3.1371 \(\int (5-x) (3+2 x) \left (2+3 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=72 \[ \frac{1}{45} (21-5 x) \left (3 x^2+2\right )^{5/2}+\frac{137}{36} x \left (3 x^2+2\right )^{3/2}+\frac{137}{12} x \sqrt{3 x^2+2}+\frac{137 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{6 \sqrt{3}} \]

[Out]

(137*x*Sqrt[2 + 3*x^2])/12 + (137*x*(2 + 3*x^2)^(3/2))/36 + ((21 - 5*x)*(2 + 3*x
^2)^(5/2))/45 + (137*ArcSinh[Sqrt[3/2]*x])/(6*Sqrt[3])

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Rubi [A]  time = 0.0587054, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{1}{45} (21-5 x) \left (3 x^2+2\right )^{5/2}+\frac{137}{36} x \left (3 x^2+2\right )^{3/2}+\frac{137}{12} x \sqrt{3 x^2+2}+\frac{137 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{6 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(137*x*Sqrt[2 + 3*x^2])/12 + (137*x*(2 + 3*x^2)^(3/2))/36 + ((21 - 5*x)*(2 + 3*x
^2)^(5/2))/45 + (137*ArcSinh[Sqrt[3/2]*x])/(6*Sqrt[3])

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Rubi in Sympy [A]  time = 5.70898, size = 65, normalized size = 0.9 \[ \frac{137 x \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{36} + \frac{137 x \sqrt{3 x^{2} + 2}}{12} + \frac{\left (- 10 x + 42\right ) \left (3 x^{2} + 2\right )^{\frac{5}{2}}}{90} + \frac{137 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{18} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

137*x*(3*x**2 + 2)**(3/2)/36 + 137*x*sqrt(3*x**2 + 2)/12 + (-10*x + 42)*(3*x**2
+ 2)**(5/2)/90 + 137*sqrt(3)*asinh(sqrt(6)*x/2)/18

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Mathematica [A]  time = 0.0673433, size = 60, normalized size = 0.83 \[ \frac{1}{60} \sqrt{3 x^2+2} \left (-60 x^5+252 x^4+605 x^3+336 x^2+1115 x+112\right )+\frac{137 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{6 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[2 + 3*x^2]*(112 + 1115*x + 336*x^2 + 605*x^3 + 252*x^4 - 60*x^5))/60 + (13
7*ArcSinh[Sqrt[3/2]*x])/(6*Sqrt[3])

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Maple [A]  time = 0.008, size = 61, normalized size = 0.9 \[{\frac{137\,x}{36} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}+{\frac{137\,x}{12}\sqrt{3\,{x}^{2}+2}}+{\frac{137\,\sqrt{3}}{18}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{7}{15} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{5}{2}}}}-{\frac{x}{9} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

137/36*x*(3*x^2+2)^(3/2)+137/12*x*(3*x^2+2)^(1/2)+137/18*arcsinh(1/2*x*6^(1/2))*
3^(1/2)+7/15*(3*x^2+2)^(5/2)-1/9*x*(3*x^2+2)^(5/2)

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Maxima [A]  time = 0.781912, size = 81, normalized size = 1.12 \[ -\frac{1}{9} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} x + \frac{7}{15} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} + \frac{137}{36} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + \frac{137}{12} \, \sqrt{3 \, x^{2} + 2} x + \frac{137}{18} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9*(3*x^2 + 2)^(5/2)*x + 7/15*(3*x^2 + 2)^(5/2) + 137/36*(3*x^2 + 2)^(3/2)*x +
137/12*sqrt(3*x^2 + 2)*x + 137/18*sqrt(3)*arcsinh(1/2*sqrt(6)*x)

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Fricas [A]  time = 0.290444, size = 97, normalized size = 1.35 \[ -\frac{1}{180} \, \sqrt{3}{\left (\sqrt{3}{\left (60 \, x^{5} - 252 \, x^{4} - 605 \, x^{3} - 336 \, x^{2} - 1115 \, x - 112\right )} \sqrt{3 \, x^{2} + 2} - 685 \, \log \left (-\sqrt{3}{\left (3 \, x^{2} + 1\right )} - 3 \, \sqrt{3 \, x^{2} + 2} x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/180*sqrt(3)*(sqrt(3)*(60*x^5 - 252*x^4 - 605*x^3 - 336*x^2 - 1115*x - 112)*sq
rt(3*x^2 + 2) - 685*log(-sqrt(3)*(3*x^2 + 1) - 3*sqrt(3*x^2 + 2)*x))

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Sympy [A]  time = 20.6555, size = 110, normalized size = 1.53 \[ - x^{5} \sqrt{3 x^{2} + 2} + \frac{21 x^{4} \sqrt{3 x^{2} + 2}}{5} + \frac{121 x^{3} \sqrt{3 x^{2} + 2}}{12} + \frac{28 x^{2} \sqrt{3 x^{2} + 2}}{5} + \frac{223 x \sqrt{3 x^{2} + 2}}{12} + \frac{28 \sqrt{3 x^{2} + 2}}{15} + \frac{137 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{18} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**5*sqrt(3*x**2 + 2) + 21*x**4*sqrt(3*x**2 + 2)/5 + 121*x**3*sqrt(3*x**2 + 2)/
12 + 28*x**2*sqrt(3*x**2 + 2)/5 + 223*x*sqrt(3*x**2 + 2)/12 + 28*sqrt(3*x**2 + 2
)/15 + 137*sqrt(3)*asinh(sqrt(6)*x/2)/18

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GIAC/XCAS [A]  time = 0.278466, size = 76, normalized size = 1.06 \[ -\frac{1}{60} \,{\left ({\left ({\left ({\left (12 \,{\left (5 \, x - 21\right )} x - 605\right )} x - 336\right )} x - 1115\right )} x - 112\right )} \sqrt{3 \, x^{2} + 2} - \frac{137}{18} \, \sqrt{3}{\rm ln}\left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/60*((((12*(5*x - 21)*x - 605)*x - 336)*x - 1115)*x - 112)*sqrt(3*x^2 + 2) - 1
37/18*sqrt(3)*ln(-sqrt(3)*x + sqrt(3*x^2 + 2))