3.1372 \(\int (5-x) \left (2+3 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=67 \[ -\frac{1}{15} \left (3 x^2+2\right )^{5/2}+\frac{5}{4} x \left (3 x^2+2\right )^{3/2}+\frac{15}{4} x \sqrt{3 x^2+2}+\frac{5}{2} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

[Out]

(15*x*Sqrt[2 + 3*x^2])/4 + (5*x*(2 + 3*x^2)^(3/2))/4 - (2 + 3*x^2)^(5/2)/15 + (5
*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/2

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Rubi [A]  time = 0.0383007, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\frac{1}{15} \left (3 x^2+2\right )^{5/2}+\frac{5}{4} x \left (3 x^2+2\right )^{3/2}+\frac{15}{4} x \sqrt{3 x^2+2}+\frac{5}{2} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

Antiderivative was successfully verified.

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

[Out]

(15*x*Sqrt[2 + 3*x^2])/4 + (5*x*(2 + 3*x^2)^(3/2))/4 - (2 + 3*x^2)^(5/2)/15 + (5
*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/2

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Rubi in Sympy [A]  time = 3.86091, size = 60, normalized size = 0.9 \[ \frac{5 x \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{4} + \frac{15 x \sqrt{3 x^{2} + 2}}{4} - \frac{\left (3 x^{2} + 2\right )^{\frac{5}{2}}}{15} + \frac{5 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*x*(3*x**2 + 2)**(3/2)/4 + 15*x*sqrt(3*x**2 + 2)/4 - (3*x**2 + 2)**(5/2)/15 + 5
*sqrt(3)*asinh(sqrt(6)*x/2)/2

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Mathematica [A]  time = 0.0332635, size = 55, normalized size = 0.82 \[ \frac{5}{2} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-\frac{1}{60} \sqrt{3 x^2+2} \left (36 x^4-225 x^3+48 x^2-375 x+16\right ) \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[2 + 3*x^2]*(16 - 375*x + 48*x^2 - 225*x^3 + 36*x^4))/60 + (5*Sqrt[3]*ArcS
inh[Sqrt[3/2]*x])/2

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Maple [A]  time = 0.006, size = 49, normalized size = 0.7 \[{\frac{5\,x}{4} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}-{\frac{1}{15} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{5}{2}}}}+{\frac{5\,\sqrt{3}}{2}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{15\,x}{4}\sqrt{3\,{x}^{2}+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)*(3*x^2+2)^(3/2),x)

[Out]

5/4*x*(3*x^2+2)^(3/2)-1/15*(3*x^2+2)^(5/2)+5/2*arcsinh(1/2*x*6^(1/2))*3^(1/2)+15
/4*x*(3*x^2+2)^(1/2)

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Maxima [A]  time = 0.780597, size = 65, normalized size = 0.97 \[ -\frac{1}{15} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} + \frac{5}{4} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + \frac{15}{4} \, \sqrt{3 \, x^{2} + 2} x + \frac{5}{2} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*(3*x^2 + 2)^(5/2) + 5/4*(3*x^2 + 2)^(3/2)*x + 15/4*sqrt(3*x^2 + 2)*x + 5/2
*sqrt(3)*arcsinh(1/2*sqrt(6)*x)

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Fricas [A]  time = 0.290231, size = 81, normalized size = 1.21 \[ -\frac{1}{60} \,{\left (36 \, x^{4} - 225 \, x^{3} + 48 \, x^{2} - 375 \, x + 16\right )} \sqrt{3 \, x^{2} + 2} + \frac{5}{4} \, \sqrt{3} \log \left (-\sqrt{3} \sqrt{3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/60*(36*x^4 - 225*x^3 + 48*x^2 - 375*x + 16)*sqrt(3*x^2 + 2) + 5/4*sqrt(3)*log
(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1)

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Sympy [A]  time = 10.556, size = 97, normalized size = 1.45 \[ - \frac{3 x^{4} \sqrt{3 x^{2} + 2}}{5} + \frac{15 x^{3} \sqrt{3 x^{2} + 2}}{4} - \frac{4 x^{2} \sqrt{3 x^{2} + 2}}{5} + \frac{25 x \sqrt{3 x^{2} + 2}}{4} - \frac{4 \sqrt{3 x^{2} + 2}}{15} + \frac{5 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*x**4*sqrt(3*x**2 + 2)/5 + 15*x**3*sqrt(3*x**2 + 2)/4 - 4*x**2*sqrt(3*x**2 + 2
)/5 + 25*x*sqrt(3*x**2 + 2)/4 - 4*sqrt(3*x**2 + 2)/15 + 5*sqrt(3)*asinh(sqrt(6)*
x/2)/2

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GIAC/XCAS [A]  time = 0.279346, size = 72, normalized size = 1.07 \[ -\frac{1}{60} \,{\left (3 \,{\left ({\left (3 \,{\left (4 \, x - 25\right )} x + 16\right )} x - 125\right )} x + 16\right )} \sqrt{3 \, x^{2} + 2} - \frac{5}{2} \, \sqrt{3}{\rm ln}\left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/60*(3*((3*(4*x - 25)*x + 16)*x - 125)*x + 16)*sqrt(3*x^2 + 2) - 5/2*sqrt(3)*l
n(-sqrt(3)*x + sqrt(3*x^2 + 2))