3.1359 \(\int (5-x) (3+2 x) \sqrt{2+3 x^2} \, dx\)

Optimal. Leaf size=56 \[ \frac{1}{18} (14-3 x) \left (3 x^2+2\right )^{3/2}+\frac{23}{3} x \sqrt{3 x^2+2}+\frac{46 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]

[Out]

(23*x*Sqrt[2 + 3*x^2])/3 + ((14 - 3*x)*(2 + 3*x^2)^(3/2))/18 + (46*ArcSinh[Sqrt[
3/2]*x])/(3*Sqrt[3])

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Rubi [A]  time = 0.0486342, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{1}{18} (14-3 x) \left (3 x^2+2\right )^{3/2}+\frac{23}{3} x \sqrt{3 x^2+2}+\frac{46 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(23*x*Sqrt[2 + 3*x^2])/3 + ((14 - 3*x)*(2 + 3*x^2)^(3/2))/18 + (46*ArcSinh[Sqrt[
3/2]*x])/(3*Sqrt[3])

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Rubi in Sympy [A]  time = 5.38462, size = 49, normalized size = 0.88 \[ \frac{23 x \sqrt{3 x^{2} + 2}}{3} + \frac{\left (- 6 x + 28\right ) \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{36} + \frac{46 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{9} \]

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]

23*x*sqrt(3*x**2 + 2)/3 + (-6*x + 28)*(3*x**2 + 2)**(3/2)/36 + 46*sqrt(3)*asinh(
sqrt(6)*x/2)/9

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Mathematica [A]  time = 0.0407873, size = 50, normalized size = 0.89 \[ \frac{1}{18} \left (92 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-\sqrt{3 x^2+2} \left (9 x^3-42 x^2-132 x-28\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-(Sqrt[2 + 3*x^2]*(-28 - 132*x - 42*x^2 + 9*x^3)) + 92*Sqrt[3]*ArcSinh[Sqrt[3/2
]*x])/18

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Maple [A]  time = 0.008, size = 49, normalized size = 0.9 \[{\frac{23\,x}{3}\sqrt{3\,{x}^{2}+2}}+{\frac{46\,\sqrt{3}}{9}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{7}{9} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}-{\frac{x}{6} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}} \]

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]

23/3*x*(3*x^2+2)^(1/2)+46/9*arcsinh(1/2*x*6^(1/2))*3^(1/2)+7/9*(3*x^2+2)^(3/2)-1
/6*x*(3*x^2+2)^(3/2)

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Maxima [A]  time = 0.776287, size = 65, normalized size = 1.16 \[ -\frac{1}{6} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + \frac{7}{9} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} + \frac{23}{3} \, \sqrt{3 \, x^{2} + 2} x + \frac{46}{9} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(3*x^2 + 2)^(3/2)*x + 7/9*(3*x^2 + 2)^(3/2) + 23/3*sqrt(3*x^2 + 2)*x + 46/9
*sqrt(3)*arcsinh(1/2*sqrt(6)*x)

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Fricas [A]  time = 0.290133, size = 84, normalized size = 1.5 \[ -\frac{1}{54} \, \sqrt{3}{\left (\sqrt{3}{\left (9 \, x^{3} - 42 \, x^{2} - 132 \, x - 28\right )} \sqrt{3 \, x^{2} + 2} - 138 \, \log \left (-\sqrt{3}{\left (3 \, x^{2} + 1\right )} - 3 \, \sqrt{3 \, x^{2} + 2} x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/54*sqrt(3)*(sqrt(3)*(9*x^3 - 42*x^2 - 132*x - 28)*sqrt(3*x^2 + 2) - 138*log(-
sqrt(3)*(3*x^2 + 1) - 3*sqrt(3*x^2 + 2)*x))

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Sympy [A]  time = 18.253, size = 94, normalized size = 1.68 \[ - \frac{3 x^{5}}{2 \sqrt{3 x^{2} + 2}} - \frac{3 x^{3}}{2 \sqrt{3 x^{2} + 2}} + \frac{15 x \sqrt{3 x^{2} + 2}}{2} - \frac{x}{3 \sqrt{3 x^{2} + 2}} + \frac{7 \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{9} + \frac{46 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{9} \]

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]

-3*x**5/(2*sqrt(3*x**2 + 2)) - 3*x**3/(2*sqrt(3*x**2 + 2)) + 15*x*sqrt(3*x**2 +
2)/2 - x/(3*sqrt(3*x**2 + 2)) + 7*(3*x**2 + 2)**(3/2)/9 + 46*sqrt(3)*asinh(sqrt(
6)*x/2)/9

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GIAC/XCAS [A]  time = 0.286304, size = 65, normalized size = 1.16 \[ -\frac{1}{18} \,{\left (3 \,{\left ({\left (3 \, x - 14\right )} x - 44\right )} x - 28\right )} \sqrt{3 \, x^{2} + 2} - \frac{46}{9} \, \sqrt{3}{\rm ln}\left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/18*(3*((3*x - 14)*x - 44)*x - 28)*sqrt(3*x^2 + 2) - 46/9*sqrt(3)*ln(-sqrt(3)*
x + sqrt(3*x^2 + 2))