3.1399 \(\int \frac{(5-x) (3+2 x)^2}{\sqrt{2+3 x^2}} \, dx\)

Optimal. Leaf size=62 \[ -\frac{1}{9} \sqrt{3 x^2+2} (2 x+3)^2+\frac{2}{27} (36 x+251) \sqrt{3 x^2+2}+\frac{127 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]

[Out]

-((3 + 2*x)^2*Sqrt[2 + 3*x^2])/9 + (2*(251 + 36*x)*Sqrt[2 + 3*x^2])/27 + (127*Ar
cSinh[Sqrt[3/2]*x])/(3*Sqrt[3])

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Rubi [A]  time = 0.0994978, antiderivative size = 62, 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 \[ -\frac{1}{9} \sqrt{3 x^2+2} (2 x+3)^2+\frac{2}{27} (36 x+251) \sqrt{3 x^2+2}+\frac{127 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

-((3 + 2*x)^2*Sqrt[2 + 3*x^2])/9 + (2*(251 + 36*x)*Sqrt[2 + 3*x^2])/27 + (127*Ar
cSinh[Sqrt[3/2]*x])/(3*Sqrt[3])

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Rubi in Sympy [A]  time = 9.59991, size = 53, normalized size = 0.85 \[ - \frac{\left (2 x + 3\right )^{2} \sqrt{3 x^{2} + 2}}{9} + \frac{\left (144 x + 1004\right ) \sqrt{3 x^{2} + 2}}{54} + \frac{127 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*x + 3)**2*sqrt(3*x**2 + 2)/9 + (144*x + 1004)*sqrt(3*x**2 + 2)/54 + 127*sqrt
(3)*asinh(sqrt(6)*x/2)/9

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Mathematica [A]  time = 0.0500697, size = 45, normalized size = 0.73 \[ \frac{1}{27} \left (381 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-\sqrt{3 x^2+2} \left (12 x^2-36 x-475\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-(Sqrt[2 + 3*x^2]*(-475 - 36*x + 12*x^2)) + 381*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/2
7

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Maple [A]  time = 0.009, size = 51, normalized size = 0.8 \[{\frac{127\,\sqrt{3}}{9}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{475}{27}\sqrt{3\,{x}^{2}+2}}+{\frac{4\,x}{3}\sqrt{3\,{x}^{2}+2}}-{\frac{4\,{x}^{2}}{9}\sqrt{3\,{x}^{2}+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

127/9*arcsinh(1/2*x*6^(1/2))*3^(1/2)+475/27*(3*x^2+2)^(1/2)+4/3*x*(3*x^2+2)^(1/2
)-4/9*x^2*(3*x^2+2)^(1/2)

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Maxima [A]  time = 0.756347, size = 68, normalized size = 1.1 \[ -\frac{4}{9} \, \sqrt{3 \, x^{2} + 2} x^{2} + \frac{4}{3} \, \sqrt{3 \, x^{2} + 2} x + \frac{127}{9} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) + \frac{475}{27} \, \sqrt{3 \, x^{2} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/9*sqrt(3*x^2 + 2)*x^2 + 4/3*sqrt(3*x^2 + 2)*x + 127/9*sqrt(3)*arcsinh(1/2*sqr
t(6)*x) + 475/27*sqrt(3*x^2 + 2)

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Fricas [A]  time = 0.274766, size = 78, normalized size = 1.26 \[ -\frac{1}{162} \, \sqrt{3}{\left (2 \, \sqrt{3}{\left (12 \, x^{2} - 36 \, x - 475\right )} \sqrt{3 \, x^{2} + 2} - 1143 \, \log \left (-\sqrt{3}{\left (3 \, x^{2} + 1\right )} - 3 \, \sqrt{3 \, x^{2} + 2} x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/162*sqrt(3)*(2*sqrt(3)*(12*x^2 - 36*x - 475)*sqrt(3*x^2 + 2) - 1143*log(-sqrt
(3)*(3*x^2 + 1) - 3*sqrt(3*x^2 + 2)*x))

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Sympy [A]  time = 1.59052, size = 63, normalized size = 1.02 \[ - \frac{4 x^{2} \sqrt{3 x^{2} + 2}}{9} + \frac{4 x \sqrt{3 x^{2} + 2}}{3} + \frac{475 \sqrt{3 x^{2} + 2}}{27} + \frac{127 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*x**2*sqrt(3*x**2 + 2)/9 + 4*x*sqrt(3*x**2 + 2)/3 + 475*sqrt(3*x**2 + 2)/27 +
127*sqrt(3)*asinh(sqrt(6)*x/2)/9

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GIAC/XCAS [A]  time = 0.293023, size = 57, normalized size = 0.92 \[ -\frac{1}{27} \,{\left (12 \,{\left (x - 3\right )} x - 475\right )} \sqrt{3 \, x^{2} + 2} - \frac{127}{9} \, \sqrt{3}{\rm ln}\left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/27*(12*(x - 3)*x - 475)*sqrt(3*x^2 + 2) - 127/9*sqrt(3)*ln(-sqrt(3)*x + sqrt(
3*x^2 + 2))