\(\int ((2-x) x (4-2 x+x^2))^{3/2} \, dx\) [55]

Optimal result

Integrand size = 19, antiderivative size = 116 \[ \int \left ((2-x) x \left (4-2 x+x^2\right )\right )^{3/2} \, dx=-\frac {2}{35} \left (13-3 (1-x)^2\right ) \sqrt {3-2 (1-x)^2-(1-x)^4} (1-x)-\frac {1}{7} \left (3-2 (1-x)^2-(1-x)^4\right )^{3/2} (1-x)+\frac {16}{5} \sqrt {3} E\left (\arcsin (1-x)\left |-\frac {1}{3}\right .\right )-\frac {176}{35} \sqrt {3} \operatorname {EllipticF}\left (\arcsin (1-x),-\frac {1}{3}\right ) \] Output:

-2/35*(13-3*(1-x)^2)*(3-2*(1-x)^2-(1-x)^4)^(1/2)*(1-x)-1/7*(3-2*(1-x)^2-(1 
-x)^4)^(3/2)*(1-x)-16/5*3^(1/2)*EllipticE(-1+x,1/3*I*3^(1/2))+176/35*3^(1/ 
2)*EllipticF(-1+x,1/3*I*3^(1/2))
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 27.18 (sec) , antiderivative size = 278, normalized size of antiderivative = 2.40 \[ \int \left ((2-x) x \left (4-2 x+x^2\right )\right )^{3/2} \, dx=\frac {\sqrt {-x \left (-8+8 x-4 x^2+x^3\right )} \left (\sqrt {\frac {4-2 x+x^2}{x^2}} \left (-224+152 x+44 x^2-228 x^3+230 x^4-116 x^5+35 x^6-5 x^7\right )+112 \sqrt {2} \left (-i+\sqrt {3}\right ) \sqrt {-\frac {i (-2+x)}{\left (-i+\sqrt {3}\right ) x}} E\left (\arcsin \left (\frac {\sqrt {i+\sqrt {3}-\frac {4 i}{x}}}{\sqrt {2} \sqrt [4]{3}}\right )|\frac {2 \sqrt {3}}{-i+\sqrt {3}}\right )+304 i \sqrt {2} \sqrt {-\frac {i (-2+x)}{\left (-i+\sqrt {3}\right ) x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {i+\sqrt {3}-\frac {4 i}{x}}}{\sqrt {2} \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{-i+\sqrt {3}}\right )\right )}{35 (-2+x) x \sqrt {\frac {4-2 x+x^2}{x^2}}} \] Input:

Integrate[((2 - x)*x*(4 - 2*x + x^2))^(3/2),x]
 

Output:

(Sqrt[-(x*(-8 + 8*x - 4*x^2 + x^3))]*(Sqrt[(4 - 2*x + x^2)/x^2]*(-224 + 15 
2*x + 44*x^2 - 228*x^3 + 230*x^4 - 116*x^5 + 35*x^6 - 5*x^7) + 112*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]) 
] + (304*I)*Sqrt[2]*Sqrt[((-I)*(-2 + x))/((-I + Sqrt[3])*x)]*EllipticF[Arc 
Sin[Sqrt[I + Sqrt[3] - (4*I)/x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-I + Sqrt 
[3])]))/(35*(-2 + x)*x*Sqrt[(4 - 2*x + x^2)/x^2])
 

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.93, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {2458, 1404, 27, 1490, 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.

\(\displaystyle \int \left ((2-x) x \left (x^2-2 x+4\right )\right )^{3/2} \, dx\)

\(\Big \downarrow \) 2458

\(\displaystyle \int \left (-(x-1)^4-2 (x-1)^2+3\right )^{3/2}d(x-1)\)

\(\Big \downarrow \) 1404

\(\displaystyle \frac {3}{7} \int 2 \left (3-(x-1)^2\right ) \sqrt {-(x-1)^4-2 (x-1)^2+3}d(x-1)+\frac {1}{7} (x-1) \left (-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {6}{7} \int \left (3-(x-1)^2\right ) \sqrt {-(x-1)^4-2 (x-1)^2+3}d(x-1)+\frac {1}{7} (x-1) \left (-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\)

\(\Big \downarrow \) 1490

\(\displaystyle \frac {6}{7} \left (\frac {1}{15} \left (13-3 (x-1)^2\right ) \sqrt {-(x-1)^4-2 (x-1)^2+3} (x-1)-\frac {1}{15} \int -\frac {8 \left (12-7 (x-1)^2\right )}{\sqrt {-(x-1)^4-2 (x-1)^2+3}}d(x-1)\right )+\frac {1}{7} (x-1) \left (-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {6}{7} \left (\frac {8}{15} \int \frac {12-7 (x-1)^2}{\sqrt {-(x-1)^4-2 (x-1)^2+3}}d(x-1)+\frac {1}{15} \left (13-3 (x-1)^2\right ) \sqrt {-(x-1)^4-2 (x-1)^2+3} (x-1)\right )+\frac {1}{7} (x-1) \left (-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\)

