3.1398 \(\int \frac{(5-x) (3+2 x)^3}{\sqrt{2+3 x^2}} \, dx\)

Optimal. Leaf size=84 \[ -\frac{1}{12} \sqrt{3 x^2+2} (2 x+3)^3+\frac{31}{36} \sqrt{3 x^2+2} (2 x+3)^2+\frac{5}{54} (171 x+809) \sqrt{3 x^2+2}+\frac{275 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]

[Out]

(31*(3 + 2*x)^2*Sqrt[2 + 3*x^2])/36 - ((3 + 2*x)^3*Sqrt[2 + 3*x^2])/12 + (5*(809
+ 171*x)*Sqrt[2 + 3*x^2])/54 + (275*ArcSinh[Sqrt[3/2]*x])/(3*Sqrt[3])

_______________________________________________________________________________________

Rubi [A]  time = 0.156513, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{1}{12} \sqrt{3 x^2+2} (2 x+3)^3+\frac{31}{36} \sqrt{3 x^2+2} (2 x+3)^2+\frac{5}{54} (171 x+809) \sqrt{3 x^2+2}+\frac{275 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(31*(3 + 2*x)^2*Sqrt[2 + 3*x^2])/36 - ((3 + 2*x)^3*Sqrt[2 + 3*x^2])/12 + (5*(809
+ 171*x)*Sqrt[2 + 3*x^2])/54 + (275*ArcSinh[Sqrt[3/2]*x])/(3*Sqrt[3])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 14.2694, size = 73, normalized size = 0.87 \[ - \frac{\left (2 x + 3\right )^{3} \sqrt{3 x^{2} + 2}}{12} + \frac{31 \left (2 x + 3\right )^{2} \sqrt{3 x^{2} + 2}}{36} + \frac{\left (10260 x + 48540\right ) \sqrt{3 x^{2} + 2}}{648} + \frac{275 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*x + 3)**3*sqrt(3*x**2 + 2)/12 + 31*(2*x + 3)**2*sqrt(3*x**2 + 2)/36 + (10260
*x + 48540)*sqrt(3*x**2 + 2)/648 + 275*sqrt(3)*asinh(sqrt(6)*x/2)/9

_______________________________________________________________________________________

Mathematica [A]  time = 0.0552019, size = 50, normalized size = 0.6 \[ \frac{1}{27} \left (825 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-\sqrt{3 x^2+2} \left (18 x^3-12 x^2-585 x-2171\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-(Sqrt[2 + 3*x^2]*(-2171 - 585*x - 12*x^2 + 18*x^3)) + 825*Sqrt[3]*ArcSinh[Sqrt
[3/2]*x])/27

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 65, normalized size = 0.8 \[{\frac{275\,\sqrt{3}}{9}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{2171}{27}\sqrt{3\,{x}^{2}+2}}+{\frac{65\,x}{3}\sqrt{3\,{x}^{2}+2}}+{\frac{4\,{x}^{2}}{9}\sqrt{3\,{x}^{2}+2}}-{\frac{2\,{x}^{3}}{3}\sqrt{3\,{x}^{2}+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

275/9*arcsinh(1/2*x*6^(1/2))*3^(1/2)+2171/27*(3*x^2+2)^(1/2)+65/3*x*(3*x^2+2)^(1
/2)+4/9*x^2*(3*x^2+2)^(1/2)-2/3*x^3*(3*x^2+2)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.764267, size = 86, normalized size = 1.02 \[ -\frac{2}{3} \, \sqrt{3 \, x^{2} + 2} x^{3} + \frac{4}{9} \, \sqrt{3 \, x^{2} + 2} x^{2} + \frac{65}{3} \, \sqrt{3 \, x^{2} + 2} x + \frac{275}{9} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) + \frac{2171}{27} \, \sqrt{3 \, x^{2} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*sqrt(3*x^2 + 2)*x^3 + 4/9*sqrt(3*x^2 + 2)*x^2 + 65/3*sqrt(3*x^2 + 2)*x + 27
5/9*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 2171/27*sqrt(3*x^2 + 2)

_______________________________________________________________________________________

Fricas [A]  time = 0.274984, size = 85, normalized size = 1.01 \[ -\frac{1}{162} \, \sqrt{3}{\left (2 \, \sqrt{3}{\left (18 \, x^{3} - 12 \, x^{2} - 585 \, x - 2171\right )} \sqrt{3 \, x^{2} + 2} - 2475 \, \log \left (-\sqrt{3}{\left (3 \, x^{2} + 1\right )} - 3 \, \sqrt{3 \, x^{2} + 2} x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/162*sqrt(3)*(2*sqrt(3)*(18*x^3 - 12*x^2 - 585*x - 2171)*sqrt(3*x^2 + 2) - 247
5*log(-sqrt(3)*(3*x^2 + 1) - 3*sqrt(3*x^2 + 2)*x))

_______________________________________________________________________________________

Sympy [A]  time = 3.3974, size = 80, normalized size = 0.95 \[ - \frac{2 x^{3} \sqrt{3 x^{2} + 2}}{3} + \frac{4 x^{2} \sqrt{3 x^{2} + 2}}{9} + \frac{65 x \sqrt{3 x^{2} + 2}}{3} + \frac{2171 \sqrt{3 x^{2} + 2}}{27} + \frac{275 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*x**3*sqrt(3*x**2 + 2)/3 + 4*x**2*sqrt(3*x**2 + 2)/9 + 65*x*sqrt(3*x**2 + 2)/3
+ 2171*sqrt(3*x**2 + 2)/27 + 275*sqrt(3)*asinh(sqrt(6)*x/2)/9

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.312317, size = 66, normalized size = 0.79 \[ -\frac{1}{27} \,{\left (3 \,{\left (2 \,{\left (3 \, x - 2\right )} x - 195\right )} x - 2171\right )} \sqrt{3 \, x^{2} + 2} - \frac{275}{9} \, \sqrt{3}{\rm ln}\left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/27*(3*(2*(3*x - 2)*x - 195)*x - 2171)*sqrt(3*x^2 + 2) - 275/9*sqrt(3)*ln(-sqr
t(3)*x + sqrt(3*x^2 + 2))