3.8 \(\int \left (3 x-4 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=57 \[ -\frac{1}{32} (3-8 x) \left (3 x-4 x^2\right )^{3/2}-\frac{27 (3-8 x) \sqrt{3 x-4 x^2}}{1024}-\frac{243 \sin ^{-1}\left (1-\frac{8 x}{3}\right )}{4096} \]

[Out]

(-27*(3 - 8*x)*Sqrt[3*x - 4*x^2])/1024 - ((3 - 8*x)*(3*x - 4*x^2)^(3/2))/32 - (2
43*ArcSin[1 - (8*x)/3])/4096

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Rubi [A]  time = 0.0331173, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{1}{32} (3-8 x) \left (3 x-4 x^2\right )^{3/2}-\frac{27 (3-8 x) \sqrt{3 x-4 x^2}}{1024}-\frac{243 \sin ^{-1}\left (1-\frac{8 x}{3}\right )}{4096} \]

Antiderivative was successfully verified.

[In]  Int[(3*x - 4*x^2)^(3/2),x]

[Out]

(-27*(3 - 8*x)*Sqrt[3*x - 4*x^2])/1024 - ((3 - 8*x)*(3*x - 4*x^2)^(3/2))/32 - (2
43*ArcSin[1 - (8*x)/3])/4096

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Rubi in Sympy [A]  time = 2.18366, size = 49, normalized size = 0.86 \[ - \frac{\left (- 8 x + 3\right ) \left (- 4 x^{2} + 3 x\right )^{\frac{3}{2}}}{32} - \frac{27 \left (- 8 x + 3\right ) \sqrt{- 4 x^{2} + 3 x}}{1024} + \frac{243 \operatorname{asin}{\left (\frac{8 x}{3} - 1 \right )}}{4096} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-8*x + 3)*(-4*x**2 + 3*x)**(3/2)/32 - 27*(-8*x + 3)*sqrt(-4*x**2 + 3*x)/1024 +
243*asin(8*x/3 - 1)/4096

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Mathematica [A]  time = 0.0714304, size = 74, normalized size = 1.3 \[ \frac{2 x \left (4096 x^4-7680 x^3+3744 x^2+108 x-243\right )+243 \sqrt{x} \sqrt{4 x-3} \log \left (2 \sqrt{x}+\sqrt{4 x-3}\right )}{2048 \sqrt{-x (4 x-3)}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x*(-243 + 108*x + 3744*x^2 - 7680*x^3 + 4096*x^4) + 243*Sqrt[x]*Sqrt[-3 + 4*x
]*Log[2*Sqrt[x] + Sqrt[-3 + 4*x]])/(2048*Sqrt[-(x*(-3 + 4*x))])

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Maple [A]  time = 0.004, size = 46, normalized size = 0.8 \[ -{\frac{3-8\,x}{32} \left ( -4\,{x}^{2}+3\,x \right ) ^{{\frac{3}{2}}}}+{\frac{243}{4096}\arcsin \left ( -1+{\frac{8\,x}{3}} \right ) }-{\frac{81-216\,x}{1024}\sqrt{-4\,{x}^{2}+3\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-4*x^2+3*x)^(3/2),x)

[Out]

-1/32*(3-8*x)*(-4*x^2+3*x)^(3/2)+243/4096*arcsin(-1+8/3*x)-27/1024*(3-8*x)*(-4*x
^2+3*x)^(1/2)

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Maxima [A]  time = 0.793911, size = 85, normalized size = 1.49 \[ \frac{1}{4} \,{\left (-4 \, x^{2} + 3 \, x\right )}^{\frac{3}{2}} x - \frac{3}{32} \,{\left (-4 \, x^{2} + 3 \, x\right )}^{\frac{3}{2}} + \frac{27}{128} \, \sqrt{-4 \, x^{2} + 3 \, x} x - \frac{81}{1024} \, \sqrt{-4 \, x^{2} + 3 \, x} - \frac{243}{4096} \, \arcsin \left (-\frac{8}{3} \, x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(-4*x^2 + 3*x)^(3/2)*x - 3/32*(-4*x^2 + 3*x)^(3/2) + 27/128*sqrt(-4*x^2 + 3*
x)*x - 81/1024*sqrt(-4*x^2 + 3*x) - 243/4096*arcsin(-8/3*x + 1)

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Fricas [A]  time = 0.210289, size = 65, normalized size = 1.14 \[ -\frac{1}{1024} \,{\left (1024 \, x^{3} - 1152 \, x^{2} + 72 \, x + 81\right )} \sqrt{-4 \, x^{2} + 3 \, x} - \frac{243}{2048} \, \arctan \left (\frac{\sqrt{-4 \, x^{2} + 3 \, x}}{2 \, x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1024*(1024*x^3 - 1152*x^2 + 72*x + 81)*sqrt(-4*x^2 + 3*x) - 243/2048*arctan(1
/2*sqrt(-4*x^2 + 3*x)/x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (- 4 x^{2} + 3 x\right )^{\frac{3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((-4*x**2 + 3*x)**(3/2), x)

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GIAC/XCAS [A]  time = 0.216115, size = 50, normalized size = 0.88 \[ -\frac{1}{1024} \,{\left (8 \,{\left (16 \,{\left (8 \, x - 9\right )} x + 9\right )} x + 81\right )} \sqrt{-4 \, x^{2} + 3 \, x} + \frac{243}{4096} \, \arcsin \left (\frac{8}{3} \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1024*(8*(16*(8*x - 9)*x + 9)*x + 81)*sqrt(-4*x^2 + 3*x) + 243/4096*arcsin(8/3
*x - 1)