3.1383 \(\int (5-x) (3+2 x)^2 \left (2+3 x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=110 \[ -\frac{1}{27} (2 x+3)^2 \left (3 x^2+2\right )^{7/2}+\frac{1}{81} (63 x+226) \left (3 x^2+2\right )^{7/2}+\frac{133}{18} x \left (3 x^2+2\right )^{5/2}+\frac{665}{36} x \left (3 x^2+2\right )^{3/2}+\frac{665}{12} x \sqrt{3 x^2+2}+\frac{665 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{6 \sqrt{3}} \]

[Out]

(665*x*Sqrt[2 + 3*x^2])/12 + (665*x*(2 + 3*x^2)^(3/2))/36 + (133*x*(2 + 3*x^2)^(
5/2))/18 - ((3 + 2*x)^2*(2 + 3*x^2)^(7/2))/27 + ((226 + 63*x)*(2 + 3*x^2)^(7/2))
/81 + (665*ArcSinh[Sqrt[3/2]*x])/(6*Sqrt[3])

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Rubi [A]  time = 0.124363, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{1}{27} (2 x+3)^2 \left (3 x^2+2\right )^{7/2}+\frac{1}{81} (63 x+226) \left (3 x^2+2\right )^{7/2}+\frac{133}{18} x \left (3 x^2+2\right )^{5/2}+\frac{665}{36} x \left (3 x^2+2\right )^{3/2}+\frac{665}{12} x \sqrt{3 x^2+2}+\frac{665 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{6 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(665*x*Sqrt[2 + 3*x^2])/12 + (665*x*(2 + 3*x^2)^(3/2))/36 + (133*x*(2 + 3*x^2)^(
5/2))/18 - ((3 + 2*x)^2*(2 + 3*x^2)^(7/2))/27 + ((226 + 63*x)*(2 + 3*x^2)^(7/2))
/81 + (665*ArcSinh[Sqrt[3/2]*x])/(6*Sqrt[3])

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Rubi in Sympy [A]  time = 10.8836, size = 99, normalized size = 0.9 \[ \frac{133 x \left (3 x^{2} + 2\right )^{\frac{5}{2}}}{18} + \frac{665 x \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{36} + \frac{665 x \sqrt{3 x^{2} + 2}}{12} - \frac{\left (2 x + 3\right )^{2} \left (3 x^{2} + 2\right )^{\frac{7}{2}}}{27} + \frac{\left (3528 x + 12656\right ) \left (3 x^{2} + 2\right )^{\frac{7}{2}}}{4536} + \frac{665 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{18} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

133*x*(3*x**2 + 2)**(5/2)/18 + 665*x*(3*x**2 + 2)**(3/2)/36 + 665*x*sqrt(3*x**2
+ 2)/12 - (2*x + 3)**2*(3*x**2 + 2)**(7/2)/27 + (3528*x + 12656)*(3*x**2 + 2)**(
7/2)/4536 + 665*sqrt(3)*asinh(sqrt(6)*x/2)/18

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Mathematica [A]  time = 0.0767111, size = 75, normalized size = 0.68 \[ \frac{1}{324} \left (11970 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-\sqrt{3 x^2+2} \left (1296 x^8-2916 x^7-18900 x^6-27378 x^5-41256 x^4-50571 x^3-28272 x^2-40365 x-6368\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-(Sqrt[2 + 3*x^2]*(-6368 - 40365*x - 28272*x^2 - 50571*x^3 - 41256*x^4 - 27378*
x^5 - 18900*x^6 - 2916*x^7 + 1296*x^8)) + 11970*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/32
4

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Maple [A]  time = 0.009, size = 87, normalized size = 0.8 \[{\frac{133\,x}{18} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{5}{2}}}}+{\frac{665\,x}{36} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}+{\frac{665\,x}{12}\sqrt{3\,{x}^{2}+2}}+{\frac{665\,\sqrt{3}}{18}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{199}{81} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{7}{2}}}}+{\frac{x}{3} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{7}{2}}}}-{\frac{4\,{x}^{2}}{27} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

133/18*x*(3*x^2+2)^(5/2)+665/36*x*(3*x^2+2)^(3/2)+665/12*x*(3*x^2+2)^(1/2)+665/1
8*arcsinh(1/2*x*6^(1/2))*3^(1/2)+199/81*(3*x^2+2)^(7/2)+1/3*x*(3*x^2+2)^(7/2)-4/
27*x^2*(3*x^2+2)^(7/2)

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Maxima [A]  time = 0.777537, size = 116, normalized size = 1.05 \[ -\frac{4}{27} \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}} x^{2} + \frac{1}{3} \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}} x + \frac{199}{81} \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}} + \frac{133}{18} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} x + \frac{665}{36} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + \frac{665}{12} \, \sqrt{3 \, x^{2} + 2} x + \frac{665}{18} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/27*(3*x^2 + 2)^(7/2)*x^2 + 1/3*(3*x^2 + 2)^(7/2)*x + 199/81*(3*x^2 + 2)^(7/2)
+ 133/18*(3*x^2 + 2)^(5/2)*x + 665/36*(3*x^2 + 2)^(3/2)*x + 665/12*sqrt(3*x^2 +
2)*x + 665/18*sqrt(3)*arcsinh(1/2*sqrt(6)*x)

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Fricas [A]  time = 0.275922, size = 117, normalized size = 1.06 \[ -\frac{1}{972} \, \sqrt{3}{\left (\sqrt{3}{\left (1296 \, x^{8} - 2916 \, x^{7} - 18900 \, x^{6} - 27378 \, x^{5} - 41256 \, x^{4} - 50571 \, x^{3} - 28272 \, x^{2} - 40365 \, x - 6368\right )} \sqrt{3 \, x^{2} + 2} - 17955 \, \log \left (-\sqrt{3}{\left (3 \, x^{2} + 1\right )} - 3 \, \sqrt{3 \, x^{2} + 2} x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/972*sqrt(3)*(sqrt(3)*(1296*x^8 - 2916*x^7 - 18900*x^6 - 27378*x^5 - 41256*x^4
- 50571*x^3 - 28272*x^2 - 40365*x - 6368)*sqrt(3*x^2 + 2) - 17955*log(-sqrt(3)*
(3*x^2 + 1) - 3*sqrt(3*x^2 + 2)*x))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.29017, size = 97, normalized size = 0.88 \[ -\frac{1}{324} \,{\left (3 \,{\left ({\left (9 \,{\left (2 \,{\left ({\left (2 \,{\left (3 \,{\left (4 \, x - 9\right )} x - 175\right )} x - 507\right )} x - 764\right )} x - 1873\right )} x - 9424\right )} x - 13455\right )} x - 6368\right )} \sqrt{3 \, x^{2} + 2} - \frac{665}{18} \, \sqrt{3}{\rm ln}\left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/324*(3*((9*(2*((2*(3*(4*x - 9)*x - 175)*x - 507)*x - 764)*x - 1873)*x - 9424)
*x - 13455)*x - 6368)*sqrt(3*x^2 + 2) - 665/18*sqrt(3)*ln(-sqrt(3)*x + sqrt(3*x^
2 + 2))