\(\int x \sqrt {3 x-4 x^2} \, dx\) [40]

Optimal result

Integrand size = 15, antiderivative size = 52 \[ \int x \sqrt {3 x-4 x^2} \, dx=-\frac {3}{128} (3-8 x) \sqrt {3 x-4 x^2}-\frac {1}{12} \left (3 x-4 x^2\right )^{3/2}-\frac {27}{512} \arcsin \left (1-\frac {8 x}{3}\right ) \]

[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)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {654, 626, 633, 222} \[ \int x \sqrt {3 x-4 x^2} \, dx=-\frac {27}{512} \arcsin \left (1-\frac {8 x}{3}\right )-\frac {1}{12} \left (3 x-4 x^2\right )^{3/2}-\frac {3}{128} (3-8 x) \sqrt {3 x-4 x^2} \]

[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 222

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 626

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 633

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 654

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*} \text {integral}& = -\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} \text {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] (verified)

Time = 0.10 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.44 \[ \int x \sqrt {3 x-4 x^2} \, dx=\frac {1}{384} \sqrt {-x (-3+4 x)} \left (-27-24 x+128 x^2\right )+\frac {27 \sqrt {-x (-3+4 x)} \log \left (-2 \sqrt {x}+\sqrt {-3+4 x}\right )}{256 \sqrt {x} \sqrt {-3+4 x}} \]

[In]

Integrate[x*Sqrt[3*x - 4*x^2],x]

[Out]

(Sqrt[-(x*(-3 + 4*x))]*(-27 - 24*x + 128*x^2))/384 + (27*Sqrt[-(x*(-3 + 4*x))]*Log[-2*Sqrt[x] + Sqrt[-3 + 4*x]
])/(256*Sqrt[x]*Sqrt[-3 + 4*x])

Maple [A] (verified)

Time = 2.10 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.73

method	result	size
risch	\(-\frac {\left (128 x^{2}-24 x -27\right ) x \left (4 x -3\right )}{384 \sqrt {-x \left (4 x -3\right )}}+\frac {27 \arcsin \left (-1+\frac {8 x}{3}\right )}{512}\)	\(38\)
default	\(-\frac {\left (-4 x^{2}+3 x \right )^{\frac {3}{2}}}{12}+\frac {27 \arcsin \left (-1+\frac {8 x}{3}\right )}{512}-\frac {3 \left (3-8 x \right ) \sqrt {-4 x^{2}+3 x}}{128}\)	\(41\)
pseudoelliptic	\(-\frac {27 \arctan \left (\frac {\sqrt {-4 x^{2}+3 x}}{2 x}\right )}{256}+\frac {\left (256 x^{2}-48 x -54\right ) \sqrt {-4 x^{2}+3 x}}{768}\)	\(44\)
meijerg	\(\frac {27 i \left (\frac {i \sqrt {\pi }\, \sqrt {x}\, \sqrt {3}\, \left (-\frac {640}{9} x^{2}+\frac {40}{3} x +15\right ) \sqrt {-\frac {4 x}{3}+1}}{90}-\frac {i \sqrt {\pi }\, \arcsin \left (\frac {2 \sqrt {3}\, \sqrt {x}}{3}\right )}{4}\right )}{64 \sqrt {\pi }}\)	\(52\)
trager	\(\left (\frac {1}{3} x^{2}-\frac {1}{16} x -\frac {9}{128}\right ) \sqrt {-4 x^{2}+3 x}+\frac {27 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+4 \sqrt {-4 x^{2}+3 x}\right )}{512}\)	\(64\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.83 \[ \int x \sqrt {3 x-4 x^2} \, dx=\frac {1}{384} \, {\left (128 \, x^{2} - 24 \, x - 27\right )} \sqrt {-4 \, x^{2} + 3 \, x} - \frac {27}{256} \, \arctan \left (\frac {\sqrt {-4 \, x^{2} + 3 \, x}}{2 \, x}\right ) \]

[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)

Sympy [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.65 \[ \int x \sqrt {3 x-4 x^2} \, dx=\sqrt {- 4 x^{2} + 3 x} \left (\frac {x^{2}}{3} - \frac {x}{16} - \frac {9}{128}\right ) + \frac {27 \operatorname {asin}{\left (\frac {8 x}{3} - 1 \right )}}{512} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.94 \[ \int x \sqrt {3 x-4 x^2} \, dx=-\frac {1}{12} \, {\left (-4 \, x^{2} + 3 \, x\right )}^{\frac {3}{2}} + \frac {3}{16} \, \sqrt {-4 \, x^{2} + 3 \, x} x - \frac {9}{128} \, \sqrt {-4 \, x^{2} + 3 \, x} - \frac {27}{512} \, \arcsin \left (-\frac {8}{3} \, x + 1\right ) \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.62 \[ \int x \sqrt {3 x-4 x^2} \, dx=\frac {1}{384} \, {\left (8 \, {\left (16 \, x - 3\right )} x - 27\right )} \sqrt {-4 \, x^{2} + 3 \, x} + \frac {27}{512} \, \arcsin \left (\frac {8}{3} \, x - 1\right ) \]

[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)

Mupad [B] (verification not implemented)

Time = 9.11 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.85 \[ \int x \sqrt {3 x-4 x^2} \, dx=-\frac {\sqrt {3\,x-4\,x^2}\,\left (-128\,x^2+24\,x+27\right )}{384}-\frac {\ln \left (x-\frac {3}{8}-\frac {\sqrt {-x\,\left (4\,x-3\right )}\,1{}\mathrm {i}}{2}\right )\,27{}\mathrm {i}}{512} \]

[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