\(\int \frac {1}{((2-x) x (4-2 x+x^2))^{5/2}} \, dx\) [59]

Optimal result

Integrand size = 19, antiderivative size = 123 \[ \int \frac {1}{\left ((2-x) x \left (4-2 x+x^2\right )\right )^{5/2}} \, dx=-\frac {\left (26+7 (1-x)^2\right ) (1-x)}{432 \sqrt {3-2 (1-x)^2-(1-x)^4}}-\frac {\left (5+(-1+x)^2\right ) (1-x)}{72 \left (3-2 (1-x)^2-(1-x)^4\right )^{3/2}}+\frac {7 E\left (\arcsin (1-x)\left |-\frac {1}{3}\right .\right )}{144 \sqrt {3}}-\frac {11 \operatorname {EllipticF}\left (\arcsin (1-x),-\frac {1}{3}\right )}{144 \sqrt {3}} \] Output:

-1/432*(26+7*(1-x)^2)*(1-x)/(3-2*(1-x)^2-(1-x)^4)^(1/2)-1/72*(5+(-1+x)^2)* 
(1-x)/(3-2*(1-x)^2-(1-x)^4)^(3/2)-7/432*3^(1/2)*EllipticE(-1+x,1/3*I*3^(1/ 
2))+11/432*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 = 23.43 (sec) , antiderivative size = 327, normalized size of antiderivative = 2.66 \[ \int \frac {1}{\left ((2-x) x \left (4-2 x+x^2\right )\right )^{5/2}} \, dx=\frac {(-2+x)^3 x^2 \left (4-2 x+x^2\right )^2 \left (-\frac {7 x \left (4-2 x+x^2\right )}{-2+x}+\frac {36+216 x-622 x^2+670 x^3-445 x^4+187 x^5-49 x^6+7 x^7}{(-2+x)^2 x \left (4-2 x+x^2\right )}+\frac {7 i \sqrt {2} x \sqrt {\frac {4-2 x+x^2}{(-2+x)^2}} E\left (\arcsin \left (\frac {\sqrt {-i+\sqrt {3}-\frac {4 i}{-2+x}}}{\sqrt {2} \sqrt [4]{3}}\right )|\frac {2 \sqrt {3}}{i+\sqrt {3}}\right )}{\sqrt {\frac {i x}{\left (i+\sqrt {3}\right ) (-2+x)}}}-19 i \sqrt {2} (-2+x) \sqrt {\frac {i x}{\left (i+\sqrt {3}\right ) (-2+x)}} \sqrt {\frac {4-2 x+x^2}{(-2+x)^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i+\sqrt {3}-\frac {4 i}{-2+x}}}{\sqrt {2} \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{i+\sqrt {3}}\right )\right )}{432 \left (-x \left (-8+8 x-4 x^2+x^3\right )\right )^{5/2}} \] Input:

Integrate[((2 - x)*x*(4 - 2*x + x^2))^(-5/2),x]
 

Output:

((-2 + x)^3*x^2*(4 - 2*x + x^2)^2*((-7*x*(4 - 2*x + x^2))/(-2 + x) + (36 + 
 216*x - 622*x^2 + 670*x^3 - 445*x^4 + 187*x^5 - 49*x^6 + 7*x^7)/((-2 + x) 
^2*x*(4 - 2*x + x^2)) + ((7*I)*Sqrt[2]*x*Sqrt[(4 - 2*x + x^2)/(-2 + x)^2]* 
EllipticE[ArcSin[Sqrt[-I + Sqrt[3] - (4*I)/(-2 + x)]/(Sqrt[2]*3^(1/4))], ( 
2*Sqrt[3])/(I + Sqrt[3])])/Sqrt[(I*x)/((I + Sqrt[3])*(-2 + x))] - (19*I)*S 
qrt[2]*(-2 + x)*Sqrt[(I*x)/((I + Sqrt[3])*(-2 + x))]*Sqrt[(4 - 2*x + x^2)/ 
(-2 + x)^2]*EllipticF[ArcSin[Sqrt[-I + Sqrt[3] - (4*I)/(-2 + x)]/(Sqrt[2]* 
3^(1/4))], (2*Sqrt[3])/(I + Sqrt[3])]))/(432*(-(x*(-8 + 8*x - 4*x^2 + x^3) 
))^(5/2))
 

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 115, 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, 1405, 27, 1492, 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.

