3.1357 \(\int (5-x) (3+2 x)^3 \sqrt{2+3 x^2} \, dx\)

Optimal. Leaf size=100 \[ -\frac{1}{18} \left (3 x^2+2\right )^{3/2} (2 x+3)^3+\frac{17}{30} \left (3 x^2+2\right )^{3/2} (2 x+3)^2+\frac{7}{270} (267 x+898) \left (3 x^2+2\right )^{3/2}+\frac{511}{9} x \sqrt{3 x^2+2}+\frac{1022 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{9 \sqrt{3}} \]

[Out]

(511*x*Sqrt[2 + 3*x^2])/9 + (17*(3 + 2*x)^2*(2 + 3*x^2)^(3/2))/30 - ((3 + 2*x)^3
*(2 + 3*x^2)^(3/2))/18 + (7*(898 + 267*x)*(2 + 3*x^2)^(3/2))/270 + (1022*ArcSinh
[Sqrt[3/2]*x])/(9*Sqrt[3])

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Rubi [A]  time = 0.158039, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{1}{18} \left (3 x^2+2\right )^{3/2} (2 x+3)^3+\frac{17}{30} \left (3 x^2+2\right )^{3/2} (2 x+3)^2+\frac{7}{270} (267 x+898) \left (3 x^2+2\right )^{3/2}+\frac{511}{9} x \sqrt{3 x^2+2}+\frac{1022 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{9 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(511*x*Sqrt[2 + 3*x^2])/9 + (17*(3 + 2*x)^2*(2 + 3*x^2)^(3/2))/30 - ((3 + 2*x)^3
*(2 + 3*x^2)^(3/2))/18 + (7*(898 + 267*x)*(2 + 3*x^2)^(3/2))/270 + (1022*ArcSinh
[Sqrt[3/2]*x])/(9*Sqrt[3])

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Rubi in Sympy [A]  time = 14.2638, size = 88, normalized size = 0.88 \[ \frac{511 x \sqrt{3 x^{2} + 2}}{9} - \frac{\left (2 x + 3\right )^{3} \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{18} + \frac{17 \left (2 x + 3\right )^{2} \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{30} + \frac{\left (67284 x + 226296\right ) \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{9720} + \frac{1022 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{27} \]

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]

511*x*sqrt(3*x**2 + 2)/9 - (2*x + 3)**3*(3*x**2 + 2)**(3/2)/18 + 17*(2*x + 3)**2
*(3*x**2 + 2)**(3/2)/30 + (67284*x + 226296)*(3*x**2 + 2)**(3/2)/9720 + 1022*sqr
t(3)*asinh(sqrt(6)*x/2)/27

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Mathematica [A]  time = 0.0595197, size = 60, normalized size = 0.6 \[ \frac{1}{270} \left (10220 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-\sqrt{3 x^2+2} \left (360 x^5-216 x^4-8445 x^3-21918 x^2-21120 x-14516\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-(Sqrt[2 + 3*x^2]*(-14516 - 21120*x - 21918*x^2 - 8445*x^3 - 216*x^4 + 360*x^5)
) + 10220*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/270

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Maple [A]  time = 0.009, size = 77, normalized size = 0.8 \[{\frac{511\,x}{9}\sqrt{3\,{x}^{2}+2}}+{\frac{1022\,\sqrt{3}}{27}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{3629}{135} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}+{\frac{193\,x}{18} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}+{\frac{4\,{x}^{2}}{15} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}-{\frac{4\,{x}^{3}}{9} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

511/9*x*(3*x^2+2)^(1/2)+1022/27*arcsinh(1/2*x*6^(1/2))*3^(1/2)+3629/135*(3*x^2+2
)^(3/2)+193/18*x*(3*x^2+2)^(3/2)+4/15*x^2*(3*x^2+2)^(3/2)-4/9*x^3*(3*x^2+2)^(3/2
)

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Maxima [A]  time = 0.775498, size = 103, normalized size = 1.03 \[ -\frac{4}{9} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x^{3} + \frac{4}{15} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x^{2} + \frac{193}{18} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + \frac{3629}{135} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} + \frac{511}{9} \, \sqrt{3 \, x^{2} + 2} x + \frac{1022}{27} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/9*(3*x^2 + 2)^(3/2)*x^3 + 4/15*(3*x^2 + 2)^(3/2)*x^2 + 193/18*(3*x^2 + 2)^(3/
2)*x + 3629/135*(3*x^2 + 2)^(3/2) + 511/9*sqrt(3*x^2 + 2)*x + 1022/27*sqrt(3)*ar
csinh(1/2*sqrt(6)*x)

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Fricas [A]  time = 0.286617, size = 97, normalized size = 0.97 \[ -\frac{1}{810} \, \sqrt{3}{\left (\sqrt{3}{\left (360 \, x^{5} - 216 \, x^{4} - 8445 \, x^{3} - 21918 \, x^{2} - 21120 \, x - 14516\right )} \sqrt{3 \, x^{2} + 2} - 15330 \, \log \left (-\sqrt{3}{\left (3 \, x^{2} + 1\right )} - 3 \, \sqrt{3 \, x^{2} + 2} x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/810*sqrt(3)*(sqrt(3)*(360*x^5 - 216*x^4 - 8445*x^3 - 21918*x^2 - 21120*x - 14
516)*sqrt(3*x^2 + 2) - 15330*log(-sqrt(3)*(3*x^2 + 1) - 3*sqrt(3*x^2 + 2)*x))

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Sympy [A]  time = 53.8152, size = 150, normalized size = 1.5 \[ - \frac{4 x^{7}}{\sqrt{3 x^{2} + 2}} + \frac{547 x^{5}}{6 \sqrt{3 x^{2} + 2}} + \frac{1705 x^{3}}{18 \sqrt{3 x^{2} + 2}} + \frac{135 x \sqrt{3 x^{2} + 2}}{2} + \frac{193 x}{9 \sqrt{3 x^{2} + 2}} + \frac{16 \sqrt{2} \left (\frac{3 x^{2}}{2} + 1\right )^{\frac{5}{2}}}{45} - \frac{16 \sqrt{2} \left (\frac{3 x^{2}}{2} + 1\right )^{\frac{3}{2}}}{27} + 27 \left (3 x^{2} + 2\right )^{\frac{3}{2}} + \frac{1022 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{27} \]

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]

-4*x**7/sqrt(3*x**2 + 2) + 547*x**5/(6*sqrt(3*x**2 + 2)) + 1705*x**3/(18*sqrt(3*
x**2 + 2)) + 135*x*sqrt(3*x**2 + 2)/2 + 193*x/(9*sqrt(3*x**2 + 2)) + 16*sqrt(2)*
(3*x**2/2 + 1)**(5/2)/45 - 16*sqrt(2)*(3*x**2/2 + 1)**(3/2)/27 + 27*(3*x**2 + 2)
**(3/2) + 1022*sqrt(3)*asinh(sqrt(6)*x/2)/27

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GIAC/XCAS [A]  time = 0.291213, size = 77, normalized size = 0.77 \[ -\frac{1}{270} \,{\left (3 \,{\left ({\left ({\left (24 \,{\left (5 \, x - 3\right )} x - 2815\right )} x - 7306\right )} x - 7040\right )} x - 14516\right )} \sqrt{3 \, x^{2} + 2} - \frac{1022}{27} \, \sqrt{3}{\rm ln}\left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/270*(3*(((24*(5*x - 3)*x - 2815)*x - 7306)*x - 7040)*x - 14516)*sqrt(3*x^2 +
2) - 1022/27*sqrt(3)*ln(-sqrt(3)*x + sqrt(3*x^2 + 2))