3.770 \(\int \sqrt {(2-x) x (4-2 x+x^2)} \, dx\)

Optimal. Leaf size=62 \[ \frac {1}{3} \sqrt {-(x-1)^4-2 (x-1)^2+3} (x-1)-\frac {4 F\left (\sin ^{-1}(1-x)|-\frac {1}{3}\right )}{\sqrt {3}}+\frac {2 E\left (\sin ^{-1}(1-x)|-\frac {1}{3}\right )}{\sqrt {3}} \]

[Out]

-2/3*EllipticE(-1+x,1/3*I*3^(1/2))*3^(1/2)+4/3*EllipticF(-1+x,1/3*I*3^(1/2))*3^(1/2)+1/3*(-1+x)*(3-2*(-1+x)^2-
(-1+x)^4)^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {1106, 1091, 1180, 524, 424, 419} \[ \frac {1}{3} \sqrt {-(x-1)^4-2 (x-1)^2+3} (x-1)-\frac {4 F\left (\sin ^{-1}(1-x)|-\frac {1}{3}\right )}{\sqrt {3}}+\frac {2 E\left (\sin ^{-1}(1-x)|-\frac {1}{3}\right )}{\sqrt {3}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[(2 - x)*x*(4 - 2*x + x^2)],x]

[Out]

(Sqrt[3 - 2*(-1 + x)^2 - (-1 + x)^4]*(-1 + x))/3 + (2*EllipticE[ArcSin[1 - x], -1/3])/Sqrt[3] - (4*EllipticF[A
rcSin[1 - x], -1/3])/Sqrt[3]

Rule 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rule 524

Int[((e_) + (f_.)*(x_)^(n_))/(Sqrt[(a_) + (b_.)*(x_)^(n_)]*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/
b, Int[Sqrt[a + b*x^n]/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/(Sqrt[a + b*x^n]*Sqrt[c + d*x^n]),
x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&  !(EqQ[n, 2] && ((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[
d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-(b/a), -(d/c)]))))))

Rule 1091

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(x*(a + b*x^2 + c*x^4)^p)/(4*p + 1), x] + Dis
t[(2*p)/(4*p + 1), Int[(2*a + b*x^2)*(a + b*x^2 + c*x^4)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4
*a*c, 0] && GtQ[p, 0] && IntegerQ[2*p]

Rule 1106

Int[(P4_)^(p_), x_Symbol] :> With[{a = Coeff[P4, x, 0], b = Coeff[P4, x, 1], c = Coeff[P4, x, 2], d = Coeff[P4
, x, 3], e = Coeff[P4, x, 4]}, Subst[Int[SimplifyIntegrand[(a + d^4/(256*e^3) - (b*d)/(8*e) + (c - (3*d^2)/(8*
e))*x^2 + e*x^4)^p, x], x], x, d/(4*e) + x] /; EqQ[d^3 - 4*c*d*e + 8*b*e^2, 0] && NeQ[d, 0]] /; FreeQ[p, x] &&
PolyQ[P4, x, 4] && NeQ[p, 2] && NeQ[p, 3]

Rule 1180

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}
, Dist[2*Sqrt[-c], Int[(d + e*x^2)/(Sqrt[b + q + 2*c*x^2]*Sqrt[-b + q - 2*c*x^2]), x], x]] /; FreeQ[{a, b, c,
d, e}, x] && GtQ[b^2 - 4*a*c, 0] && LtQ[c, 0]