\(\Big \downarrow \) 1494

\(\displaystyle \frac {6}{7} \left (\frac {16}{15} \int \frac {12-7 (x-1)^2}{2 \sqrt {1-(x-1)^2} \sqrt {(x-1)^2+3}}d(x-1)+\frac {1}{15} \left (13-3 (x-1)^2\right ) \sqrt {-(x-1)^4-2 (x-1)^2+3} (x-1)\right )+\frac {1}{7} (x-1) \left (-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {6}{7} \left (\frac {8}{15} \int \frac {12-7 (x-1)^2}{\sqrt {1-(x-1)^2} \sqrt {(x-1)^2+3}}d(x-1)+\frac {1}{15} \left (13-3 (x-1)^2\right ) \sqrt {-(x-1)^4-2 (x-1)^2+3} (x-1)\right )+\frac {1}{7} (x-1) \left (-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {6}{7} \left (\frac {8}{15} \left (33 \int \frac {1}{\sqrt {1-(x-1)^2} \sqrt {(x-1)^2+3}}d(x-1)-7 \int \frac {\sqrt {(x-1)^2+3}}{\sqrt {1-(x-1)^2}}d(x-1)\right )+\frac {1}{15} \left (13-3 (x-1)^2\right ) \sqrt {-(x-1)^4-2 (x-1)^2+3} (x-1)\right )+\frac {1}{7} (x-1) \left (-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {6}{7} \left (\frac {8}{15} \left (-7 \int \frac {\sqrt {(x-1)^2+3}}{\sqrt {1-(x-1)^2}}d(x-1)-11 \sqrt {3} \operatorname {EllipticF}\left (\arcsin (1-x),-\frac {1}{3}\right )\right )+\frac {1}{15} \left (13-3 (x-1)^2\right ) \sqrt {-(x-1)^4-2 (x-1)^2+3} (x-1)\right )+\frac {1}{7} (x-1) \left (-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {6}{7} \left (\frac {8}{15} \left (7 \sqrt {3} E\left (\arcsin (1-x)\left |-\frac {1}{3}\right .\right )-11 \sqrt {3} \operatorname {EllipticF}\left (\arcsin (1-x),-\frac {1}{3}\right )\right )+\frac {1}{15} \left (13-3 (x-1)^2\right ) \sqrt {-(x-1)^4-2 (x-1)^2+3} (x-1)\right )+\frac {1}{7} (x-1) \left (-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\)

Input:

Int[((2 - x)*x*(4 - 2*x + x^2))^(3/2),x]
 

Output:

((3 - 2*(-1 + x)^2 - (-1 + x)^4)^(3/2)*(-1 + x))/7 + (6*(((13 - 3*(-1 + x) 
^2)*Sqrt[3 - 2*(-1 + x)^2 - (-1 + x)^4]*(-1 + x))/15 + (8*(7*Sqrt[3]*Ellip 
ticE[ArcSin[1 - x], -1/3] - 11*Sqrt[3]*EllipticF[ArcSin[1 - x], -1/3]))/15 
))/7
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 321

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])
 

rule 327

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]
 

rule 399

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])))))
 

rule 1404

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]
 

rule 1490

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb 
ol] :> Simp[x*(2*b*e*p + c*d*(4*p + 3) + c*e*(4*p + 1)*x^2)*((a + b*x^2 + c 
*x^4)^p/(c*(4*p + 1)*(4*p + 3))), x] + Simp[2*(p/(c*(4*p + 1)*(4*p + 3))) 
 Int[Simp[2*a*c*d*(4*p + 3) - a*b*e + (2*a*c*e*(4*p + 1) + b*c*d*(4*p + 3) 
- b^2*e*(2*p + 1))*x^2, x]*(a + b*x^2 + c*x^4)^(p - 1), x], x] /; FreeQ[{a, 
 b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && 
 GtQ[p, 0] && FractionQ[p] && IntegerQ[2*p]
 

rule 1494

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]
 

rule 2458

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]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 953 vs. \(2 (100 ) = 200\).

Time = 1.59 (sec) , antiderivative size = 954, normalized size of antiderivative = 8.22

method	result	size
risch	\(\text {Expression too large to display}\)	\(954\)
default	\(\text {Expression too large to display}\)	\(1050\)
elliptic	\(\text {Expression too large to display}\)	\(1050\)

Input:

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

Output:

1/35*(5*x^5-25*x^4+66*x^3-98*x^2+32*x+20)*x*(x-2)*(x^2-2*x+4)/(-x*(x-2)*(x 
^2-2*x+4))^(1/2)+32/7*(-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))+64/5*( 
-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*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)))-16/5*(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))*Ellipti 
cF(((-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^...
                                                                                    
