3.1356 \(\int (5-x) (3+2 x)^4 \sqrt{2+3 x^2} \, dx\)

Optimal. Leaf size=122 \[ -\frac{1}{21} \left (3 x^2+2\right )^{3/2} (2 x+3)^4+\frac{29}{63} \left (3 x^2+2\right )^{3/2} (2 x+3)^3+\frac{923}{315} \left (3 x^2+2\right )^{3/2} (2 x+3)^2+\frac{2}{405} (4599 x+13781) \left (3 x^2+2\right )^{3/2}+\frac{2341}{18} x \sqrt{3 x^2+2}+\frac{2341 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{9 \sqrt{3}} \]

[Out]

(2341*x*Sqrt[2 + 3*x^2])/18 + (923*(3 + 2*x)^2*(2 + 3*x^2)^(3/2))/315 + (29*(3 +
2*x)^3*(2 + 3*x^2)^(3/2))/63 - ((3 + 2*x)^4*(2 + 3*x^2)^(3/2))/21 + (2*(13781 +
4599*x)*(2 + 3*x^2)^(3/2))/405 + (2341*ArcSinh[Sqrt[3/2]*x])/(9*Sqrt[3])

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Rubi [A]  time = 0.215558, antiderivative size = 122, 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 \[ -\frac{1}{21} \left (3 x^2+2\right )^{3/2} (2 x+3)^4+\frac{29}{63} \left (3 x^2+2\right )^{3/2} (2 x+3)^3+\frac{923}{315} \left (3 x^2+2\right )^{3/2} (2 x+3)^2+\frac{2}{405} (4599 x+13781) \left (3 x^2+2\right )^{3/2}+\frac{2341}{18} x \sqrt{3 x^2+2}+\frac{2341 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{9 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(5 - x)*(3 + 2*x)^4*Sqrt[2 + 3*x^2],x]

[Out]

(2341*x*Sqrt[2 + 3*x^2])/18 + (923*(3 + 2*x)^2*(2 + 3*x^2)^(3/2))/315 + (29*(3 +
2*x)^3*(2 + 3*x^2)^(3/2))/63 - ((3 + 2*x)^4*(2 + 3*x^2)^(3/2))/21 + (2*(13781 +
4599*x)*(2 + 3*x^2)^(3/2))/405 + (2341*ArcSinh[Sqrt[3/2]*x])/(9*Sqrt[3])

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Rubi in Sympy [A]  time = 19.0974, size = 109, normalized size = 0.89 \[ \frac{2341 x \sqrt{3 x^{2} + 2}}{18} - \frac{\left (2 x + 3\right )^{4} \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{21} + \frac{29 \left (2 x + 3\right )^{3} \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{63} + \frac{923 \left (2 x + 3\right )^{2} \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{315} + \frac{\left (4635792 x + 13891248\right ) \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{204120} + \frac{2341 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{27} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)*(3+2*x)**4*(3*x**2+2)**(1/2),x)

[Out]

2341*x*sqrt(3*x**2 + 2)/18 - (2*x + 3)**4*(3*x**2 + 2)**(3/2)/21 + 29*(2*x + 3)*
*3*(3*x**2 + 2)**(3/2)/63 + 923*(2*x + 3)**2*(3*x**2 + 2)**(3/2)/315 + (4635792*
x + 13891248)*(3*x**2 + 2)**(3/2)/204120 + 2341*sqrt(3)*asinh(sqrt(6)*x/2)/27

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Mathematica [A]  time = 0.0685634, size = 65, normalized size = 0.53 \[ \frac{491610 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-\sqrt{3 x^2+2} \left (12960 x^6+15120 x^5-297648 x^4-1222200 x^3-1956174 x^2-1558935 x-1167988\right )}{5670} \]

Antiderivative was successfully verified.

[In]  Integrate[(5 - x)*(3 + 2*x)^4*Sqrt[2 + 3*x^2],x]

[Out]

(-(Sqrt[2 + 3*x^2]*(-1167988 - 1558935*x - 1956174*x^2 - 1222200*x^3 - 297648*x^
4 + 15120*x^5 + 12960*x^6)) + 491610*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/5670

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Maple [A]  time = 0.019, size = 91, normalized size = 0.8 \[{\frac{2341\,x}{18}\sqrt{3\,{x}^{2}+2}}+{\frac{2341\,\sqrt{3}}{27}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{291997}{2835} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}+{\frac{652\,x}{9} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}+{\frac{5672\,{x}^{2}}{315} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}-{\frac{8\,{x}^{3}}{9} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}-{\frac{16\,{x}^{4}}{21} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)*(2*x+3)^4*(3*x^2+2)^(1/2),x)

[Out]