Rubi steps

\begin {align*} \int \sqrt {(2-x) x \left (4-2 x+x^2\right )} \, dx &=\operatorname {Subst}\left (\int \sqrt {3-2 x^2-x^4} \, dx,x,-1+x\right )\\ &=\frac {1}{3} \sqrt {3-2 (-1+x)^2-(-1+x)^4} (-1+x)+\frac {1}{3} \operatorname {Subst}\left (\int \frac {6-2 x^2}{\sqrt {3-2 x^2-x^4}} \, dx,x,-1+x\right )\\ &=\frac {1}{3} \sqrt {3-2 (-1+x)^2-(-1+x)^4} (-1+x)+\frac {2}{3} \operatorname {Subst}\left (\int \frac {6-2 x^2}{\sqrt {2-2 x^2} \sqrt {6+2 x^2}} \, dx,x,-1+x\right )\\ &=\frac {1}{3} \sqrt {3-2 (-1+x)^2-(-1+x)^4} (-1+x)-\frac {2}{3} \operatorname {Subst}\left (\int \frac {\sqrt {6+2 x^2}}{\sqrt {2-2 x^2}} \, dx,x,-1+x\right )+8 \operatorname {Subst}\left (\int \frac {1}{\sqrt {2-2 x^2} \sqrt {6+2 x^2}} \, dx,x,-1+x\right )\\ &=\frac {1}{3} \sqrt {3-2 (-1+x)^2-(-1+x)^4} (-1+x)+\frac {2 E\left (\sin ^{-1}(1-x)|-\frac {1}{3}\right )}{\sqrt {3}}-\frac {4 F\left (\sin ^{-1}(1-x)|-\frac {1}{3}\right )}{\sqrt {3}}\\ \end {align*}

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Mathematica [C]  time = 0.84, size = 256, normalized size = 4.13 \[ \frac {\sqrt {-x \left (x^3-4 x^2+8 x-8\right )} \left (\sqrt {\frac {x^2-2 x+4}{x^2}} \left (x^3-3 x^2+4 x-4\right )+8 i \sqrt {2} \sqrt {-\frac {i (x-2)}{\left (\sqrt {3}-i\right ) x}} F\left (\sin ^{-1}\left (\frac {\sqrt {\sqrt {3}+i-\frac {4 i}{x}}}{\sqrt {2} \sqrt [4]{3}}\right )|\frac {2 \sqrt {3}}{-i+\sqrt {3}}\right )+2 \sqrt {2} \left (\sqrt {3}-i\right ) \sqrt {-\frac {i (x-2)}{\left (\sqrt {3}-i\right ) x}} E\left (\sin ^{-1}\left (\frac {\sqrt {\sqrt {3}+i-\frac {4 i}{x}}}{\sqrt {2} \sqrt [4]{3}}\right )|\frac {2 \sqrt {3}}{-i+\sqrt {3}}\right )\right )}{3 (x-2) x \sqrt {\frac {x^2-2 x+4}{x^2}}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sqrt[(2 - x)*x*(4 - 2*x + x^2)],x]

[Out]

(Sqrt[-(x*(-8 + 8*x - 4*x^2 + x^3))]*(Sqrt[(4 - 2*x + x^2)/x^2]*(-4 + 4*x - 3*x^2 + x^3) + 2*Sqrt[2]*(-I + Sqr
t[3])*Sqrt[((-I)*(-2 + x))/((-I + Sqrt[3])*x)]*EllipticE[ArcSin[Sqrt[I + Sqrt[3] - (4*I)/x]/(Sqrt[2]*3^(1/4))]
, (2*Sqrt[3])/(-I + Sqrt[3])] + (8*I)*Sqrt[2]*Sqrt[((-I)*(-2 + x))/((-I + Sqrt[3])*x)]*EllipticF[ArcSin[Sqrt[I
+ Sqrt[3] - (4*I)/x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-I + Sqrt[3])]))/(3*(-2 + x)*x*Sqrt[(4 - 2*x + x^2)/x^2
])

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fricas [F]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(-x^4 + 4*x^3 - 8*x^2 + 8*x), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {-{\left (x^{2} - 2 \, x + 4\right )} {\left (x - 2\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-(x^2 - 2*x + 4)*(x - 2)*x), x)