                                                                                    
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.78 \[ \int \left ((2-x) x \left (4-2 x+x^2\right )\right )^{3/2} \, dx=-\frac {112 \, {\left (-i \, x + i\right )} E(\arcsin \left (\frac {1}{x - 1}\right )\,|\,-3) + 80 \, {\left (-i \, x + i\right )} F(\arcsin \left (\frac {1}{x - 1}\right )\,|\,-3) + {\left (5 \, x^{6} - 30 \, x^{5} + 91 \, x^{4} - 164 \, x^{3} + 130 \, x^{2} - 12 \, x - 132\right )} \sqrt {-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x}}{35 \, {\left (x - 1\right )}} \] Input:

integrate(((2-x)*x*(x^2-2*x+4))^(3/2),x, algorithm="fricas")
 

Output:

-1/35*(112*(-I*x + I)*elliptic_e(arcsin(1/(x - 1)), -3) + 80*(-I*x + I)*el 
liptic_f(arcsin(1/(x - 1)), -3) + (5*x^6 - 30*x^5 + 91*x^4 - 164*x^3 + 130 
*x^2 - 12*x - 132)*sqrt(-x^4 + 4*x^3 - 8*x^2 + 8*x))/(x - 1)
 

Sympy [F]

\[ \int \left ((2-x) x \left (4-2 x+x^2\right )\right )^{3/2} \, dx=\int \left (x \left (2 - x\right ) \left (x^{2} - 2 x + 4\right )\right )^{\frac {3}{2}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \left ((2-x) x \left (4-2 x+x^2\right )\right )^{3/2} \, dx=\int { \left (-{\left (x^{2} - 2 \, x + 4\right )} {\left (x - 2\right )} x\right )^{\frac {3}{2}} \,d x } \] Input:

integrate(((2-x)*x*(x^2-2*x+4))^(3/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \left ((2-x) x \left (4-2 x+x^2\right )\right )^{3/2} \, dx=\int { \left (-{\left (x^{2} - 2 \, x + 4\right )} {\left (x - 2\right )} x\right )^{\frac {3}{2}} \,d x } \] Input:

integrate(((2-x)*x*(x^2-2*x+4))^(3/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \left ((2-x) x \left (4-2 x+x^2\right )\right )^{3/2} \, dx=\int {\left (-x\,\left (x-2\right )\,\left (x^2-2\,x+4\right )\right )}^{3/2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \left ((2-x) x \left (4-2 x+x^2\right )\right )^{3/2} \, dx=-\frac {\sqrt {x}\, \sqrt {-x^{3}+4 x^{2}-8 x +8}\, x^{5}}{7}+\frac {5 \sqrt {x}\, \sqrt {-x^{3}+4 x^{2}-8 x +8}\, x^{4}}{7}-\frac {66 \sqrt {x}\, \sqrt {-x^{3}+4 x^{2}-8 x +8}\, x^{3}}{35}+\frac {14 \sqrt {x}\, \sqrt {-x^{3}+4 x^{2}-8 x +8}\, x^{2}}{5}-\frac {32 \sqrt {x}\, \sqrt {-x^{3}+4 x^{2}-8 x +8}\, x}{35}-\frac {116 \sqrt {x}\, \sqrt {-x^{3}+4 x^{2}-8 x +8}}{105}+\frac {16 \left (\int \frac {\sqrt {x}\, \sqrt {-x^{3}+4 x^{2}-8 x +8}\, x^{2}}{x^{3}-4 x^{2}+8 x -8}d x \right )}{15}-\frac {464 \left (\int \frac {\sqrt {x}\, \sqrt {-x^{3}+4 x^{2}-8 x +8}}{x^{4}-4 x^{3}+8 x^{2}-8 x}d x \right )}{105}-\frac {32 \left (\int \frac {\sqrt {x}\, \sqrt {-x^{3}+4 x^{2}-8 x +8}}{x^{3}-4 x^{2}+8 x -8}d x \right )}{15} \] Input:

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

Output:

( - 15*sqrt(x)*sqrt( - x**3 + 4*x**2 - 8*x + 8)*x**5 + 75*sqrt(x)*sqrt( - 
x**3 + 4*x**2 - 8*x + 8)*x**4 - 198*sqrt(x)*sqrt( - x**3 + 4*x**2 - 8*x + 
8)*x**3 + 294*sqrt(x)*sqrt( - x**3 + 4*x**2 - 8*x + 8)*x**2 - 96*sqrt(x)*s 
qrt( - x**3 + 4*x**2 - 8*x + 8)*x - 116*sqrt(x)*sqrt( - x**3 + 4*x**2 - 8* 
x + 8) + 112*int((sqrt(x)*sqrt( - x**3 + 4*x**2 - 8*x + 8)*x**2)/(x**3 - 4 
*x**2 + 8*x - 8),x) - 464*int((sqrt(x)*sqrt( - x**3 + 4*x**2 - 8*x + 8))/( 
x**4 - 4*x**3 + 8*x**2 - 8*x),x) - 224*int((sqrt(x)*sqrt( - x**3 + 4*x**2 
- 8*x + 8))/(x**3 - 4*x**2 + 8*x - 8),x))/105