\(\displaystyle \int \frac {1}{\left ((2-x) x \left (x^2-2 x+4\right )\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 2458

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

\(\Big \downarrow \) 1405

\(\displaystyle \frac {\left ((x-1)^2+5\right ) (x-1)}{72 \left (-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}-\frac {1}{144} \int -\frac {2 \left (3 (x-1)^2+19\right )}{\left (-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}d(x-1)\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{72} \int \frac {3 (x-1)^2+19}{\left (-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}d(x-1)+\frac {\left ((x-1)^2+5\right ) (x-1)}{72 \left (-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}\)

\(\Big \downarrow \) 1492

\(\displaystyle \frac {1}{72} \left (\frac {\left (7 (x-1)^2+26\right ) (x-1)}{6 \sqrt {-(x-1)^4-2 (x-1)^2+3}}-\frac {1}{48} \int -\frac {8 \left (12-7 (x-1)^2\right )}{\sqrt {-(x-1)^4-2 (x-1)^2+3}}d(x-1)\right )+\frac {\left ((x-1)^2+5\right ) (x-1)}{72 \left (-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{72} \left (\frac {1}{6} \int \frac {12-7 (x-1)^2}{\sqrt {-(x-1)^4-2 (x-1)^2+3}}d(x-1)+\frac {\left (7 (x-1)^2+26\right ) (x-1)}{6 \sqrt {-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {\left ((x-1)^2+5\right ) (x-1)}{72 \left (-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}\)

\(\Big \downarrow \) 1494

\(\displaystyle \frac {1}{72} \left (\frac {1}{3} \int \frac {12-7 (x-1)^2}{2 \sqrt {1-(x-1)^2} \sqrt {(x-1)^2+3}}d(x-1)+\frac {\left (7 (x-1)^2+26\right ) (x-1)}{6 \sqrt {-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {\left ((x-1)^2+5\right ) (x-1)}{72 \left (-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{72} \left (\frac {1}{6} \int \frac {12-7 (x-1)^2}{\sqrt {1-(x-1)^2} \sqrt {(x-1)^2+3}}d(x-1)+\frac {\left (7 (x-1)^2+26\right ) (x-1)}{6 \sqrt {-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {\left ((x-1)^2+5\right ) (x-1)}{72 \left (-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {1}{72} \left (\frac {1}{6} \left (33 \int \frac {1}{\sqrt {1-(x-1)^2} \sqrt {(x-1)^2+3}}d(x-1)-7 \int \frac {\sqrt {(x-1)^2+3}}{\sqrt {1-(x-1)^2}}d(x-1)\right )+\frac {\left (7 (x-1)^2+26\right ) (x-1)}{6 \sqrt {-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {\left ((x-1)^2+5\right ) (x-1)}{72 \left (-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {1}{72} \left (\frac {1}{6} \left (-7 \int \frac {\sqrt {(x-1)^2+3}}{\sqrt {1-(x-1)^2}}d(x-1)-11 \sqrt {3} \operatorname {EllipticF}\left (\arcsin (1-x),-\frac {1}{3}\right )\right )+\frac {\left (7 (x-1)^2+26\right ) (x-1)}{6 \sqrt {-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {\left ((x-1)^2+5\right ) (x-1)}{72 \left (-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {1}{72} \left (\frac {1}{6} \left (7 \sqrt {3} E\left (\arcsin (1-x)\left |-\frac {1}{3}\right .\right )-11 \sqrt {3} \operatorname {EllipticF}\left (\arcsin (1-x),-\frac {1}{3}\right )\right )+\frac {\left (7 (x-1)^2+26\right ) (x-1)}{6 \sqrt {-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {\left ((x-1)^2+5\right ) (x-1)}{72 \left (-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}\)

Input:

Int[((2 - x)*x*(4 - 2*x + x^2))^(-5/2),x]
 

Output:

((5 + (-1 + x)^2)*(-1 + x))/(72*(3 - 2*(-1 + x)^2 - (-1 + x)^4)^(3/2)) + ( 
((26 + 7*(-1 + x)^2)*(-1 + x))/(6*Sqrt[3 - 2*(-1 + x)^2 - (-1 + x)^4]) + ( 
7*Sqrt[3]*EllipticE[ArcSin[1 - x], -1/3] - 11*Sqrt[3]*EllipticF[ArcSin[1 - 
 x], -1/3])/6)/72
 

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 1492

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb 
ol] :> Simp[x*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*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[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 
7)*(d*b - 2*a*e)*c*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] && 
 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. 1038 vs. \(2 (107 ) = 214\).

Time = 0.65 (sec) , antiderivative size = 1039, normalized size of antiderivative = 8.45

method	result	size
default	\(\text {Expression too large to display}\)	\(1039\)
elliptic	\(\text {Expression too large to display}\)	\(1039\)

Input:

int(1/((2-x)*x*(x^2-2*x+4))^(5/2),x,method=_RETURNVERBOSE)
 

Output:

(1/36+1/288*x^2-1/96*x)*(-x^4+4*x^3-8*x^2+8*x)^(1/2)/(x^3-4*x^2+8*x-8)^2+2 
*x*(53/3456+5/1728*x^2-19/4608*x)/(-x*(x^3-4*x^2+8*x-8))^(1/2)-1/768*(-x^4 
+4*x^3-8*x^2+8*x)^(1/2)/x^2-1/96*(-x^3+4*x^2-8*x+8)/(x*(-x^3+4*x^2-8*x+8)) 
^(1/2)+5/216*(-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))+7/108*(-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*Elliptic 
F(((-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)))-7/432*(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...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 195 vs. \(2 (91) = 182\).

