3.1400 \(\int \frac{(5-x) (3+2 x)}{\sqrt{2+3 x^2}} \, dx\)

Optimal. Leaf size=40 \[ \frac{1}{3} \sqrt{3 x^2+2} (7-x)+\frac{47 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]

[Out]

((7 - x)*Sqrt[2 + 3*x^2])/3 + (47*ArcSinh[Sqrt[3/2]*x])/(3*Sqrt[3])

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Rubi [A]  time = 0.0459979, 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 \[ \frac{1}{3} \sqrt{3 x^2+2} (7-x)+\frac{47 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

((7 - x)*Sqrt[2 + 3*x^2])/3 + (47*ArcSinh[Sqrt[3/2]*x])/(3*Sqrt[3])

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Rubi in Sympy [A]  time = 5.2838, size = 34, normalized size = 0.85 \[ \frac{\left (- 2 x + 14\right ) \sqrt{3 x^{2} + 2}}{6} + \frac{47 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)*(3+2*x)/(3*x**2+2)**(1/2),x)

[Out]

(-2*x + 14)*sqrt(3*x**2 + 2)/6 + 47*sqrt(3)*asinh(sqrt(6)*x/2)/9

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Mathematica [A]  time = 0.0357418, size = 38, normalized size = 0.95 \[ \frac{1}{9} \left (47 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-3 (x-7) \sqrt{3 x^2+2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-3*(-7 + x)*Sqrt[2 + 3*x^2] + 47*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/9

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Maple [A]  time = 0.008, size = 37, normalized size = 0.9 \[{\frac{47\,\sqrt{3}}{9}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{7}{3}\sqrt{3\,{x}^{2}+2}}-{\frac{x}{3}\sqrt{3\,{x}^{2}+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)*(2*x+3)/(3*x^2+2)^(1/2),x)

[Out]

47/9*arcsinh(1/2*x*6^(1/2))*3^(1/2)+7/3*(3*x^2+2)^(1/2)-1/3*x*(3*x^2+2)^(1/2)

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Maxima [A]  time = 0.754916, size = 49, normalized size = 1.22 \[ -\frac{1}{3} \, \sqrt{3 \, x^{2} + 2} x + \frac{47}{9} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) + \frac{7}{3} \, \sqrt{3 \, x^{2} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*sqrt(3*x^2 + 2)*x + 47/9*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 7/3*sqrt(3*x^2 +
2)

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Fricas [A]  time = 0.273574, size = 69, normalized size = 1.72 \[ -\frac{1}{18} \, \sqrt{3}{\left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 2}{\left (x - 7\right )} - 47 \, \log \left (-\sqrt{3}{\left (3 \, x^{2} + 1\right )} - 3 \, \sqrt{3 \, x^{2} + 2} x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/18*sqrt(3)*(2*sqrt(3)*sqrt(3*x^2 + 2)*(x - 7) - 47*log(-sqrt(3)*(3*x^2 + 1) -
3*sqrt(3*x^2 + 2)*x))

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Sympy [A]  time = 0.798894, size = 44, normalized size = 1.1 \[ - \frac{x \sqrt{3 x^{2} + 2}}{3} + \frac{7 \sqrt{3 x^{2} + 2}}{3} + \frac{47 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5-x)*(3+2*x)/(3*x**2+2)**(1/2),x)

[Out]

-x*sqrt(3*x**2 + 2)/3 + 7*sqrt(3*x**2 + 2)/3 + 47*sqrt(3)*asinh(sqrt(6)*x/2)/9

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GIAC/XCAS [A]  time = 0.297445, size = 50, normalized size = 1.25 \[ -\frac{1}{3} \, \sqrt{3 \, x^{2} + 2}{\left (x - 7\right )} - \frac{47}{9} \, \sqrt{3}{\rm ln}\left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*sqrt(3*x^2 + 2)*(x - 7) - 47/9*sqrt(3)*ln(-sqrt(3)*x + sqrt(3*x^2 + 2))