\(\int \frac {1}{(8 x-8 x^2+4 x^3-x^4)^{3/2}} \, dx\) [53]

Optimal result

Integrand size = 23, antiderivative size = 79 \[ \int \frac {1}{\left (8 x-8 x^2+4 x^3-x^4\right )^{3/2}} \, dx=-\frac {\left (5+(-1+x)^2\right ) (1-x)}{24 \sqrt {3-2 (1-x)^2-(1-x)^4}}+\frac {E\left (\arcsin (1-x)\left |-\frac {1}{3}\right .\right )}{8 \sqrt {3}}-\frac {\operatorname {EllipticF}\left (\arcsin (1-x),-\frac {1}{3}\right )}{4 \sqrt {3}} \] Output:

-1/24*(5+(-1+x)^2)*(1-x)/(3-2*(1-x)^2-(1-x)^4)^(1/2)-1/24*3^(1/2)*Elliptic 
E(-1+x,1/3*I*3^(1/2))+1/12*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 = 23.15 (sec) , antiderivative size = 261, normalized size of antiderivative = 3.30 \[ \int \frac {1}{\left (8 x-8 x^2+4 x^3-x^4\right )^{3/2}} \, dx=\frac {\sqrt {-x \left (-8+8 x-4 x^2+x^3\right )} \left (\frac {\sqrt {2} \left (-i+\sqrt {3}\right ) \sqrt {-\frac {i (-2+x)}{\left (-i+\sqrt {3}\right ) x}} E\left (\arcsin \left (\frac {\sqrt {i+\sqrt {3}-\frac {4 i}{x}}}{\sqrt {2} \sqrt [4]{3}}\right )|\frac {2 \sqrt {3}}{-i+\sqrt {3}}\right )}{\sqrt {\frac {4-2 x+x^2}{x^2}}}-\frac {2+x^2-4 i \sqrt {2} \sqrt {-\frac {i (-2+x)}{\left (-i+\sqrt {3}\right ) x}} x^2 \sqrt {\frac {4-2 x+x^2}{x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {i+\sqrt {3}-\frac {4 i}{x}}}{\sqrt {2} \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{-i+\sqrt {3}}\right )}{4-2 x+x^2}\right )}{24 (-2+x) x} \] Input:

Integrate[(8*x - 8*x^2 + 4*x^3 - x^4)^(-3/2),x]
 

Output:

(Sqrt[-(x*(-8 + 8*x - 4*x^2 + x^3))]*((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])])/Sqrt[(4 - 2*x + x^2)/x^2] 
 - (2 + x^2 - (4*I)*Sqrt[2]*Sqrt[((-I)*(-2 + x))/((-I + Sqrt[3])*x)]*x^2*S 
qrt[(4 - 2*x + x^2)/x^2]*EllipticF[ArcSin[Sqrt[I + Sqrt[3] - (4*I)/x]/(Sqr 
t[2]*3^(1/4))], (2*Sqrt[3])/(-I + Sqrt[3])])/(4 - 2*x + x^2)))/(24*(-2 + x 
)*x)
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.92, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {2458, 1405, 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.

\(\displaystyle \int \frac {1}{\left (-x^4+4 x^3-8 x^2+8 x\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 2458

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

\(\Big \downarrow \) 1405

\(\displaystyle \frac {\left ((x-1)^2+5\right ) (x-1)}{24 \sqrt {-(x-1)^4-2 (x-1)^2+3}}-\frac {1}{48} \int -\frac {2 \left (3-(x-1)^2\right )}{\sqrt {-(x-1)^4-2 (x-1)^2+3}}d(x-1)\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{24} \int \frac {3-(x-1)^2}{\sqrt {-(x-1)^4-2 (x-1)^2+3}}d(x-1)+\frac {\left ((x-1)^2+5\right ) (x-1)}{24 \sqrt {-(x-1)^4-2 (x-1)^2+3}}\)

\(\Big \downarrow \) 1494

\(\displaystyle \frac {1}{12} \int \frac {3-(x-1)^2}{2 \sqrt {1-(x-1)^2} \sqrt {(x-1)^2+3}}d(x-1)+\frac {\left ((x-1)^2+5\right ) (x-1)}{24 \sqrt {-(x-1)^4-2 (x-1)^2+3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{24} \int \frac {3-(x-1)^2}{\sqrt {1-(x-1)^2} \sqrt {(x-1)^2+3}}d(x-1)+\frac {\left ((x-1)^2+5\right ) (x-1)}{24 \sqrt {-(x-1)^4-2 (x-1)^2+3}}\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {1}{24} \left (6 \int \frac {1}{\sqrt {1-(x-1)^2} \sqrt {(x-1)^2+3}}d(x-1)-\int \frac {\sqrt {(x-1)^2+3}}{\sqrt {1-(x-1)^2}}d(x-1)\right )+\frac {\left ((x-1)^2+5\right ) (x-1)}{24 \sqrt {-(x-1)^4-2 (x-1)^2+3}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {1}{24} \left (-\int \frac {\sqrt {(x-1)^2+3}}{\sqrt {1-(x-1)^2}}d(x-1)-2 \sqrt {3} \operatorname {EllipticF}\left (\arcsin (1-x),-\frac {1}{3}\right )\right )+\frac {\left ((x-1)^2+5\right ) (x-1)}{24 \sqrt {-(x-1)^4-2 (x-1)^2+3}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {1}{24} \left (\sqrt {3} E\left (\arcsin (1-x)\left |-\frac {1}{3}\right .\right )-2 \sqrt {3} \operatorname {EllipticF}\left (\arcsin (1-x),-\frac {1}{3}\right )\right )+\frac {\left ((x-1)^2+5\right ) (x-1)}{24 \sqrt {-(x-1)^4-2 (x-1)^2+3}}\)

