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]

-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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Rubi [A]  time = 0.01, antiderivative size = 52, normalized size of antiderivative = 1.00, 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 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]

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 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]

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*}

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Mathematica [A]  time = 0.04, 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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fricas [A]  time = 0.83, size = 43, normalized size = 0.83 \[ \frac {1}{384} \, {\left (128 \, x^{2} - 24 \, x - 27\right )} \sqrt {-4 \, x^{2} + 3 \, x} - \frac {27}{256} \, \arctan \left (\frac {\sqrt {-4 \, x^{2} + 3 \, x}}{2 \, x}\right ) \]

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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giac [A]  time = 0.29, size = 32, normalized size = 0.62 \[ \frac {1}{384} \, {\left (8 \, {\left (16 \, x - 3\right )} x - 27\right )} \sqrt {-4 \, x^{2} + 3 \, x} + \frac {27}{512} \, \arcsin \left (\frac {8}{3} \, x - 1\right ) \]

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)

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maple [A]  time = 0.04, size = 41, normalized size = 0.79 \[ \frac {27 \arcsin \left (\frac {8 x}{3}-1\right )}{512}-\frac {\left (-4 x^{2}+3 x \right )^{\frac {3}{2}}}{12}-\frac {3 \left (-8 x +3\right ) \sqrt {-4 x^{2}+3 x}}{128} \]

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(8/3*x-1)-3/128*(-8*x+3)*(-4*x^2+3*x)^(1/2)

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maxima [A]  time = 2.95, size = 49, normalized size = 0.94 \[ -\frac {1}{12} \, {\left (-4 \, x^{2} + 3 \, x\right )}^{\frac {3}{2}} + \frac {3}{16} \, \sqrt {-4 \, x^{2} + 3 \, x} x - \frac {9}{128} \, \sqrt {-4 \, x^{2} + 3 \, x} - \frac {27}{512} \, \arcsin \left (-\frac {8}{3} \, x + 1\right ) \]

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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mupad [B]  time = 0.20, size = 44, normalized size = 0.85 \[ -\frac {\sqrt {3\,x-4\,x^2}\,\left (-128\,x^2+24\,x+27\right )}{384}-\frac {\ln \left (x-\frac {3}{8}-\frac {\sqrt {-x\,\left (4\,x-3\right )}\,1{}\mathrm {i}}{2}\right )\,27{}\mathrm {i}}{512} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(3*x - 4*x^2)^(1/2),x)

[Out]

- (log(x - ((-x*(4*x - 3))^(1/2)*1i)/2 - 3/8)*27i)/512 - ((3*x - 4*x^2)^(1/2)*(24*x - 128*x^2 + 27))/384

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \sqrt {- x \left (4 x - 3\right )}\, dx \]

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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