\(\int (3 x-4 x^2)^{3/2} \, dx\) [8]

Optimal result

Integrand size = 13, antiderivative size = 57 \[ \int \left (3 x-4 x^2\right )^{3/2} \, dx=-\frac {27 (3-8 x) \sqrt {3 x-4 x^2}}{1024}-\frac {1}{32} (3-8 x) \left (3 x-4 x^2\right )^{3/2}-\frac {243 \arcsin \left (1-\frac {8 x}{3}\right )}{4096} \]

[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)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {626, 633, 222} \[ \int \left (3 x-4 x^2\right )^{3/2} \, dx=-\frac {243 \arcsin \left (1-\frac {8 x}{3}\right )}{4096}-\frac {1}{32} (3-8 x) \left (3 x-4 x^2\right )^{3/2}-\frac {27 (3-8 x) \sqrt {3 x-4 x^2}}{1024} \]

[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 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]

Rubi steps \begin{align*} \text {integral}& = -\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 \text {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] (verified)

Time = 0.11 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.44 \[ \int \left (3 x-4 x^2\right )^{3/2} \, dx=\frac {\sqrt {-x (-3+4 x)} \left (-2 \sqrt {x} \sqrt {-3+4 x} \left (81+72 x-1152 x^2+1024 x^3\right )+243 \log \left (-2 \sqrt {x}+\sqrt {-3+4 x}\right )\right )}{2048 \sqrt {x} \sqrt {-3+4 x}} \]

[In]

Integrate[(3*x - 4*x^2)^(3/2),x]

[Out]

(Sqrt[-(x*(-3 + 4*x))]*(-2*Sqrt[x]*Sqrt[-3 + 4*x]*(81 + 72*x - 1152*x^2 + 1024*x^3) + 243*Log[-2*Sqrt[x] + Sqr
t[-3 + 4*x]]))/(2048*Sqrt[x]*Sqrt[-3 + 4*x])

Maple [A] (verified)

Time = 2.43 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.75

method	result	size
risch	\(\frac {\left (1024 x^{3}-1152 x^{2}+72 x +81\right ) x \left (4 x -3\right )}{1024 \sqrt {-x \left (4 x -3\right )}}+\frac {243 \arcsin \left (-1+\frac {8 x}{3}\right )}{4096}\)	\(43\)
default	\(-\frac {\left (3-8 x \right ) \left (-4 x^{2}+3 x \right )^{\frac {3}{2}}}{32}+\frac {243 \arcsin \left (-1+\frac {8 x}{3}\right )}{4096}-\frac {27 \left (3-8 x \right ) \sqrt {-4 x^{2}+3 x}}{1024}\)	\(46\)
pseudoelliptic	\(-\frac {243 \arctan \left (\frac {\sqrt {-4 x^{2}+3 x}}{2 x}\right )}{2048}+\frac {\left (-1024 x^{3}+1152 x^{2}-72 x -81\right ) \sqrt {-4 x^{2}+3 x}}{1024}\)	\(49\)
meijerg	\(-\frac {243 i \left (-\frac {i \sqrt {\pi }\, \sqrt {x}\, \sqrt {3}\, \left (\frac {5120}{27} x^{3}-\frac {640}{3} x^{2}+\frac {40}{3} x +15\right ) \sqrt {-\frac {4 x}{3}+1}}{360}+\frac {i \sqrt {\pi }\, \arcsin \left (\frac {2 \sqrt {3}\, \sqrt {x}}{3}\right )}{16}\right )}{128 \sqrt {\pi }}\)	\(57\)
trager	\(\left (-x^{3}+\frac {9}{8} x^{2}-\frac {9}{128} x -\frac {81}{1024}\right ) \sqrt {-4 x^{2}+3 x}+\frac {243 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +4 \sqrt {-4 x^{2}+3 x}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right )}{4096}\)	\(69\)

[In]

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

[Out]

1/1024*(1024*x^3-1152*x^2+72*x+81)*x*(4*x-3)/(-x*(4*x-3))^(1/2)+243/4096*arcsin(-1+8/3*x)

Fricas [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.84 \[ \int \left (3 x-4 x^2\right )^{3/2} \, dx=-\frac {1}{1024} \, {\left (1024 \, x^{3} - 1152 \, x^{2} + 72 \, x + 81\right )} \sqrt {-4 \, x^{2} + 3 \, x} - \frac {243}{2048} \, \arctan \left (\frac {\sqrt {-4 \, x^{2} + 3 \, x}}{2 \, x}\right ) \]

[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)

Sympy [A] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.19 \[ \int \left (3 x-4 x^2\right )^{3/2} \, dx=3 \sqrt {- 4 x^{2} + 3 x} \left (\frac {x^{2}}{3} - \frac {x}{16} - \frac {9}{128}\right ) - 4 \sqrt {- 4 x^{2} + 3 x} \left (\frac {x^{3}}{4} - \frac {x^{2}}{32} - \frac {15 x}{512} - \frac {135}{4096}\right ) + \frac {243 \operatorname {asin}{\left (\frac {8 x}{3} - 1 \right )}}{4096} \]

[In]

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

[Out]

3*sqrt(-4*x**2 + 3*x)*(x**2/3 - x/16 - 9/128) - 4*sqrt(-4*x**2 + 3*x)*(x**3/4 - x**2/32 - 15*x/512 - 135/4096)
+ 243*asin(8*x/3 - 1)/4096

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.11 \[ \int \left (3 x-4 x^2\right )^{3/2} \, dx=\frac {1}{4} \, {\left (-4 \, x^{2} + 3 \, x\right )}^{\frac {3}{2}} x - \frac {3}{32} \, {\left (-4 \, x^{2} + 3 \, x\right )}^{\frac {3}{2}} + \frac {27}{128} \, \sqrt {-4 \, x^{2} + 3 \, x} x - \frac {81}{1024} \, \sqrt {-4 \, x^{2} + 3 \, x} - \frac {243}{4096} \, \arcsin \left (-\frac {8}{3} \, x + 1\right ) \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.65 \[ \int \left (3 x-4 x^2\right )^{3/2} \, dx=-\frac {1}{1024} \, {\left (8 \, {\left (16 \, {\left (8 \, x - 9\right )} x + 9\right )} x + 81\right )} \sqrt {-4 \, x^{2} + 3 \, x} + \frac {243}{4096} \, \arcsin \left (\frac {8}{3} \, x - 1\right ) \]

[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)

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.79 \[ \int \left (3 x-4 x^2\right )^{3/2} \, dx=\frac {243\,\mathrm {asin}\left (\frac {8\,x}{3}-1\right )}{4096}+\frac {\left (4\,x-\frac {3}{2}\right )\,{\left (3\,x-4\,x^2\right )}^{3/2}}{16}+\frac {27\,\left (\frac {x}{2}-\frac {3}{16}\right )\,\sqrt {3\,x-4\,x^2}}{64} \]

[In]

int((3*x - 4*x^2)^(3/2),x)

[Out]

(243*asin((8*x)/3 - 1))/4096 + ((4*x - 3/2)*(3*x - 4*x^2)^(3/2))/16 + (27*(x/2 - 3/16)*(3*x - 4*x^2)^(1/2))/64