Input:

Int[(8*x - 8*x^2 + 4*x^3 - x^4)^(-3/2),x]
 

Output:

((5 + (-1 + x)^2)*(-1 + x))/(24*Sqrt[3 - 2*(-1 + x)^2 - (-1 + x)^4]) + (Sq 
rt[3]*EllipticE[ArcSin[1 - x], -1/3] - 2*Sqrt[3]*EllipticF[ArcSin[1 - x], 
-1/3])/24
 

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 1405

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-x)*(b^2 
- 2*a*c + b*c*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c)) 
), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c))   Int[(b^2 - 2*a*c + 2*(p + 1)*( 
b^2 - 4*a*c) + b*c*(4*p + 7)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; Fr 
eeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && 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. 931 vs. \(2 (67 ) = 134\).

Time = 1.27 (sec) , antiderivative size = 932, normalized size of antiderivative = 11.80

method	result	size
risch	\(\text {Expression too large to display}\)	\(932\)
default	\(\text {Expression too large to display}\)	\(963\)
elliptic	\(\text {Expression too large to display}\)	\(963\)

Input:

int(1/(-x^4+4*x^3-8*x^2+8*x)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/24*(x^3-3*x^2+8*x-6)/(-x*(x^3-4*x^2+8*x-8))^(1/2)-1/24*(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)+1/6*(-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))+1/6*(-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(...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (59) = 118\).

Time = 0.08 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.51 \[ \int \frac {1}{\left (8 x-8 x^2+4 x^3-x^4\right )^{3/2}} \, dx=-\frac {5 \, \sqrt {2} {\left (x^{4} - 4 \, x^{3} + 8 \, x^{2} - 8 \, x\right )} {\rm weierstrassPInverse}\left (-\frac {2}{3}, \frac {7}{54}, -\frac {x - 3}{3 \, x}\right ) - 6 \, \sqrt {2} {\left (x^{4} - 4 \, x^{3} + 8 \, x^{2} - 8 \, x\right )} {\rm weierstrassZeta}\left (-\frac {2}{3}, \frac {7}{54}, {\rm weierstrassPInverse}\left (-\frac {2}{3}, \frac {7}{54}, -\frac {x - 3}{3 \, x}\right )\right ) + 3 \, \sqrt {-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x} {\left (x^{2} + 2\right )}}{72 \, {\left (x^{4} - 4 \, x^{3} + 8 \, x^{2} - 8 \, x\right )}} \] Input:

integrate(1/(-x^4+4*x^3-8*x^2+8*x)^(3/2),x, algorithm="fricas")
 

Output:

-1/72*(5*sqrt(2)*(x^4 - 4*x^3 + 8*x^2 - 8*x)*weierstrassPInverse(-2/3, 7/5 
4, -1/3*(x - 3)/x) - 6*sqrt(2)*(x^4 - 4*x^3 + 8*x^2 - 8*x)*weierstrassZeta 
(-2/3, 7/54, weierstrassPInverse(-2/3, 7/54, -1/3*(x - 3)/x)) + 3*sqrt(-x^ 
4 + 4*x^3 - 8*x^2 + 8*x)*(x^2 + 2))/(x^4 - 4*x^3 + 8*x^2 - 8*x)
 

Sympy [F]

\[ \int \frac {1}{\left (8 x-8 x^2+4 x^3-x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (- x^{4} + 4 x^{3} - 8 x^{2} + 8 x\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(1/(-x**4+4*x**3-8*x**2+8*x)**(3/2),x)
 

Output:

Integral((-x**4 + 4*x**3 - 8*x**2 + 8*x)**(-3/2), x)
 

Maxima [F]

\[ \int \frac {1}{\left (8 x-8 x^2+4 x^3-x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(-x^4+4*x^3-8*x^2+8*x)^(3/2),x, algorithm="maxima")
 

Output:

integrate((-x^4 + 4*x^3 - 8*x^2 + 8*x)^(-3/2), x)
 

Giac [F]

\[ \int \frac {1}{\left (8 x-8 x^2+4 x^3-x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(-x^4+4*x^3-8*x^2+8*x)^(3/2),x, algorithm="giac")
 

