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

Optimal. Leaf size=51 \[ -\frac{1}{8} (1-2 x) \left (x-x^2\right )^{3/2}-\frac{3}{64} (1-2 x) \sqrt{x-x^2}-\frac{3}{128} \sin ^{-1}(1-2 x) \]

[Out]

(-3*(1 - 2*x)*Sqrt[x - x^2])/64 - ((1 - 2*x)*(x - x^2)^(3/2))/8 - (3*ArcSin[1 - 2*x])/128

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Rubi [A]  time = 0.0101865, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {612, 619, 216} \[ -\frac{1}{8} (1-2 x) \left (x-x^2\right )^{3/2}-\frac{3}{64} (1-2 x) \sqrt{x-x^2}-\frac{3}{128} \sin ^{-1}(1-2 x) \]

Antiderivative was successfully verified.

[In]

Int[(x - x^2)^(3/2),x]

[Out]

(-3*(1 - 2*x)*Sqrt[x - x^2])/64 - ((1 - 2*x)*(x - x^2)^(3/2))/8 - (3*ArcSin[1 - 2*x])/128

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 (x-x^2\right )^{3/2} \, dx &=-\frac{1}{8} (1-2 x) \left (x-x^2\right )^{3/2}+\frac{3}{16} \int \sqrt{x-x^2} \, dx\\ &=-\frac{3}{64} (1-2 x) \sqrt{x-x^2}-\frac{1}{8} (1-2 x) \left (x-x^2\right )^{3/2}+\frac{3}{128} \int \frac{1}{\sqrt{x-x^2}} \, dx\\ &=-\frac{3}{64} (1-2 x) \sqrt{x-x^2}-\frac{1}{8} (1-2 x) \left (x-x^2\right )^{3/2}-\frac{3}{128} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2}} \, dx,x,1-2 x\right )\\ &=-\frac{3}{64} (1-2 x) \sqrt{x-x^2}-\frac{1}{8} (1-2 x) \left (x-x^2\right )^{3/2}-\frac{3}{128} \sin ^{-1}(1-2 x)\\ \end{align*}

Mathematica [A]  time = 0.0669618, size = 44, normalized size = 0.86 \[ \frac{1}{64} \left (-\sqrt{-(x-1) x} \left (16 x^3-24 x^2+2 x+3\right )-3 \sin ^{-1}\left (\sqrt{1-x}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(x - x^2)^(3/2),x]

[Out]

(-(Sqrt[-((-1 + x)*x)]*(3 + 2*x - 24*x^2 + 16*x^3)) - 3*ArcSin[Sqrt[1 - x]])/64

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Maple [A]  time = 0.052, size = 42, normalized size = 0.8 \begin{align*} -{\frac{1-2\,x}{8} \left ( -{x}^{2}+x \right ) ^{{\frac{3}{2}}}}+{\frac{3\,\arcsin \left ( 2\,x-1 \right ) }{128}}-{\frac{3-6\,x}{64}\sqrt{-{x}^{2}+x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^2+x)^(3/2),x)

[Out]

-1/8*(1-2*x)*(-x^2+x)^(3/2)+3/128*arcsin(2*x-1)-3/64*(1-2*x)*(-x^2+x)^(1/2)

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Maxima [A]  time = 2.57555, size = 74, normalized size = 1.45 \begin{align*} \frac{1}{4} \,{\left (-x^{2} + x\right )}^{\frac{3}{2}} x - \frac{1}{8} \,{\left (-x^{2} + x\right )}^{\frac{3}{2}} + \frac{3}{32} \, \sqrt{-x^{2} + x} x - \frac{3}{64} \, \sqrt{-x^{2} + x} + \frac{3}{128} \, \arcsin \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+x)^(3/2),x, algorithm="maxima")

[Out]

1/4*(-x^2 + x)^(3/2)*x - 1/8*(-x^2 + x)^(3/2) + 3/32*sqrt(-x^2 + x)*x - 3/64*sqrt(-x^2 + x) + 3/128*arcsin(2*x
- 1)

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Fricas [A]  time = 2.09505, size = 111, normalized size = 2.18 \begin{align*} -\frac{1}{64} \,{\left (16 \, x^{3} - 24 \, x^{2} + 2 \, x + 3\right )} \sqrt{-x^{2} + x} - \frac{3}{64} \, \arctan \left (\frac{\sqrt{-x^{2} + x}}{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+x)^(3/2),x, algorithm="fricas")

[Out]

-1/64*(16*x^3 - 24*x^2 + 2*x + 3)*sqrt(-x^2 + x) - 3/64*arctan(sqrt(-x^2 + x)/x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x**2+x)**(3/2),x)

[Out]

Integral((-x**2 + x)**(3/2), x)

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Giac [A]  time = 1.37434, size = 47, normalized size = 0.92 \begin{align*} -\frac{1}{64} \,{\left (2 \,{\left (4 \,{\left (2 \, x - 3\right )} x + 1\right )} x + 3\right )} \sqrt{-x^{2} + x} + \frac{3}{128} \, \arcsin \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+x)^(3/2),x, algorithm="giac")

[Out]

-1/64*(2*(4*(2*x - 3)*x + 1)*x + 3)*sqrt(-x^2 + x) + 3/128*arcsin(2*x - 1)