3.1360 \(\int (5-x) \sqrt{2+3 x^2} \, dx\)

Optimal. Leaf size=49 \[ -\frac{1}{9} \left (3 x^2+2\right )^{3/2}+\frac{5}{2} x \sqrt{3 x^2+2}+\frac{5 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{\sqrt{3}} \]

[Out]

(5*x*Sqrt[2 + 3*x^2])/2 - (2 + 3*x^2)^(3/2)/9 + (5*ArcSinh[Sqrt[3/2]*x])/Sqrt[3]

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Rubi [A]  time = 0.028841, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\frac{1}{9} \left (3 x^2+2\right )^{3/2}+\frac{5}{2} x \sqrt{3 x^2+2}+\frac{5 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(5 - x)*Sqrt[2 + 3*x^2],x]

[Out]

(5*x*Sqrt[2 + 3*x^2])/2 - (2 + 3*x^2)^(3/2)/9 + (5*ArcSinh[Sqrt[3/2]*x])/Sqrt[3]

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Rubi in Sympy [A]  time = 3.50409, size = 44, normalized size = 0.9 \[ \frac{5 x \sqrt{3 x^{2} + 2}}{2} - \frac{\left (3 x^{2} + 2\right )^{\frac{3}{2}}}{9} + \frac{5 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*x*sqrt(3*x**2 + 2)/2 - (3*x**2 + 2)**(3/2)/9 + 5*sqrt(3)*asinh(sqrt(6)*x/2)/3

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Mathematica [A]  time = 0.0264658, size = 43, normalized size = 0.88 \[ \frac{5 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{\sqrt{3}}-\frac{1}{18} \sqrt{3 x^2+2} \left (6 x^2-45 x+4\right ) \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[2 + 3*x^2]*(4 - 45*x + 6*x^2))/18 + (5*ArcSinh[Sqrt[3/2]*x])/Sqrt[3]

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Maple [A]  time = 0.006, size = 37, normalized size = 0.8 \[ -{\frac{1}{9} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}+{\frac{5\,\sqrt{3}}{3}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{5\,x}{2}\sqrt{3\,{x}^{2}+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)*(3*x^2+2)^(1/2),x)

[Out]

-1/9*(3*x^2+2)^(3/2)+5/3*arcsinh(1/2*x*6^(1/2))*3^(1/2)+5/2*x*(3*x^2+2)^(1/2)

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Maxima [A]  time = 0.769663, size = 49, normalized size = 1. \[ -\frac{1}{9} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} + \frac{5}{2} \, \sqrt{3 \, x^{2} + 2} x + \frac{5}{3} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9*(3*x^2 + 2)^(3/2) + 5/2*sqrt(3*x^2 + 2)*x + 5/3*sqrt(3)*arcsinh(1/2*sqrt(6)
*x)

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Fricas [A]  time = 0.289113, size = 77, normalized size = 1.57 \[ -\frac{1}{54} \, \sqrt{3}{\left (\sqrt{3}{\left (6 \, x^{2} - 45 \, x + 4\right )} \sqrt{3 \, x^{2} + 2} - 45 \, \log \left (-\sqrt{3}{\left (3 \, x^{2} + 1\right )} - 3 \, \sqrt{3 \, x^{2} + 2} x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/54*sqrt(3)*(sqrt(3)*(6*x^2 - 45*x + 4)*sqrt(3*x^2 + 2) - 45*log(-sqrt(3)*(3*x
^2 + 1) - 3*sqrt(3*x^2 + 2)*x))

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Sympy [A]  time = 0.695119, size = 61, normalized size = 1.24 \[ - \frac{x^{2} \sqrt{3 x^{2} + 2}}{3} + \frac{5 x \sqrt{3 x^{2} + 2}}{2} - \frac{2 \sqrt{3 x^{2} + 2}}{9} + \frac{5 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**2*sqrt(3*x**2 + 2)/3 + 5*x*sqrt(3*x**2 + 2)/2 - 2*sqrt(3*x**2 + 2)/9 + 5*sqr
t(3)*asinh(sqrt(6)*x/2)/3

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GIAC/XCAS [A]  time = 0.285582, size = 59, normalized size = 1.2 \[ -\frac{1}{18} \,{\left (3 \,{\left (2 \, x - 15\right )} x + 4\right )} \sqrt{3 \, x^{2} + 2} - \frac{5}{3} \, \sqrt{3}{\rm ln}\left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/18*(3*(2*x - 15)*x + 4)*sqrt(3*x^2 + 2) - 5/3*sqrt(3)*ln(-sqrt(3)*x + sqrt(3*
x^2 + 2))