3.1370 \(\int (5-x) (3+2 x)^2 \left (2+3 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=94 \[ -\frac{1}{21} (2 x+3)^2 \left (3 x^2+2\right )^{5/2}+\frac{2}{315} (160 x+611) \left (3 x^2+2\right )^{5/2}+\frac{397}{36} x \left (3 x^2+2\right )^{3/2}+\frac{397}{12} x \sqrt{3 x^2+2}+\frac{397 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{6 \sqrt{3}} \]

[Out]

(397*x*Sqrt[2 + 3*x^2])/12 + (397*x*(2 + 3*x^2)^(3/2))/36 - ((3 + 2*x)^2*(2 + 3*
x^2)^(5/2))/21 + (2*(611 + 160*x)*(2 + 3*x^2)^(5/2))/315 + (397*ArcSinh[Sqrt[3/2
]*x])/(6*Sqrt[3])

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Rubi [A]  time = 0.113835, antiderivative size = 94, 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 \[ -\frac{1}{21} (2 x+3)^2 \left (3 x^2+2\right )^{5/2}+\frac{2}{315} (160 x+611) \left (3 x^2+2\right )^{5/2}+\frac{397}{36} x \left (3 x^2+2\right )^{3/2}+\frac{397}{12} x \sqrt{3 x^2+2}+\frac{397 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{6 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(397*x*Sqrt[2 + 3*x^2])/12 + (397*x*(2 + 3*x^2)^(3/2))/36 - ((3 + 2*x)^2*(2 + 3*
x^2)^(5/2))/21 + (2*(611 + 160*x)*(2 + 3*x^2)^(5/2))/315 + (397*ArcSinh[Sqrt[3/2
]*x])/(6*Sqrt[3])

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Rubi in Sympy [A]  time = 10.1932, size = 83, normalized size = 0.88 \[ \frac{397 x \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{36} + \frac{397 x \sqrt{3 x^{2} + 2}}{12} - \frac{\left (2 x + 3\right )^{2} \left (3 x^{2} + 2\right )^{\frac{5}{2}}}{21} + \frac{\left (1920 x + 7332\right ) \left (3 x^{2} + 2\right )^{\frac{5}{2}}}{1890} + \frac{397 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{18} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

397*x*(3*x**2 + 2)**(3/2)/36 + 397*x*sqrt(3*x**2 + 2)/12 - (2*x + 3)**2*(3*x**2
+ 2)**(5/2)/21 + (1920*x + 7332)*(3*x**2 + 2)**(5/2)/1890 + 397*sqrt(3)*asinh(sq
rt(6)*x/2)/18

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Mathematica [A]  time = 0.0655754, size = 65, normalized size = 0.69 \[ \frac{27790 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-\sqrt{3 x^2+2} \left (2160 x^6-5040 x^5-36252 x^4-48405 x^3-51216 x^2-71715 x-17392\right )}{1260} \]

Antiderivative was successfully verified.

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

[Out]

(-(Sqrt[2 + 3*x^2]*(-17392 - 71715*x - 51216*x^2 - 48405*x^3 - 36252*x^4 - 5040*
x^5 + 2160*x^6)) + 27790*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/1260

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Maple [A]  time = 0.009, size = 75, normalized size = 0.8 \[{\frac{397\,x}{36} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}+{\frac{397\,x}{12}\sqrt{3\,{x}^{2}+2}}+{\frac{397\,\sqrt{3}}{18}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{1087}{315} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{5}{2}}}}+{\frac{4\,x}{9} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{5}{2}}}}-{\frac{4\,{x}^{2}}{21} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

397/36*x*(3*x^2+2)^(3/2)+397/12*x*(3*x^2+2)^(1/2)+397/18*arcsinh(1/2*x*6^(1/2))*
3^(1/2)+1087/315*(3*x^2+2)^(5/2)+4/9*x*(3*x^2+2)^(5/2)-4/21*x^2*(3*x^2+2)^(5/2)

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Maxima [A]  time = 0.770039, size = 100, normalized size = 1.06 \[ -\frac{4}{21} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} x^{2} + \frac{4}{9} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} x + \frac{1087}{315} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} + \frac{397}{36} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + \frac{397}{12} \, \sqrt{3 \, x^{2} + 2} x + \frac{397}{18} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/21*(3*x^2 + 2)^(5/2)*x^2 + 4/9*(3*x^2 + 2)^(5/2)*x + 1087/315*(3*x^2 + 2)^(5/
2) + 397/36*(3*x^2 + 2)^(3/2)*x + 397/12*sqrt(3*x^2 + 2)*x + 397/18*sqrt(3)*arcs
inh(1/2*sqrt(6)*x)

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Fricas [A]  time = 0.291996, size = 104, normalized size = 1.11 \[ -\frac{1}{3780} \, \sqrt{3}{\left (\sqrt{3}{\left (2160 \, x^{6} - 5040 \, x^{5} - 36252 \, x^{4} - 48405 \, x^{3} - 51216 \, x^{2} - 71715 \, x - 17392\right )} \sqrt{3 \, x^{2} + 2} - 41685 \, \log \left (-\sqrt{3}{\left (3 \, x^{2} + 1\right )} - 3 \, \sqrt{3 \, x^{2} + 2} x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3780*sqrt(3)*(sqrt(3)*(2160*x^6 - 5040*x^5 - 36252*x^4 - 48405*x^3 - 51216*x^
2 - 71715*x - 17392)*sqrt(3*x^2 + 2) - 41685*log(-sqrt(3)*(3*x^2 + 1) - 3*sqrt(3
*x^2 + 2)*x))

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Sympy [A]  time = 36.4523, size = 129, normalized size = 1.37 \[ - \frac{12 x^{6} \sqrt{3 x^{2} + 2}}{7} + 4 x^{5} \sqrt{3 x^{2} + 2} + \frac{1007 x^{4} \sqrt{3 x^{2} + 2}}{35} + \frac{461 x^{3} \sqrt{3 x^{2} + 2}}{12} + \frac{4268 x^{2} \sqrt{3 x^{2} + 2}}{105} + \frac{683 x \sqrt{3 x^{2} + 2}}{12} + \frac{4348 \sqrt{3 x^{2} + 2}}{315} + \frac{397 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{18} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-12*x**6*sqrt(3*x**2 + 2)/7 + 4*x**5*sqrt(3*x**2 + 2) + 1007*x**4*sqrt(3*x**2 +
2)/35 + 461*x**3*sqrt(3*x**2 + 2)/12 + 4268*x**2*sqrt(3*x**2 + 2)/105 + 683*x*sq
rt(3*x**2 + 2)/12 + 4348*sqrt(3*x**2 + 2)/315 + 397*sqrt(3)*asinh(sqrt(6)*x/2)/1
8

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GIAC/XCAS [A]  time = 0.275954, size = 84, normalized size = 0.89 \[ -\frac{1}{1260} \,{\left (3 \,{\left ({\left ({\left (12 \,{\left (20 \,{\left (3 \, x - 7\right )} x - 1007\right )} x - 16135\right )} x - 17072\right )} x - 23905\right )} x - 17392\right )} \sqrt{3 \, x^{2} + 2} - \frac{397}{18} \, \sqrt{3}{\rm ln}\left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1260*(3*(((12*(20*(3*x - 7)*x - 1007)*x - 16135)*x - 17072)*x - 23905)*x - 17
392)*sqrt(3*x^2 + 2) - 397/18*sqrt(3)*ln(-sqrt(3)*x + sqrt(3*x^2 + 2))