3.1411 \(\int \frac{(5-x) (3+2 x)}{\left (2+3 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=40 \[ -\frac{7 (2-7 x)}{6 \sqrt{3 x^2+2}}-\frac{2 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]

[Out]

(-7*(2 - 7*x))/(6*Sqrt[2 + 3*x^2]) - (2*ArcSinh[Sqrt[3/2]*x])/(3*Sqrt[3])

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Rubi [A]  time = 0.0430496, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{7 (2-7 x)}{6 \sqrt{3 x^2+2}}-\frac{2 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(-7*(2 - 7*x))/(6*Sqrt[2 + 3*x^2]) - (2*ArcSinh[Sqrt[3/2]*x])/(3*Sqrt[3])

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Rubi in Sympy [A]  time = 5.15276, size = 36, normalized size = 0.9 \[ - \frac{- 49 x + 14}{6 \sqrt{3 x^{2} + 2}} - \frac{2 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-49*x + 14)/(6*sqrt(3*x**2 + 2)) - 2*sqrt(3)*asinh(sqrt(6)*x/2)/9

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Mathematica [A]  time = 0.042607, size = 40, normalized size = 1. \[ \frac{7 (7 x-2)}{6 \sqrt{3 x^2+2}}-\frac{2 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(7*(-2 + 7*x))/(6*Sqrt[2 + 3*x^2]) - (2*ArcSinh[Sqrt[3/2]*x])/(3*Sqrt[3])

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Maple [A]  time = 0.008, size = 37, normalized size = 0.9 \[{\frac{49\,x}{6}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}-{\frac{7}{3}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}-{\frac{2\,\sqrt{3}}{9}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

49/6*x/(3*x^2+2)^(1/2)-7/3/(3*x^2+2)^(1/2)-2/9*arcsinh(1/2*x*6^(1/2))*3^(1/2)

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Maxima [A]  time = 0.754281, size = 49, normalized size = 1.22 \[ -\frac{2}{9} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) + \frac{49 \, x}{6 \, \sqrt{3 \, x^{2} + 2}} - \frac{7}{3 \, \sqrt{3 \, x^{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/9*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 49/6*x/sqrt(3*x^2 + 2) - 7/3/sqrt(3*x^2 +
2)

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Fricas [A]  time = 0.275498, size = 93, normalized size = 2.32 \[ \frac{\sqrt{3}{\left (7 \, \sqrt{3} \sqrt{3 \, x^{2} + 2}{\left (7 \, x - 2\right )} + 2 \,{\left (3 \, x^{2} + 2\right )} \log \left (-\sqrt{3}{\left (3 \, x^{2} + 1\right )} + 3 \, \sqrt{3 \, x^{2} + 2} x\right )\right )}}{18 \,{\left (3 \, x^{2} + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/18*sqrt(3)*(7*sqrt(3)*sqrt(3*x^2 + 2)*(7*x - 2) + 2*(3*x^2 + 2)*log(-sqrt(3)*(
3*x^2 + 1) + 3*sqrt(3*x^2 + 2)*x))/(3*x^2 + 2)

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Sympy [A]  time = 46.9244, size = 99, normalized size = 2.48 \[ - \frac{6 \sqrt{3} x^{2} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{27 x^{2} + 18} + \frac{6 x \sqrt{3 x^{2} + 2}}{27 x^{2} + 18} + \frac{15 x}{2 \sqrt{3 x^{2} + 2}} - \frac{4 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{27 x^{2} + 18} - \frac{7}{3 \sqrt{3 x^{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-6*sqrt(3)*x**2*asinh(sqrt(6)*x/2)/(27*x**2 + 18) + 6*x*sqrt(3*x**2 + 2)/(27*x**
2 + 18) + 15*x/(2*sqrt(3*x**2 + 2)) - 4*sqrt(3)*asinh(sqrt(6)*x/2)/(27*x**2 + 18
) - 7/(3*sqrt(3*x**2 + 2))

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GIAC/XCAS [A]  time = 0.286515, size = 53, normalized size = 1.32 \[ \frac{2}{9} \, \sqrt{3}{\rm ln}\left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) + \frac{7 \,{\left (7 \, x - 2\right )}}{6 \, \sqrt{3 \, x^{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9*sqrt(3)*ln(-sqrt(3)*x + sqrt(3*x^2 + 2)) + 7/6*(7*x - 2)/sqrt(3*x^2 + 2)