Output:

integrate((-x^4 + 4*x^3 - 8*x^2 + 8*x)^(-3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (8 x-8 x^2+4 x^3-x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (-x^4+4\,x^3-8\,x^2+8\,x\right )}^{3/2}} \,d x \] Input:

int(1/(8*x - 8*x^2 + 4*x^3 - x^4)^(3/2),x)
 

Output:

int(1/(8*x - 8*x^2 + 4*x^3 - x^4)^(3/2), x)
 

Reduce [F]

\[ \int \frac {1}{\left (8 x-8 x^2+4 x^3-x^4\right )^{3/2}} \, dx=\frac {\sqrt {-x^{3}+4 x^{2}-8 x +8}\, x^{2}-2 \sqrt {-x^{3}+4 x^{2}-8 x +8}\, x +\sqrt {-x^{3}+4 x^{2}-8 x +8}-2 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {-x^{3}+4 x^{2}-8 x +8}\, x}{x^{6}-8 x^{5}+32 x^{4}-80 x^{3}+128 x^{2}-128 x +64}d x \right ) x^{3}+8 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {-x^{3}+4 x^{2}-8 x +8}\, x}{x^{6}-8 x^{5}+32 x^{4}-80 x^{3}+128 x^{2}-128 x +64}d x \right ) x^{2}-16 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {-x^{3}+4 x^{2}-8 x +8}\, x}{x^{6}-8 x^{5}+32 x^{4}-80 x^{3}+128 x^{2}-128 x +64}d x \right ) x +16 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {-x^{3}+4 x^{2}-8 x +8}\, x}{x^{6}-8 x^{5}+32 x^{4}-80 x^{3}+128 x^{2}-128 x +64}d x \right )+6 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {-x^{3}+4 x^{2}-8 x +8}}{x^{6}-8 x^{5}+32 x^{4}-80 x^{3}+128 x^{2}-128 x +64}d x \right ) x^{3}-24 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {-x^{3}+4 x^{2}-8 x +8}}{x^{6}-8 x^{5}+32 x^{4}-80 x^{3}+128 x^{2}-128 x +64}d x \right ) x^{2}+48 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {-x^{3}+4 x^{2}-8 x +8}}{x^{6}-8 x^{5}+32 x^{4}-80 x^{3}+128 x^{2}-128 x +64}d x \right ) x -48 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {-x^{3}+4 x^{2}-8 x +8}}{x^{6}-8 x^{5}+32 x^{4}-80 x^{3}+128 x^{2}-128 x +64}d x \right )}{4 \sqrt {x}\, \left (x^{3}-4 x^{2}+8 x -8\right )} \] Input:

int(1/(-x^4+4*x^3-8*x^2+8*x)^(3/2),x)
 

Output:

(sqrt( - x**3 + 4*x**2 - 8*x + 8)*x**2 - 2*sqrt( - x**3 + 4*x**2 - 8*x + 8 
)*x + sqrt( - x**3 + 4*x**2 - 8*x + 8) - 2*sqrt(x)*int((sqrt(x)*sqrt( - x* 
*3 + 4*x**2 - 8*x + 8)*x)/(x**6 - 8*x**5 + 32*x**4 - 80*x**3 + 128*x**2 - 
128*x + 64),x)*x**3 + 8*sqrt(x)*int((sqrt(x)*sqrt( - x**3 + 4*x**2 - 8*x + 
 8)*x)/(x**6 - 8*x**5 + 32*x**4 - 80*x**3 + 128*x**2 - 128*x + 64),x)*x**2 
 - 16*sqrt(x)*int((sqrt(x)*sqrt( - x**3 + 4*x**2 - 8*x + 8)*x)/(x**6 - 8*x 
**5 + 32*x**4 - 80*x**3 + 128*x**2 - 128*x + 64),x)*x + 16*sqrt(x)*int((sq 
rt(x)*sqrt( - x**3 + 4*x**2 - 8*x + 8)*x)/(x**6 - 8*x**5 + 32*x**4 - 80*x* 
*3 + 128*x**2 - 128*x + 64),x) + 6*sqrt(x)*int((sqrt(x)*sqrt( - x**3 + 4*x 
**2 - 8*x + 8))/(x**6 - 8*x**5 + 32*x**4 - 80*x**3 + 128*x**2 - 128*x + 64 
),x)*x**3 - 24*sqrt(x)*int((sqrt(x)*sqrt( - x**3 + 4*x**2 - 8*x + 8))/(x** 
6 - 8*x**5 + 32*x**4 - 80*x**3 + 128*x**2 - 128*x + 64),x)*x**2 + 48*sqrt( 
x)*int((sqrt(x)*sqrt( - x**3 + 4*x**2 - 8*x + 8))/(x**6 - 8*x**5 + 32*x**4 
 - 80*x**3 + 128*x**2 - 128*x + 64),x)*x - 48*sqrt(x)*int((sqrt(x)*sqrt( - 
 x**3 + 4*x**2 - 8*x + 8))/(x**6 - 8*x**5 + 32*x**4 - 80*x**3 + 128*x**2 - 
 128*x + 64),x))/(4*sqrt(x)*(x**3 - 4*x**2 + 8*x - 8))