3.2508 \(\int \frac{(5-x) (3+2 x)}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=62 \[ -\frac{2 (139 x+121)}{3 \sqrt{3 x^2+5 x+2}}-\frac{2 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{3 \sqrt{3}} \]

[Out]

(-2*(121 + 139*x))/(3*Sqrt[2 + 5*x + 3*x^2]) - (2*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*S
qrt[2 + 5*x + 3*x^2])])/(3*Sqrt[3])

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Rubi [A]  time = 0.0700811, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ -\frac{2 (139 x+121)}{3 \sqrt{3 x^2+5 x+2}}-\frac{2 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(121 + 139*x))/(3*Sqrt[2 + 5*x + 3*x^2]) - (2*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*S
qrt[2 + 5*x + 3*x^2])])/(3*Sqrt[3])

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Rubi in Sympy [A]  time = 8.95299, size = 56, normalized size = 0.9 \[ - \frac{2 \left (139 x + 121\right )}{3 \sqrt{3 x^{2} + 5 x + 2}} - \frac{2 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(139*x + 121)/(3*sqrt(3*x**2 + 5*x + 2)) - 2*sqrt(3)*atanh(sqrt(3)*(6*x + 5)/
(6*sqrt(3*x**2 + 5*x + 2)))/9

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Mathematica [A]  time = 0.0928223, size = 53, normalized size = 0.85 \[ -\frac{2}{9} \left (\frac{417 x+363}{\sqrt{3 x^2+5 x+2}}+\sqrt{3} \log \left (2 \sqrt{9 x^2+15 x+6}+6 x+5\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-2*((363 + 417*x)/Sqrt[2 + 5*x + 3*x^2] + Sqrt[3]*Log[5 + 6*x + 2*Sqrt[6 + 15*x
+ 9*x^2]]))/9

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Maple [A]  time = 0.008, size = 79, normalized size = 1.3 \[ -{\frac{700+840\,x}{9}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}}-{\frac{26}{9}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}}+{\frac{2\,x}{3}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}}-{\frac{2\,\sqrt{3}}{9}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-140/9*(5+6*x)/(3*x^2+5*x+2)^(1/2)-26/9/(3*x^2+5*x+2)^(1/2)+2/3*x/(3*x^2+5*x+2)^
(1/2)-2/9*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 = 0.7974, size = 78, normalized size = 1.26 \[ -\frac{2}{9} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac{278 \, x}{3 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{242}{3 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/9*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) - 278/3*x/sqrt(3*x^2
+ 5*x + 2) - 242/3/sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 0.283292, size = 117, normalized size = 1.89 \[ -\frac{\sqrt{3}{\left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (139 \, x + 121\right )} -{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\sqrt{3}{\left (72 \, x^{2} + 120 \, x + 49\right )} - 12 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )}\right )\right )}}{9 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9*sqrt(3)*(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(139*x + 121) - (3*x^2 + 5*x + 2)*
log(sqrt(3)*(72*x^2 + 120*x + 49) - 12*sqrt(3*x^2 + 5*x + 2)*(6*x + 5)))/(3*x^2
+ 5*x + 2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{7 x}{3 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 5 x \sqrt{3 x^{2} + 5 x + 2} + 2 \sqrt{3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac{2 x^{2}}{3 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 5 x \sqrt{3 x^{2} + 5 x + 2} + 2 \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac{15}{3 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 5 x \sqrt{3 x^{2} + 5 x + 2} + 2 \sqrt{3 x^{2} + 5 x + 2}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(-7*x/(3*x**2*sqrt(3*x**2 + 5*x + 2) + 5*x*sqrt(3*x**2 + 5*x + 2) + 2*s
qrt(3*x**2 + 5*x + 2)), x) - Integral(2*x**2/(3*x**2*sqrt(3*x**2 + 5*x + 2) + 5*
x*sqrt(3*x**2 + 5*x + 2) + 2*sqrt(3*x**2 + 5*x + 2)), x) - Integral(-15/(3*x**2*
sqrt(3*x**2 + 5*x + 2) + 5*x*sqrt(3*x**2 + 5*x + 2) + 2*sqrt(3*x**2 + 5*x + 2)),
x)

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GIAC/XCAS [A]  time = 0.278178, size = 73, normalized size = 1.18 \[ \frac{2}{9} \, \sqrt{3}{\rm ln}\left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) - \frac{2 \,{\left (139 \, x + 121\right )}}{3 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9*sqrt(3)*ln(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5)) - 2/3*(1
39*x + 121)/sqrt(3*x^2 + 5*x + 2)