Integrand size = 19, antiderivative size = 62 \[ \int \sqrt {(2-x) x \left (4-2 x+x^2\right )} \, dx=\frac {1}{3} \sqrt {3-2 (-1+x)^2-(-1+x)^4} (-1+x)+\frac {2 E\left (\arcsin (1-x)\left |-\frac {1}{3}\right .\right )}{\sqrt {3}}-\frac {4 \operatorname {EllipticF}\left (\arcsin (1-x),-\frac {1}{3}\right )}{\sqrt {3}} \]
-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)
Result contains complex when optimal does not.
Time = 18.91 (sec) , antiderivative size = 256, normalized size of antiderivative = 4.13 \[ \int \sqrt {(2-x) x \left (4-2 x+x^2\right )} \, dx=\frac {\sqrt {-x \left (-8+8 x-4 x^2+x^3\right )} \left (\sqrt {\frac {4-2 x+x^2}{x^2}} \left (-4+4 x-3 x^2+x^3\right )+2 \sqrt {2} \left (-i+\sqrt {3}\right ) \sqrt {-\frac {i (-2+x)}{\left (-i+\sqrt {3}\right ) x}} E\left (\arcsin \left (\frac {\sqrt {i+\sqrt {3}-\frac {4 i}{x}}}{\sqrt {2} \sqrt [4]{3}}\right )|\frac {2 \sqrt {3}}{-i+\sqrt {3}}\right )+8 i \sqrt {2} \sqrt {-\frac {i (-2+x)}{\left (-i+\sqrt {3}\right ) x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {i+\sqrt {3}-\frac {4 i}{x}}}{\sqrt {2} \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{-i+\sqrt {3}}\right )\right )}{3 (-2+x) x \sqrt {\frac {4-2 x+x^2}{x^2}}} \]
(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 + Sqrt[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 + Sq rt[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])
Time = 0.22 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {2458, 1404, 27, 1494, 27, 399, 321, 327}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \sqrt {(2-x) x \left (x^2-2 x+4\right )} \, dx\) |
| \(\Big \downarrow \) 2458 |
| \(\displaystyle \int \sqrt {-(x-1)^4-2 (x-1)^2+3}d(x-1)\) |
| \(\Big \downarrow \) 1404 |
| \(\displaystyle \frac {1}{3} \int \frac {2 \left (3-(x-1)^2\right )}{\sqrt {-(x-1)^4-2 (x-1)^2+3}}d(x-1)+\frac {1}{3} \sqrt {-(x-1)^4-2 (x-1)^2+3} (x-1)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{3} \int \frac {3-(x-1)^2}{\sqrt {-(x-1)^4-2 (x-1)^2+3}}d(x-1)+\frac {1}{3} \sqrt {-(x-1)^4-2 (x-1)^2+3} (x-1)\) |
| \(\Big \downarrow \) 1494 |
| \(\displaystyle \frac {4}{3} \int \frac {3-(x-1)^2}{2 \sqrt {1-(x-1)^2} \sqrt {(x-1)^2+3}}d(x-1)+\frac {1}{3} \sqrt {-(x-1)^4-2 (x-1)^2+3} (x-1)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{3} \int \frac {3-(x-1)^2}{\sqrt {1-(x-1)^2} \sqrt {(x-1)^2+3}}d(x-1)+\frac {1}{3} \sqrt {-(x-1)^4-2 (x-1)^2+3} (x-1)\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {2}{3} \left (6 \int \frac {1}{\sqrt {1-(x-1)^2} \sqrt {(x-1)^2+3}}d(x-1)-\int \frac {\sqrt {(x-1)^2+3}}{\sqrt {1-(x-1)^2}}d(x-1)\right )+\frac {1}{3} \sqrt {-(x-1)^4-2 (x-1)^2+3} (x-1)\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {2}{3} \left (-\int \frac {\sqrt {(x-1)^2+3}}{\sqrt {1-(x-1)^2}}d(x-1)-2 \sqrt {3} \operatorname {EllipticF}\left (\arcsin (1-x),-\frac {1}{3}\right )\right )+\frac {1}{3} \sqrt {-(x-1)^4-2 (x-1)^2+3} (x-1)\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {2}{3} \left (\sqrt {3} E\left (\arcsin (1-x)\left |-\frac {1}{3}\right .\right )-2 \sqrt {3} \operatorname {EllipticF}\left (\arcsin (1-x),-\frac {1}{3}\right )\right )+\frac {1}{3} \sqrt {-(x-1)^4-2 (x-1)^2+3} (x-1)\) |
(Sqrt[3 - 2*(-1 + x)^2 - (-1 + x)^4]*(-1 + x))/3 + (2*(Sqrt[3]*EllipticE[A rcSin[1 - x], -1/3] - 2*Sqrt[3]*EllipticF[ArcSin[1 - x], -1/3]))/3
3.8.70.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) ^2]), x_Symbol] :> Simp[f/b Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/b Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr eeQ[{a, b, c, d, e, f}, x] && !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
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] + Simp[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]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*Sqrt[-c] Int[(d + e*x^2)/(Sqr t[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]
Int[(Pn_)^(p_.), x_Symbol] :> With[{S = Coeff[Pn, x, Expon[Pn, x] - 1]/(Exp on[Pn, x]*Coeff[Pn, x, Expon[Pn, x]])}, Subst[Int[ExpandToSum[Pn /. x -> x - S, x]^p, x], x, x + S] /; BinomialQ[Pn /. x -> x - S, x] || (IntegerQ[Exp on[Pn, x]/2] && TrinomialQ[Pn /. x -> x - S, x])] /; FreeQ[p, x] && PolyQ[P n, x] && GtQ[Expon[Pn, x], 2] && NeQ[Coeff[Pn, x, Expon[Pn, x] - 1], 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 931 vs. \(2 (54 ) = 108\).
