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

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 - (243*ArcSin[1 - (8*x)/3])/4096

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Rubi [A]  time = 0.0128725, 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, Rules used = {612, 619, 216} \[ -\frac{1}{32} (3-8 x) \left (3 x-4 x^2\right )^{3/2}-\frac{27 (3-8 x) \sqrt{3 x-4 x^2}}{1024}-\frac{243 \sin ^{-1}\left (1-\frac{8 x}{3}\right )}{4096} \]

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 - (243*ArcSin[1 - (8*x)/3])/4096

Rule 612

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int \left (3 x-4 x^2\right )^{3/2} \, dx &=-\frac{1}{32} (3-8 x) \left (3 x-4 x^2\right )^{3/2}+\frac{27}{64} \int \sqrt{3 x-4 x^2} \, dx\\ &=-\frac{27 (3-8 x) \sqrt{3 x-4 x^2}}{1024}-\frac{1}{32} (3-8 x) \left (3 x-4 x^2\right )^{3/2}+\frac{243 \int \frac{1}{\sqrt{3 x-4 x^2}} \, dx}{2048}\\ &=-\frac{27 (3-8 x) \sqrt{3 x-4 x^2}}{1024}-\frac{1}{32} (3-8 x) \left (3 x-4 x^2\right )^{3/2}-\frac{81 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{9}}} \, dx,x,3-8 x\right )}{4096}\\ &=-\frac{27 (3-8 x) \sqrt{3 x-4 x^2}}{1024}-\frac{1}{32} (3-8 x) \left (3 x-4 x^2\right )^{3/2}-\frac{243 \sin ^{-1}\left (1-\frac{8 x}{3}\right )}{4096}\\ \end{align*}

Mathematica [A]  time = 0.0508758, size = 68, normalized size = 1.19 \[ \frac{2 x \left (4096 x^4-7680 x^3+3744 x^2+108 x-243\right )-243 \sqrt{3-4 x} \sqrt{x} \sin ^{-1}\left (\sqrt{1-\frac{4 x}{3}}\right )}{2048 \sqrt{-x (4 x-3)}} \]

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

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Maple [A]  time = 0.044, size = 46, normalized size = 0.8 \begin{align*} -{\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}} \end{align*}

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 = 1.82384, size = 85, normalized size = 1.49 \begin{align*} \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 ) \end{align*}

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 = 2.18747, size = 143, normalized size = 2.51 \begin{align*} -\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 ) \end{align*}

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. \begin{align*} \int \left (- 4 x^{2} + 3 x\right )^{\frac{3}{2}}\, dx \end{align*}

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 [A]  time = 1.35534, size = 50, normalized size = 0.88 \begin{align*} -\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 ) \end{align*}

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)