3.1410 \(\int \frac{(5-x) (3+2 x)^2}{\left (2+3 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=60 \[ -\frac{7 (2-7 x) (2 x+3)}{6 \sqrt{3 x^2+2}}-\frac{53}{9} \sqrt{3 x^2+2}+\frac{8 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]

[Out]

(-7*(2 - 7*x)*(3 + 2*x))/(6*Sqrt[2 + 3*x^2]) - (53*Sqrt[2 + 3*x^2])/9 + (8*ArcSi
nh[Sqrt[3/2]*x])/(3*Sqrt[3])

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Rubi [A]  time = 0.0795225, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{7 (2-7 x) (2 x+3)}{6 \sqrt{3 x^2+2}}-\frac{53}{9} \sqrt{3 x^2+2}+\frac{8 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(-7*(2 - 7*x)*(3 + 2*x))/(6*Sqrt[2 + 3*x^2]) - (53*Sqrt[2 + 3*x^2])/9 + (8*ArcSi
nh[Sqrt[3/2]*x])/(3*Sqrt[3])

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Rubi in Sympy [A]  time = 8.2769, size = 53, normalized size = 0.88 \[ - \frac{\left (- 98 x + 28\right ) \left (2 x + 3\right )}{12 \sqrt{3 x^{2} + 2}} - \frac{53 \sqrt{3 x^{2} + 2}}{9} + \frac{8 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-98*x + 28)*(2*x + 3)/(12*sqrt(3*x**2 + 2)) - 53*sqrt(3*x**2 + 2)/9 + 8*sqrt(3
)*asinh(sqrt(6)*x/2)/9

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Mathematica [A]  time = 0.059737, size = 58, normalized size = 0.97 \[ -\frac{\sqrt{3 x^2+2} \left (24 x^2-357 x+338\right )-16 \sqrt{3} \left (3 x^2+2\right ) \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{54 x^2+36} \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[2 + 3*x^2]*(338 - 357*x + 24*x^2) - 16*Sqrt[3]*(2 + 3*x^2)*ArcSinh[Sqrt[
3/2]*x])/(36 + 54*x^2))

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Maple [A]  time = 0.01, size = 51, normalized size = 0.9 \[{\frac{119\,x}{6}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}-{\frac{169}{9}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}+{\frac{8\,\sqrt{3}}{9}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }-{\frac{4\,{x}^{2}}{3}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

119/6*x/(3*x^2+2)^(1/2)-169/9/(3*x^2+2)^(1/2)+8/9*arcsinh(1/2*x*6^(1/2))*3^(1/2)
-4/3*x^2/(3*x^2+2)^(1/2)

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Maxima [A]  time = 0.756258, size = 68, normalized size = 1.13 \[ -\frac{4 \, x^{2}}{3 \, \sqrt{3 \, x^{2} + 2}} + \frac{8}{9} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) + \frac{119 \, x}{6 \, \sqrt{3 \, x^{2} + 2}} - \frac{169}{9 \, \sqrt{3 \, x^{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/3*x^2/sqrt(3*x^2 + 2) + 8/9*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 119/6*x/sqrt(3*x
^2 + 2) - 169/9/sqrt(3*x^2 + 2)

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Fricas [A]  time = 0.275259, size = 99, normalized size = 1.65 \[ -\frac{\sqrt{3}{\left (\sqrt{3}{\left (24 \, x^{2} - 357 \, x + 338\right )} \sqrt{3 \, x^{2} + 2} - 24 \,{\left (3 \, x^{2} + 2\right )} \log \left (-\sqrt{3}{\left (3 \, x^{2} + 1\right )} - 3 \, \sqrt{3 \, x^{2} + 2} x\right )\right )}}{54 \,{\left (3 \, x^{2} + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/54*sqrt(3)*(sqrt(3)*(24*x^2 - 357*x + 338)*sqrt(3*x^2 + 2) - 24*(3*x^2 + 2)*l
og(-sqrt(3)*(3*x^2 + 1) - 3*sqrt(3*x^2 + 2)*x))/(3*x^2 + 2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{51 x}{3 x^{2} \sqrt{3 x^{2} + 2} + 2 \sqrt{3 x^{2} + 2}}\right )\, dx - \int \left (- \frac{8 x^{2}}{3 x^{2} \sqrt{3 x^{2} + 2} + 2 \sqrt{3 x^{2} + 2}}\right )\, dx - \int \frac{4 x^{3}}{3 x^{2} \sqrt{3 x^{2} + 2} + 2 \sqrt{3 x^{2} + 2}}\, dx - \int \left (- \frac{45}{3 x^{2} \sqrt{3 x^{2} + 2} + 2 \sqrt{3 x^{2} + 2}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(-51*x/(3*x**2*sqrt(3*x**2 + 2) + 2*sqrt(3*x**2 + 2)), x) - Integral(-8
*x**2/(3*x**2*sqrt(3*x**2 + 2) + 2*sqrt(3*x**2 + 2)), x) - Integral(4*x**3/(3*x*
*2*sqrt(3*x**2 + 2) + 2*sqrt(3*x**2 + 2)), x) - Integral(-45/(3*x**2*sqrt(3*x**2
+ 2) + 2*sqrt(3*x**2 + 2)), x)

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GIAC/XCAS [A]  time = 0.303687, size = 59, normalized size = 0.98 \[ -\frac{8}{9} \, \sqrt{3}{\rm ln}\left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) - \frac{3 \,{\left (8 \, x - 119\right )} x + 338}{18 \, \sqrt{3 \, x^{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-8/9*sqrt(3)*ln(-sqrt(3)*x + sqrt(3*x^2 + 2)) - 1/18*(3*(8*x - 119)*x + 338)/sqr
t(3*x^2 + 2)