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maple [B]  time = 0.08, size = 946, normalized size = 15.26 \[ \frac {\sqrt {-x^{4}+4 x^{3}-8 x^{2}+8 x}\, x}{3}+\frac {8 \left (-1-i \sqrt {3}\right ) \sqrt {\frac {\left (i \sqrt {3}-1\right ) x}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}\, \left (x -2\right )^{2} \sqrt {\frac {x -1+i \sqrt {3}}{\left (1-i \sqrt {3}\right ) \left (x -2\right )}}\, \sqrt {\frac {x -1-i \sqrt {3}}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}\, \EllipticF \left (\sqrt {\frac {\left (i \sqrt {3}-1\right ) x}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}, \sqrt {\frac {\left (1+i \sqrt {3}\right ) \left (-1-i \sqrt {3}\right )}{\left (i \sqrt {3}-1\right ) \left (1-i \sqrt {3}\right )}}\right )}{3 \left (i \sqrt {3}-1\right ) \sqrt {-\left (x -2\right ) \left (x -1+i \sqrt {3}\right ) \left (x -1-i \sqrt {3}\right ) x}}-\frac {\sqrt {-x^{4}+4 x^{3}-8 x^{2}+8 x}}{3}+\frac {8 \left (-1-i \sqrt {3}\right ) \sqrt {\frac {\left (i \sqrt {3}-1\right ) x}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}\, \left (x -2\right )^{2} \sqrt {\frac {x -1+i \sqrt {3}}{\left (1-i \sqrt {3}\right ) \left (x -2\right )}}\, \sqrt {\frac {x -1-i \sqrt {3}}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}\, \left (2 \EllipticF \left (\sqrt {\frac {\left (i \sqrt {3}-1\right ) x}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}, \sqrt {\frac {\left (1+i \sqrt {3}\right ) \left (-1-i \sqrt {3}\right )}{\left (i \sqrt {3}-1\right ) \left (1-i \sqrt {3}\right )}}\right )-2 \EllipticPi \left (\sqrt {\frac {\left (i \sqrt {3}-1\right ) x}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}, \frac {1+i \sqrt {3}}{i \sqrt {3}-1}, \sqrt {\frac {\left (1+i \sqrt {3}\right ) \left (-1-i \sqrt {3}\right )}{\left (i \sqrt {3}-1\right ) \left (1-i \sqrt {3}\right )}}\right )\right )}{3 \left (i \sqrt {3}-1\right ) \sqrt {-\left (x -2\right ) \left (x -1+i \sqrt {3}\right ) \left (x -1-i \sqrt {3}\right ) x}}-\frac {2 \left (\left (x -1+i \sqrt {3}\right ) \left (x -1-i \sqrt {3}\right ) x +2 \left (-1-i \sqrt {3}\right ) \sqrt {\frac {\left (i \sqrt {3}-1\right ) x}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}\, \left (x -2\right )^{2} \sqrt {\frac {x -1+i \sqrt {3}}{\left (1-i \sqrt {3}\right ) \left (x -2\right )}}\, \sqrt {\frac {x -1-i \sqrt {3}}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}\, \left (\frac {\left (i \sqrt {3}-1\right ) \EllipticE \left (\sqrt {\frac {\left (i \sqrt {3}-1\right ) x}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}, \sqrt {\frac {\left (1+i \sqrt {3}\right ) \left (-1-i \sqrt {3}\right )}{\left (i \sqrt {3}-1\right ) \left (1-i \sqrt {3}\right )}}\right )}{2}+\frac {\left (6+2 i \sqrt {3}\right ) \EllipticF \left (\sqrt {\frac {\left (i \sqrt {3}-1\right ) x}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}, \sqrt {\frac {\left (1+i \sqrt {3}\right ) \left (-1-i \sqrt {3}\right )}{\left (i \sqrt {3}-1\right ) \left (1-i \sqrt {3}\right )}}\right )}{2 i \sqrt {3}-2}-\frac {4 \EllipticPi \left (\sqrt {\frac {\left (i \sqrt {3}-1\right ) x}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}, \frac {-1-i \sqrt {3}}{1-i \sqrt {3}}, \sqrt {\frac {\left (1+i \sqrt {3}\right ) \left (-1-i \sqrt {3}\right )}{\left (i \sqrt {3}-1\right ) \left (1-i \sqrt {3}\right )}}\right )}{i \sqrt {3}-1}\right )\right )}{3 \sqrt {-\left (x -2\right ) \left (x -1+i \sqrt {3}\right ) \left (x -1-i \sqrt {3}\right ) x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(-x^4+4*x^3-8*x^2+8*x)^(1/2)*x-1/3*(-x^4+4*x^3-8*x^2+8*x)^(1/2)+8/3*(-1-I*3^(1/2))*((I*3^(1/2)-1)/(1+I*3^(
1/2))/(x-2)*x)^(1/2)*(x-2)^2*((x-1+I*3^(1/2))/(1-I*3^(1/2))/(x-2))^(1/2)*((x-1-I*3^(1/2))/(1+I*3^(1/2))/(x-2))
^(1/2)/(I*3^(1/2)-1)/(-(x-2)*(x-1+I*3^(1/2))*(x-1-I*3^(1/2))*x)^(1/2)*EllipticF(((I*3^(1/2)-1)/(1+I*3^(1/2))/(
x-2)*x)^(1/2),((1+I*3^(1/2))*(-1-I*3^(1/2))/(I*3^(1/2)-1)/(1-I*3^(1/2)))^(1/2))+8/3*(-1-I*3^(1/2))*((I*3^(1/2)
-1)/(1+I*3^(1/2))/(x-2)*x)^(1/2)*(x-2)^2*((x-1+I*3^(1/2))/(1-I*3^(1/2))/(x-2))^(1/2)*((x-1-I*3^(1/2))/(1+I*3^(
1/2))/(x-2))^(1/2)/(I*3^(1/2)-1)/(-(x-2)*(x-1+I*3^(1/2))*(x-1-I*3^(1/2))*x)^(1/2)*(2*EllipticF(((I*3^(1/2)-1)/
(1+I*3^(1/2))/(x-2)*x)^(1/2),((1+I*3^(1/2))*(-1-I*3^(1/2))/(I*3^(1/2)-1)/(1-I*3^(1/2)))^(1/2))-2*EllipticPi(((
I*3^(1/2)-1)/(1+I*3^(1/2))/(x-2)*x)^(1/2),(1+I*3^(1/2))/(I*3^(1/2)-1),((1+I*3^(1/2))*(-1-I*3^(1/2))/(I*3^(1/2)
-1)/(1-I*3^(1/2)))^(1/2)))-2/3*((x-1+I*3^(1/2))*(x-1-I*3^(1/2))*x+2*(-1-I*3^(1/2))*((I*3^(1/2)-1)/(1+I*3^(1/2)
)/(x-2)*x)^(1/2)*(x-2)^2*((x-1+I*3^(1/2))/(1-I*3^(1/2))/(x-2))^(1/2)*((x-1-I*3^(1/2))/(1+I*3^(1/2))/(x-2))^(1/
2)*(1/2*(6+2*I*3^(1/2))/(I*3^(1/2)-1)*EllipticF(((I*3^(1/2)-1)/(1+I*3^(1/2))/(x-2)*x)^(1/2),((1+I*3^(1/2))*(-1
-I*3^(1/2))/(I*3^(1/2)-1)/(1-I*3^(1/2)))^(1/2))+1/2*(I*3^(1/2)-1)*EllipticE(((I*3^(1/2)-1)/(1+I*3^(1/2))/(x-2)
*x)^(1/2),((1+I*3^(1/2))*(-1-I*3^(1/2))/(I*3^(1/2)-1)/(1-I*3^(1/2)))^(1/2))-4/(I*3^(1/2)-1)*EllipticPi(((I*3^(
1/2)-1)/(1+I*3^(1/2))/(x-2)*x)^(1/2),(-1-I*3^(1/2))/(1-I*3^(1/2)),((1+I*3^(1/2))*(-1-I*3^(1/2))/(I*3^(1/2)-1)/
(1-I*3^(1/2)))^(1/2))))/(-(x-2)*(x-1+I*3^(1/2))*(x-1-I*3^(1/2))*x)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {-{\left (x^{2} - 2 \, x + 4\right )} {\left (x - 2\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-(x^2 - 2*x + 4)*(x - 2)*x), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \sqrt {-x\,\left (x-2\right )\,\left (x^2-2\,x+4\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {x \left (2 - x\right ) \left (x^{2} - 2 x + 4\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(x*(2 - x)*(x**2 - 2*x + 4)), x)

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