3.1358 \(\int (5-x) (3+2 x)^2 \sqrt{2+3 x^2} \, dx\)

Optimal. Leaf size=78 \[ -\frac{1}{15} \left (3 x^2+2\right )^{3/2} (2 x+3)^2+\frac{2}{135} (99 x+431) \left (3 x^2+2\right )^{3/2}+\frac{131}{6} x \sqrt{3 x^2+2}+\frac{131 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]

[Out]

(131*x*Sqrt[2 + 3*x^2])/6 - ((3 + 2*x)^2*(2 + 3*x^2)^(3/2))/15 + (2*(431 + 99*x)
*(2 + 3*x^2)^(3/2))/135 + (131*ArcSinh[Sqrt[3/2]*x])/(3*Sqrt[3])

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Rubi [A]  time = 0.0996811, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{1}{15} \left (3 x^2+2\right )^{3/2} (2 x+3)^2+\frac{2}{135} (99 x+431) \left (3 x^2+2\right )^{3/2}+\frac{131}{6} x \sqrt{3 x^2+2}+\frac{131 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(131*x*Sqrt[2 + 3*x^2])/6 - ((3 + 2*x)^2*(2 + 3*x^2)^(3/2))/15 + (2*(431 + 99*x)
*(2 + 3*x^2)^(3/2))/135 + (131*ArcSinh[Sqrt[3/2]*x])/(3*Sqrt[3])

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Rubi in Sympy [A]  time = 9.76079, size = 68, normalized size = 0.87 \[ \frac{131 x \sqrt{3 x^{2} + 2}}{6} - \frac{\left (2 x + 3\right )^{2} \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{15} + \frac{\left (792 x + 3448\right ) \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{540} + \frac{131 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{9} \]

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]

131*x*sqrt(3*x**2 + 2)/6 - (2*x + 3)**2*(3*x**2 + 2)**(3/2)/15 + (792*x + 3448)*
(3*x**2 + 2)**(3/2)/540 + 131*sqrt(3)*asinh(sqrt(6)*x/2)/9

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Mathematica [A]  time = 0.0537085, size = 55, normalized size = 0.71 \[ \frac{1}{270} \left (3930 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-\sqrt{3 x^2+2} \left (216 x^4-540 x^3-4542 x^2-6255 x-3124\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-(Sqrt[2 + 3*x^2]*(-3124 - 6255*x - 4542*x^2 - 540*x^3 + 216*x^4)) + 3930*Sqrt[
3]*ArcSinh[Sqrt[3/2]*x])/270

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Maple [A]  time = 0.01, size = 63, normalized size = 0.8 \[{\frac{131\,x}{6}\sqrt{3\,{x}^{2}+2}}+{\frac{131\,\sqrt{3}}{9}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{781}{135} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}+{\frac{2\,x}{3} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}-{\frac{4\,{x}^{2}}{15} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

131/6*x*(3*x^2+2)^(1/2)+131/9*arcsinh(1/2*x*6^(1/2))*3^(1/2)+781/135*(3*x^2+2)^(
3/2)+2/3*x*(3*x^2+2)^(3/2)-4/15*x^2*(3*x^2+2)^(3/2)

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Maxima [A]  time = 0.776897, size = 84, normalized size = 1.08 \[ -\frac{4}{15} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x^{2} + \frac{2}{3} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + \frac{781}{135} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} + \frac{131}{6} \, \sqrt{3 \, x^{2} + 2} x + \frac{131}{9} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/15*(3*x^2 + 2)^(3/2)*x^2 + 2/3*(3*x^2 + 2)^(3/2)*x + 781/135*(3*x^2 + 2)^(3/2
) + 131/6*sqrt(3*x^2 + 2)*x + 131/9*sqrt(3)*arcsinh(1/2*sqrt(6)*x)

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Fricas [A]  time = 0.288471, size = 90, normalized size = 1.15 \[ -\frac{1}{810} \, \sqrt{3}{\left (\sqrt{3}{\left (216 \, x^{4} - 540 \, x^{3} - 4542 \, x^{2} - 6255 \, x - 3124\right )} \sqrt{3 \, x^{2} + 2} - 5895 \, \log \left (-\sqrt{3}{\left (3 \, x^{2} + 1\right )} - 3 \, \sqrt{3 \, x^{2} + 2} x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/810*sqrt(3)*(sqrt(3)*(216*x^4 - 540*x^3 - 4542*x^2 - 6255*x - 3124)*sqrt(3*x^
2 + 2) - 5895*log(-sqrt(3)*(3*x^2 + 1) - 3*sqrt(3*x^2 + 2)*x))

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Sympy [A]  time = 2.97995, size = 95, normalized size = 1.22 \[ - \frac{4 x^{4} \sqrt{3 x^{2} + 2}}{5} + 2 x^{3} \sqrt{3 x^{2} + 2} + \frac{757 x^{2} \sqrt{3 x^{2} + 2}}{45} + \frac{139 x \sqrt{3 x^{2} + 2}}{6} + \frac{1562 \sqrt{3 x^{2} + 2}}{135} + \frac{131 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{9} \]

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**4*sqrt(3*x**2 + 2)/5 + 2*x**3*sqrt(3*x**2 + 2) + 757*x**2*sqrt(3*x**2 + 2)
/45 + 139*x*sqrt(3*x**2 + 2)/6 + 1562*sqrt(3*x**2 + 2)/135 + 131*sqrt(3)*asinh(s
qrt(6)*x/2)/9

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GIAC/XCAS [A]  time = 0.286673, size = 73, normalized size = 0.94 \[ -\frac{1}{270} \,{\left (3 \,{\left (2 \,{\left (18 \,{\left (2 \, x - 5\right )} x - 757\right )} x - 2085\right )} x - 3124\right )} \sqrt{3 \, x^{2} + 2} - \frac{131}{9} \, \sqrt{3}{\rm ln}\left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/270*(3*(2*(18*(2*x - 5)*x - 757)*x - 2085)*x - 3124)*sqrt(3*x^2 + 2) - 131/9*
sqrt(3)*ln(-sqrt(3)*x + sqrt(3*x^2 + 2))