### 3.40 $$\int x \sqrt{3 x-4 x^2} \, dx$$

Optimal. Leaf size=52 $-\frac{1}{12} \left (3 x-4 x^2\right )^{3/2}-\frac{3}{128} (3-8 x) \sqrt{3 x-4 x^2}-\frac{27}{512} \sin ^{-1}\left (1-\frac{8 x}{3}\right )$

[Out]

(-3*(3 - 8*x)*Sqrt[3*x - 4*x^2])/128 - (3*x - 4*x^2)^(3/2)/12 - (27*ArcSin[1 - (8*x)/3])/512

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Rubi [A]  time = 0.0145858, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.267, Rules used = {640, 612, 619, 216} $-\frac{1}{12} \left (3 x-4 x^2\right )^{3/2}-\frac{3}{128} (3-8 x) \sqrt{3 x-4 x^2}-\frac{27}{512} \sin ^{-1}\left (1-\frac{8 x}{3}\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[x*Sqrt[3*x - 4*x^2],x]

[Out]

(-3*(3 - 8*x)*Sqrt[3*x - 4*x^2])/128 - (3*x - 4*x^2)^(3/2)/12 - (27*ArcSin[1 - (8*x)/3])/512

Rule 640

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

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 x \sqrt{3 x-4 x^2} \, dx &=-\frac{1}{12} \left (3 x-4 x^2\right )^{3/2}+\frac{3}{8} \int \sqrt{3 x-4 x^2} \, dx\\ &=-\frac{3}{128} (3-8 x) \sqrt{3 x-4 x^2}-\frac{1}{12} \left (3 x-4 x^2\right )^{3/2}+\frac{27}{256} \int \frac{1}{\sqrt{3 x-4 x^2}} \, dx\\ &=-\frac{3}{128} (3-8 x) \sqrt{3 x-4 x^2}-\frac{1}{12} \left (3 x-4 x^2\right )^{3/2}-\frac{9}{512} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{9}}} \, dx,x,3-8 x\right )\\ &=-\frac{3}{128} (3-8 x) \sqrt{3 x-4 x^2}-\frac{1}{12} \left (3 x-4 x^2\right )^{3/2}-\frac{27}{512} \sin ^{-1}\left (1-\frac{8 x}{3}\right )\\ \end{align*}

Mathematica [A]  time = 0.0434005, size = 63, normalized size = 1.21 $\frac{2 x \left (-512 x^3+480 x^2+36 x-81\right )-81 \sqrt{3-4 x} \sqrt{x} \sin ^{-1}\left (\sqrt{1-\frac{4 x}{3}}\right )}{768 \sqrt{-x (4 x-3)}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x*Sqrt[3*x - 4*x^2],x]

[Out]

(2*x*(-81 + 36*x + 480*x^2 - 512*x^3) - 81*Sqrt[3 - 4*x]*Sqrt[x]*ArcSin[Sqrt[1 - (4*x)/3]])/(768*Sqrt[-(x*(-3
+ 4*x))])

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Maple [A]  time = 0.044, size = 41, normalized size = 0.8 \begin{align*} -{\frac{1}{12} \left ( -4\,{x}^{2}+3\,x \right ) ^{{\frac{3}{2}}}}+{\frac{27}{512}\arcsin \left ( -1+{\frac{8\,x}{3}} \right ) }-{\frac{9-24\,x}{128}\sqrt{-4\,{x}^{2}+3\,x}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x*(-4*x^2+3*x)^(1/2),x)

[Out]

-1/12*(-4*x^2+3*x)^(3/2)+27/512*arcsin(-1+8/3*x)-3/128*(3-8*x)*(-4*x^2+3*x)^(1/2)

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Maxima [A]  time = 1.81818, size = 66, normalized size = 1.27 \begin{align*} -\frac{1}{12} \,{\left (-4 \, x^{2} + 3 \, x\right )}^{\frac{3}{2}} + \frac{3}{16} \, \sqrt{-4 \, x^{2} + 3 \, x} x - \frac{9}{128} \, \sqrt{-4 \, x^{2} + 3 \, x} - \frac{27}{512} \, \arcsin \left (-\frac{8}{3} \, x + 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-4*x^2+3*x)^(1/2),x, algorithm="maxima")

[Out]

-1/12*(-4*x^2 + 3*x)^(3/2) + 3/16*sqrt(-4*x^2 + 3*x)*x - 9/128*sqrt(-4*x^2 + 3*x) - 27/512*arcsin(-8/3*x + 1)

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Fricas [A]  time = 1.96582, size = 122, normalized size = 2.35 \begin{align*} \frac{1}{384} \,{\left (128 \, x^{2} - 24 \, x - 27\right )} \sqrt{-4 \, x^{2} + 3 \, x} - \frac{27}{256} \, \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(x*(-4*x^2+3*x)^(1/2),x, algorithm="fricas")

[Out]

1/384*(128*x^2 - 24*x - 27)*sqrt(-4*x^2 + 3*x) - 27/256*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 x \sqrt{- x \left (4 x - 3\right )}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-4*x**2+3*x)**(1/2),x)

[Out]

Integral(x*sqrt(-x*(4*x - 3)), x)

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Giac [A]  time = 1.21139, size = 43, normalized size = 0.83 \begin{align*} \frac{1}{384} \,{\left (8 \,{\left (16 \, x - 3\right )} x - 27\right )} \sqrt{-4 \, x^{2} + 3 \, x} + \frac{27}{512} \, \arcsin \left (\frac{8}{3} \, x - 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-4*x^2+3*x)^(1/2),x, algorithm="giac")

[Out]

1/384*(8*(16*x - 3)*x - 27)*sqrt(-4*x^2 + 3*x) + 27/512*arcsin(8/3*x - 1)