2341/18*x*(3*x^2+2)^(1/2)+2341/27*arcsinh(1/2*x*6^(1/2))*3^(1/2)+291997/2835*(3*
x^2+2)^(3/2)+652/9*x*(3*x^2+2)^(3/2)+5672/315*x^2*(3*x^2+2)^(3/2)-8/9*x^3*(3*x^2
+2)^(3/2)-16/21*x^4*(3*x^2+2)^(3/2)

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Maxima [A]  time = 0.775105, size = 122, normalized size = 1. \[ -\frac{16}{21} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x^{4} - \frac{8}{9} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x^{3} + \frac{5672}{315} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x^{2} + \frac{652}{9} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + \frac{291997}{2835} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} + \frac{2341}{18} \, \sqrt{3 \, x^{2} + 2} x + \frac{2341}{27} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(3*x^2 + 2)*(2*x + 3)^4*(x - 5),x, algorithm="maxima")

[Out]

-16/21*(3*x^2 + 2)^(3/2)*x^4 - 8/9*(3*x^2 + 2)^(3/2)*x^3 + 5672/315*(3*x^2 + 2)^
(3/2)*x^2 + 652/9*(3*x^2 + 2)^(3/2)*x + 291997/2835*(3*x^2 + 2)^(3/2) + 2341/18*
sqrt(3*x^2 + 2)*x + 2341/27*sqrt(3)*arcsinh(1/2*sqrt(6)*x)

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Fricas [A]  time = 0.287131, size = 104, normalized size = 0.85 \[ -\frac{1}{17010} \, \sqrt{3}{\left (\sqrt{3}{\left (12960 \, x^{6} + 15120 \, x^{5} - 297648 \, x^{4} - 1222200 \, x^{3} - 1956174 \, x^{2} - 1558935 \, x - 1167988\right )} \sqrt{3 \, x^{2} + 2} - 737415 \, \log \left (-\sqrt{3}{\left (3 \, x^{2} + 1\right )} - 3 \, \sqrt{3 \, x^{2} + 2} x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(3*x^2 + 2)*(2*x + 3)^4*(x - 5),x, algorithm="fricas")

[Out]

-1/17010*sqrt(3)*(sqrt(3)*(12960*x^6 + 15120*x^5 - 297648*x^4 - 1222200*x^3 - 19
56174*x^2 - 1558935*x - 1167988)*sqrt(3*x^2 + 2) - 737415*log(-sqrt(3)*(3*x^2 +
1) - 3*sqrt(3*x^2 + 2)*x))

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Sympy [A]  time = 11.5872, size = 131, normalized size = 1.07 \[ - \frac{16 x^{6} \sqrt{3 x^{2} + 2}}{7} - \frac{8 x^{5} \sqrt{3 x^{2} + 2}}{3} + \frac{5512 x^{4} \sqrt{3 x^{2} + 2}}{105} + \frac{1940 x^{3} \sqrt{3 x^{2} + 2}}{9} + \frac{326029 x^{2} \sqrt{3 x^{2} + 2}}{945} + \frac{4949 x \sqrt{3 x^{2} + 2}}{18} + \frac{583994 \sqrt{3 x^{2} + 2}}{2835} + \frac{2341 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{27} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5-x)*(3+2*x)**4*(3*x**2+2)**(1/2),x)

[Out]

-16*x**6*sqrt(3*x**2 + 2)/7 - 8*x**5*sqrt(3*x**2 + 2)/3 + 5512*x**4*sqrt(3*x**2
+ 2)/105 + 1940*x**3*sqrt(3*x**2 + 2)/9 + 326029*x**2*sqrt(3*x**2 + 2)/945 + 494
9*x*sqrt(3*x**2 + 2)/18 + 583994*sqrt(3*x**2 + 2)/2835 + 2341*sqrt(3)*asinh(sqrt
(6)*x/2)/27

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GIAC/XCAS [A]  time = 0.27699, size = 86, normalized size = 0.7 \[ -\frac{1}{5670} \,{\left (3 \,{\left (2 \,{\left (12 \,{\left (6 \,{\left (5 \,{\left (6 \, x + 7\right )} x - 689\right )} x - 16975\right )} x - 326029\right )} x - 519645\right )} x - 1167988\right )} \sqrt{3 \, x^{2} + 2} - \frac{2341}{27} \, \sqrt{3}{\rm ln}\left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(3*x^2 + 2)*(2*x + 3)^4*(x - 5),x, algorithm="giac")

[Out]

-1/5670*(3*(2*(12*(6*(5*(6*x + 7)*x - 689)*x - 16975)*x - 326029)*x - 519645)*x
- 1167988)*sqrt(3*x^2 + 2) - 2341/27*sqrt(3)*ln(-sqrt(3)*x + sqrt(3*x^2 + 2))