16448 \, x^{3} + 41074 \, x^{2} + 33443 \, x + 8835\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + 2 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (\sqrt{3}{\left (72 \, x^{2} + 120 \, x + 49\right )} - 12 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )}\right )\right )}}{27 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/27*sqrt(3)*(sqrt(3)*(16448*x^3 + 41074*x^2 + 33443*x + 8835)*sqrt(3*x^2 + 5*x
+ 2) + 2*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*log(sqrt(3)*(72*x^2 + 120*x + 49)
- 12*sqrt(3*x^2 + 5*x + 2)*(6*x + 5)))/(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{243 x}{9 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 20 x \sqrt{3 x^{2} + 5 x + 2} + 4 \sqrt{3 x^{2} + 5 x + 2}}\right )\, dx - \int \left (- \frac{126 x^{2}}{9 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 20 x \sqrt{3 x^{2} + 5 x + 2} + 4 \sqrt{3 x^{2} + 5 x + 2}}\right )\, dx - \int \left (- \frac{4 x^{3}}{9 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 20 x \sqrt{3 x^{2} + 5 x + 2} + 4 \sqrt{3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac{8 x^{4}}{9 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 20 x \sqrt{3 x^{2} + 5 x + 2} + 4 \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac{135}{9 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 20 x \sqrt{3 x^{2} + 5 x + 2} + 4 \sqrt{3 x^{2} + 5 x + 2}}\right )\, dx \]

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(-243*x/(9*x**4*sqrt(3*x**2 + 5*x + 2) + 30*x**3*sqrt(3*x**2 + 5*x + 2)
+ 37*x**2*sqrt(3*x**2 + 5*x + 2) + 20*x*sqrt(3*x**2 + 5*x + 2) + 4*sqrt(3*x**2
+ 5*x + 2)), x) - Integral(-126*x**2/(9*x**4*sqrt(3*x**2 + 5*x + 2) + 30*x**3*sq
rt(3*x**2 + 5*x + 2) + 37*x**2*sqrt(3*x**2 + 5*x + 2) + 20*x*sqrt(3*x**2 + 5*x +
2) + 4*sqrt(3*x**2 + 5*x + 2)), x) - Integral(-4*x**3/(9*x**4*sqrt(3*x**2 + 5*x
+ 2) + 30*x**3*sqrt(3*x**2 + 5*x + 2) + 37*x**2*sqrt(3*x**2 + 5*x + 2) + 20*x*s
qrt(3*x**2 + 5*x + 2) + 4*sqrt(3*x**2 + 5*x + 2)), x) - Integral(8*x**4/(9*x**4*
sqrt(3*x**2 + 5*x + 2) + 30*x**3*sqrt(3*x**2 + 5*x + 2) + 37*x**2*sqrt(3*x**2 +
5*x + 2) + 20*x*sqrt(3*x**2 + 5*x + 2) + 4*sqrt(3*x**2 + 5*x + 2)), x) - Integra
l(-135/(9*x**4*sqrt(3*x**2 + 5*x + 2) + 30*x**3*sqrt(3*x**2 + 5*x + 2) + 37*x**2
*sqrt(3*x**2 + 5*x + 2) + 20*x*sqrt(3*x**2 + 5*x + 2) + 4*sqrt(3*x**2 + 5*x + 2)
), x)

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GIAC/XCAS [A]  time = 0.290244, size = 85, normalized size = 0.92 \[ \frac{8}{27} \, \sqrt{3}{\rm ln}\left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) + \frac{2 \,{\left ({\left (2 \,{\left (8224 \, x + 20537\right )} x + 33443\right )} x + 8835\right )}}{9 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8/27*sqrt(3)*ln(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5)) + 2/9*(
(2*(8224*x + 20537)*x + 33443)*x + 8835)/(3*x^2 + 5*x + 2)^(3/2)