3.1154 \(\int (3-6 x)^{3/2} (2+4 x)^{3/2} \, dx\)

Optimal. Leaf size=74 \[ 3 \sqrt{\frac{3}{2}} (1-2 x)^{3/2} x (2 x+1)^{3/2}+\frac{9}{2} \sqrt{\frac{3}{2}} \sqrt{1-2 x} x \sqrt{2 x+1}+\frac{9}{4} \sqrt{\frac{3}{2}} \sin ^{-1}(2 x) \]

[Out]

(9*Sqrt[3/2]*Sqrt[1 - 2*x]*x*Sqrt[1 + 2*x])/2 + 3*Sqrt[3/2]*(1 - 2*x)^(3/2)*x*(1
+ 2*x)^(3/2) + (9*Sqrt[3/2]*ArcSin[2*x])/4

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Rubi [A]  time = 0.0525844, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ 3 \sqrt{\frac{3}{2}} (1-2 x)^{3/2} x (2 x+1)^{3/2}+\frac{9}{2} \sqrt{\frac{3}{2}} \sqrt{1-2 x} x \sqrt{2 x+1}+\frac{9}{4} \sqrt{\frac{3}{2}} \sin ^{-1}(2 x) \]

Antiderivative was successfully verified.

[In]  Int[(3 - 6*x)^(3/2)*(2 + 4*x)^(3/2),x]

[Out]

(9*Sqrt[3/2]*Sqrt[1 - 2*x]*x*Sqrt[1 + 2*x])/2 + 3*Sqrt[3/2]*(1 - 2*x)^(3/2)*x*(1
+ 2*x)^(3/2) + (9*Sqrt[3/2]*ArcSin[2*x])/4

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Rubi in Sympy [A]  time = 6.53922, size = 54, normalized size = 0.73 \[ \frac{x \left (- 6 x + 3\right )^{\frac{3}{2}} \left (4 x + 2\right )^{\frac{3}{2}}}{4} + \frac{9 x \sqrt{- 6 x + 3} \sqrt{4 x + 2}}{4} + \frac{9 \sqrt{6} \operatorname{asin}{\left (2 x \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3-6*x)**(3/2)*(2+4*x)**(3/2),x)

[Out]

x*(-6*x + 3)**(3/2)*(4*x + 2)**(3/2)/4 + 9*x*sqrt(-6*x + 3)*sqrt(4*x + 2)/4 + 9*
sqrt(6)*asin(2*x)/8

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Mathematica [A]  time = 0.0744172, size = 46, normalized size = 0.62 \[ -\frac{3}{2} \sqrt{\frac{3}{2}} \left (x \sqrt{1-4 x^2} \left (8 x^2-5\right )+3 \sin ^{-1}\left (\sqrt{\frac{1}{2}-x}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(3 - 6*x)^(3/2)*(2 + 4*x)^(3/2),x]

[Out]

(-3*Sqrt[3/2]*(x*Sqrt[1 - 4*x^2]*(-5 + 8*x^2) + 3*ArcSin[Sqrt[1/2 - x]]))/2

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Maple [B]  time = 0.007, size = 102, normalized size = 1.4 \[{\frac{1}{16} \left ( 3-6\,x \right ) ^{{\frac{3}{2}}} \left ( 2+4\,x \right ) ^{{\frac{5}{2}}}}+{\frac{3}{16} \left ( 2+4\,x \right ) ^{{\frac{5}{2}}}\sqrt{3-6\,x}}-{\frac{3}{16} \left ( 2+4\,x \right ) ^{{\frac{3}{2}}}\sqrt{3-6\,x}}-{\frac{9}{8}\sqrt{3-6\,x}\sqrt{2+4\,x}}+{\frac{9\,\arcsin \left ( 2\,x \right ) \sqrt{6}}{8}\sqrt{ \left ( 2+4\,x \right ) \left ( 3-6\,x \right ) }{\frac{1}{\sqrt{3-6\,x}}}{\frac{1}{\sqrt{2+4\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3-6*x)^(3/2)*(2+4*x)^(3/2),x)

[Out]

1/16*(3-6*x)^(3/2)*(2+4*x)^(5/2)+3/16*(2+4*x)^(5/2)*(3-6*x)^(1/2)-3/16*(2+4*x)^(
3/2)*(3-6*x)^(1/2)-9/8*(3-6*x)^(1/2)*(2+4*x)^(1/2)+9/8*((2+4*x)*(3-6*x))^(1/2)/(
2+4*x)^(1/2)/(3-6*x)^(1/2)*6^(1/2)*arcsin(2*x)

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Maxima [A]  time = 1.50349, size = 46, normalized size = 0.62 \[ \frac{1}{4} \,{\left (-24 \, x^{2} + 6\right )}^{\frac{3}{2}} x + \frac{9}{4} \, \sqrt{-24 \, x^{2} + 6} x + \frac{9}{8} \, \sqrt{6} \arcsin \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x + 2)^(3/2)*(-6*x + 3)^(3/2),x, algorithm="maxima")

[Out]

1/4*(-24*x^2 + 6)^(3/2)*x + 9/4*sqrt(-24*x^2 + 6)*x + 9/8*sqrt(6)*arcsin(2*x)

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Fricas [A]  time = 0.211253, size = 86, normalized size = 1.16 \[ -\frac{3}{8} \, \sqrt{2}{\left (\sqrt{2}{\left (8 \, x^{3} - 5 \, x\right )} \sqrt{4 \, x + 2} \sqrt{-6 \, x + 3} + 3 \, \sqrt{3} \arctan \left (\frac{\sqrt{3} \sqrt{2} \sqrt{4 \, x + 2} \sqrt{-6 \, x + 3}}{12 \, x}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x + 2)^(3/2)*(-6*x + 3)^(3/2),x, algorithm="fricas")

[Out]

-3/8*sqrt(2)*(sqrt(2)*(8*x^3 - 5*x)*sqrt(4*x + 2)*sqrt(-6*x + 3) + 3*sqrt(3)*arc
tan(1/12*sqrt(3)*sqrt(2)*sqrt(4*x + 2)*sqrt(-6*x + 3)/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((3-6*x)**(3/2)*(2+4*x)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.231057, size = 103, normalized size = 1.39 \[ -\frac{3}{8} \, \sqrt{3} \sqrt{2}{\left ({\left ({\left (4 \,{\left (2 \, x + 1\right )}{\left (x - 1\right )} + 5\right )}{\left (2 \, x + 1\right )} - 1\right )} \sqrt{2 \, x + 1} \sqrt{-2 \, x + 1} - 8 \, \sqrt{2 \, x + 1} x \sqrt{-2 \, x + 1} - 6 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{2 \, x + 1}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x + 2)^(3/2)*(-6*x + 3)^(3/2),x, algorithm="giac")

[Out]

-3/8*sqrt(3)*sqrt(2)*(((4*(2*x + 1)*(x - 1) + 5)*(2*x + 1) - 1)*sqrt(2*x + 1)*sq
rt(-2*x + 1) - 8*sqrt(2*x + 1)*x*sqrt(-2*x + 1) - 6*arcsin(1/2*sqrt(2)*sqrt(2*x
+ 1)))