Time = 1.95 (sec) , antiderivative size = 932, normalized size of antiderivative = 15.03
| method | result | size |
| risch | \(-\frac {\left (x -1\right ) x \left (x -2\right ) \left (x^{2}-2 x +4\right )}{3 \sqrt {-x \left (x -2\right ) \left (x^{2}-2 x +4\right )}}+\frac {8 \left (-1-i \sqrt {3}\right ) \sqrt {\frac {\left (-1+i \sqrt {3}\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 )}}\, F\left (\sqrt {\frac {\left (-1+i \sqrt {3}\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 (-1+i \sqrt {3}\right ) \left (1-i \sqrt {3}\right )}}\right )}{3 \left (-1+i \sqrt {3}\right ) \sqrt {-x \left (x -2\right ) \left (x -1+i \sqrt {3}\right ) \left (x -1-i \sqrt {3}\right )}}-\frac {2 \left (x \left (x -1+i \sqrt {3}\right ) \left (x -1-i \sqrt {3}\right )+2 \left (-1-i \sqrt {3}\right ) \sqrt {\frac {\left (-1+i \sqrt {3}\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 (6+2 i \sqrt {3}\right ) F\left (\sqrt {\frac {\left (-1+i \sqrt {3}\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 (-1+i \sqrt {3}\right ) \left (1-i \sqrt {3}\right )}}\right )}{-2+2 i \sqrt {3}}+\frac {\left (-1+i \sqrt {3}\right ) E\left (\sqrt {\frac {\left (-1+i \sqrt {3}\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 (-1+i \sqrt {3}\right ) \left (1-i \sqrt {3}\right )}}\right )}{2}-\frac {4 \Pi \left (\sqrt {\frac {\left (-1+i \sqrt {3}\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 (-1+i \sqrt {3}\right ) \left (1-i \sqrt {3}\right )}}\right )}{-1+i \sqrt {3}}\right )\right )}{3 \sqrt {-x \left (x -2\right ) \left (x -1+i \sqrt {3}\right ) \left (x -1-i \sqrt {3}\right )}}+\frac {8 \left (-1-i \sqrt {3}\right ) \sqrt {\frac {\left (-1+i \sqrt {3}\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 F\left (\sqrt {\frac {\left (-1+i \sqrt {3}\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 (-1+i \sqrt {3}\right ) \left (1-i \sqrt {3}\right )}}\right )-2 \Pi \left (\sqrt {\frac {\left (-1+i \sqrt {3}\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 (-1+i \sqrt {3}\right ) \left (1-i \sqrt {3}\right )}}\right )\right )}{3 \left (-1+i \sqrt {3}\right ) \sqrt {-x \left (x -2\right ) \left (x -1+i \sqrt {3}\right ) \left (x -1-i \sqrt {3}\right )}}\) | \(932\) |
| default | \(\frac {x \sqrt {-x^{4}+4 x^{3}-8 x^{2}+8 x}}{3}-\frac {\sqrt {-x^{4}+4 x^{3}-8 x^{2}+8 x}}{3}+\frac {8 \left (-1-i \sqrt {3}\right ) \sqrt {\frac {\left (-1+i \sqrt {3}\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 )}}\, F\left (\sqrt {\frac {\left (-1+i \sqrt {3}\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 (-1+i \sqrt {3}\right ) \left (1-i \sqrt {3}\right )}}\right )}{3 \left (-1+i \sqrt {3}\right ) \sqrt {-x \left (x -2\right ) \left (x -1+i \sqrt {3}\right ) \left (x -1-i \sqrt {3}\right )}}+\frac {8 \left (-1-i \sqrt {3}\right ) \sqrt {\frac {\left (-1+i \sqrt {3}\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 F\left (\sqrt {\frac {\left (-1+i \sqrt {3}\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 (-1+i \sqrt {3}\right ) \left (1-i \sqrt {3}\right )}}\right )-2 \Pi \left (\sqrt {\frac {\left (-1+i \sqrt {3}\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 (-1+i \sqrt {3}\right ) \left (1-i \sqrt {3}\right )}}\right )\right )}{3 \left (-1+i \sqrt {3}\right ) \sqrt {-x \left (x -2\right ) \left (x -1+i \sqrt {3}\right ) \left (x -1-i \sqrt {3}\right )}}-\frac {2 \left (x \left (x -1+i \sqrt {3}\right ) \left (x -1-i \sqrt {3}\right )+2 \left (-1-i \sqrt {3}\right ) \sqrt {\frac {\left (-1+i \sqrt {3}\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 (6+2 i \sqrt {3}\right ) F\left (\sqrt {\frac {\left (-1+i \sqrt {3}\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 (-1+i \sqrt {3}\right ) \left (1-i \sqrt {3}\right )}}\right )}{-2+2 i \sqrt {3}}+\frac {\left (-1+i \sqrt {3}\right ) E\left (\sqrt {\frac {\left (-1+i \sqrt {3}\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 (-1+i \sqrt {3}\right ) \left (1-i \sqrt {3}\right )}}\right )}{2}-\frac {4 \Pi \left (\sqrt {\frac {\left (-1+i \sqrt {3}\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 (-1+i \sqrt {3}\right ) \left (1-i \sqrt {3}\right )}}\right )}{-1+i \sqrt {3}}\right )\right )}{3 \sqrt {-x \left (x -2\right ) \left (x -1+i \sqrt {3}\right ) \left (x -1-i \sqrt {3}\right )}}\) | \(946\) |
| elliptic | \(\frac {x \sqrt {-x^{4}+4 x^{3}-8 x^{2}+8 x}}{3}-\frac {\sqrt {-x^{4}+4 x^{3}-8 x^{2}+8 x}}{3}+\frac {8 \left (-1-i \sqrt {3}\right ) \sqrt {\frac {\left (-1+i \sqrt {3}\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 )}}\, F\left (\sqrt {\frac {\left (-1+i \sqrt {3}\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 (-1+i \sqrt {3}\right ) \left (1-i \sqrt {3}\right )}}\right )}{3 \left (-1+i \sqrt {3}\right ) \sqrt {-x \left (x -2\right ) \left (x -1+i \sqrt {3}\right ) \left (x -1-i \sqrt {3}\right )}}+\frac {8 \left (-1-i \sqrt {3}\right ) \sqrt {\frac {\left (-1+i \sqrt {3}\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 F\left (\sqrt {\frac {\left (-1+i \sqrt {3}\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 (-1+i \sqrt {3}\right ) \left (1-i \sqrt {3}\right )}}\right )-2 \Pi \left (\sqrt {\frac {\left (-1+i \sqrt {3}\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 (-1+i \sqrt {3}\right ) \left (1-i \sqrt {3}\right )}}\right )\right )}{3 \left (-1+i \sqrt {3}\right ) \sqrt {-x \left (x -2\right ) \left (x -1+i \sqrt {3}\right ) \left (x -1-i \sqrt {3}\right )}}-\frac {2 \left (x \left (x -1+i \sqrt {3}\right ) \left (x -1-i \sqrt {3}\right )+2 \left (-1-i \sqrt {3}\right ) \sqrt {\frac {\left (-1+i \sqrt {3}\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 (6+2 i \sqrt {3}\right ) F\left (\sqrt {\frac {\left (-1+i \sqrt {3}\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 (-1+i \sqrt {3}\right ) \left (1-i \sqrt {3}\right )}}\right )}{-2+2 i \sqrt {3}}+\frac {\left (-1+i \sqrt {3}\right ) E\left (\sqrt {\frac {\left (-1+i \sqrt {3}\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 (-1+i \sqrt {3}\right ) \left (1-i \sqrt {3}\right )}}\right )}{2}-\frac {4 \Pi \left (\sqrt {\frac {\left (-1+i \sqrt {3}\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 (-1+i \sqrt {3}\right ) \left (1-i \sqrt {3}\right )}}\right )}{-1+i \sqrt {3}}\right )\right )}{3 \sqrt {-x \left (x -2\right ) \left (x -1+i \sqrt {3}\right ) \left (x -1-i \sqrt {3}\right )}}\) | \(946\) |
-1/3*(x-1)*x*(x-2)*(x^2-2*x+4)/(-x*(x-2)*(x^2-2*x+4))^(1/2)+8/3*(-1-I*3^(1 /2))*((-1+I*3^(1/2))*x/(1+I*3^(1/2))/(x-2))^(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+I*3^(1/2))/(-x*(x-2)*(x-1+I*3^(1/2))*(x-1-I*3^(1/2)))^(1/2)*EllipticF((( -1+I*3^(1/2))*x/(1+I*3^(1/2))/(x-2))^(1/2),((1+I*3^(1/2))*(-1-I*3^(1/2))/( -1+I*3^(1/2))/(1-I*3^(1/2)))^(1/2))-2/3*(x*(x-1+I*3^(1/2))*(x-1-I*3^(1/2)) +2*(-1-I*3^(1/2))*((-1+I*3^(1/2))*x/(1+I*3^(1/2))/(x-2))^(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))/(-1+I*3^(1/2))*EllipticF(((-1+I*3^(1/2))*x /(1+I*3^(1/2))/(x-2))^(1/2),((1+I*3^(1/2))*(-1-I*3^(1/2))/(-1+I*3^(1/2))/( 1-I*3^(1/2)))^(1/2))+1/2*(-1+I*3^(1/2))*EllipticE(((-1+I*3^(1/2))*x/(1+I*3 ^(1/2))/(x-2))^(1/2),((1+I*3^(1/2))*(-1-I*3^(1/2))/(-1+I*3^(1/2))/(1-I*3^( 1/2)))^(1/2))-4/(-1+I*3^(1/2))*EllipticPi(((-1+I*3^(1/2))*x/(1+I*3^(1/2))/ (x-2))^(1/2),(-1-I*3^(1/2))/(1-I*3^(1/2)),((1+I*3^(1/2))*(-1-I*3^(1/2))/(- 1+I*3^(1/2))/(1-I*3^(1/2)))^(1/2))))/(-x*(x-2)*(x-1+I*3^(1/2))*(x-1-I*3^(1 /2)))^(1/2)+8/3*(-1-I*3^(1/2))*((-1+I*3^(1/2))*x/(1+I*3^(1/2))/(x-2))^(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+I*3^(1/2))/(-x*(x-2)*(x-1+I*3^(1/2))*(x-1-I*3^ (1/2)))^(1/2)*(2*EllipticF(((-1+I*3^(1/2))*x/(1+I*3^(1/2))/(x-2))^(1/2),(( 1+I*3^(1/2))*(-1-I*3^(1/2))/(-1+I*3^(1/2))/(1-I*3^(1/2)))^(1/2))-2*Elli...
Time = 0.10 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.11 \[ \int \sqrt {(2-x) x \left (4-2 x+x^2\right )} \, dx=-\frac {2 \, {\left (-i \, x + i\right )} E(\arcsin \left (\frac {1}{x - 1}\right )\,|\,-3) + 4 \, {\left (-i \, x + i\right )} F(\arcsin \left (\frac {1}{x - 1}\right )\,|\,-3) - \sqrt {-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x} {\left (x^{2} - 2 \, x + 3\right )}}{3 \, {\left (x - 1\right )}} \]
-1/3*(2*(-I*x + I)*elliptic_e(arcsin(1/(x - 1)), -3) + 4*(-I*x + I)*ellipt ic_f(arcsin(1/(x - 1)), -3) - sqrt(-x^4 + 4*x^3 - 8*x^2 + 8*x)*(x^2 - 2*x + 3))/(x - 1)
\[ \int \sqrt {(2-x) x \left (4-2 x+x^2\right )} \, dx=\int \sqrt {x \left (2 - x\right ) \left (x^{2} - 2 x + 4\right )}\, dx \]
\[ \int \sqrt {(2-x) x \left (4-2 x+x^2\right )} \, dx=\int { \sqrt {-{\left (x^{2} - 2 \, x + 4\right )} {\left (x - 2\right )} x} \,d x } \]
\[ \int \sqrt {(2-x) x \left (4-2 x+x^2\right )} \, dx=\int { \sqrt {-{\left (x^{2} - 2 \, x + 4\right )} {\left (x - 2\right )} x} \,d x } \]
Timed out. \[ \int \sqrt {(2-x) x \left (4-2 x+x^2\right )} \, dx=\int \sqrt {-x\,\left (x-2\right )\,\left (x^2-2\,x+4\right )} \,d x \]