Time = 0.07 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.59 \[ \int \frac {1}{\left ((2-x) x \left (4-2 x+x^2\right )\right )^{5/2}} \, dx=-\frac {43 \, \sqrt {2} {\left (x^{8} - 8 \, x^{7} + 32 \, x^{6} - 80 \, x^{5} + 128 \, x^{4} - 128 \, x^{3} + 64 \, x^{2}\right )} {\rm weierstrassPInverse}\left (-\frac {2}{3}, \frac {7}{54}, -\frac {x - 3}{3 \, x}\right ) - 84 \, \sqrt {2} {\left (x^{8} - 8 \, x^{7} + 32 \, x^{6} - 80 \, x^{5} + 128 \, x^{4} - 128 \, x^{3} + 64 \, x^{2}\right )} {\rm weierstrassZeta}\left (-\frac {2}{3}, \frac {7}{54}, {\rm weierstrassPInverse}\left (-\frac {2}{3}, \frac {7}{54}, -\frac {x - 3}{3 \, x}\right )\right ) + 6 \, {\left (7 \, x^{6} - 37 \, x^{5} + 115 \, x^{4} - 226 \, x^{3} + 274 \, x^{2} - 232 \, x + 36\right )} \sqrt {-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x}}{2592 \, {\left (x^{8} - 8 \, x^{7} + 32 \, x^{6} - 80 \, x^{5} + 128 \, x^{4} - 128 \, x^{3} + 64 \, x^{2}\right )}} \] Input:

integrate(1/((2-x)*x*(x^2-2*x+4))^(5/2),x, algorithm="fricas")
 

Output:

-1/2592*(43*sqrt(2)*(x^8 - 8*x^7 + 32*x^6 - 80*x^5 + 128*x^4 - 128*x^3 + 6 
4*x^2)*weierstrassPInverse(-2/3, 7/54, -1/3*(x - 3)/x) - 84*sqrt(2)*(x^8 - 
 8*x^7 + 32*x^6 - 80*x^5 + 128*x^4 - 128*x^3 + 64*x^2)*weierstrassZeta(-2/ 
3, 7/54, weierstrassPInverse(-2/3, 7/54, -1/3*(x - 3)/x)) + 6*(7*x^6 - 37* 
x^5 + 115*x^4 - 226*x^3 + 274*x^2 - 232*x + 36)*sqrt(-x^4 + 4*x^3 - 8*x^2 
+ 8*x))/(x^8 - 8*x^7 + 32*x^6 - 80*x^5 + 128*x^4 - 128*x^3 + 64*x^2)
 

Sympy [F]

\[ \int \frac {1}{\left ((2-x) x \left (4-2 x+x^2\right )\right )^{5/2}} \, dx=\int \frac {1}{\left (x \left (2 - x\right ) \left (x^{2} - 2 x + 4\right )\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(1/((2-x)*x*(x**2-2*x+4))**(5/2),x)
 

Output:

Integral((x*(2 - x)*(x**2 - 2*x + 4))**(-5/2), x)
 

Maxima [F]

\[ \int \frac {1}{\left ((2-x) x \left (4-2 x+x^2\right )\right )^{5/2}} \, dx=\int { \frac {1}{\left (-{\left (x^{2} - 2 \, x + 4\right )} {\left (x - 2\right )} x\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/((2-x)*x*(x^2-2*x+4))^(5/2),x, algorithm="maxima")
 

Output:

integrate((-(x^2 - 2*x + 4)*(x - 2)*x)^(-5/2), x)
 

Giac [F]

\[ \int \frac {1}{\left ((2-x) x \left (4-2 x+x^2\right )\right )^{5/2}} \, dx=\int { \frac {1}{\left (-{\left (x^{2} - 2 \, x + 4\right )} {\left (x - 2\right )} x\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/((2-x)*x*(x^2-2*x+4))^(5/2),x, algorithm="giac")
 

Output:

integrate((-(x^2 - 2*x + 4)*(x - 2)*x)^(-5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left ((2-x) x \left (4-2 x+x^2\right )\right )^{5/2}} \, dx=\int \frac {1}{{\left (-x\,\left (x-2\right )\,\left (x^2-2\,x+4\right )\right )}^{5/2}} \,d x \] Input:

int(1/(-x*(x - 2)*(x^2 - 2*x + 4))^(5/2),x)
 

Output:

int(1/(-x*(x - 2)*(x^2 - 2*x + 4))^(5/2), x)
 

Reduce [F]

\[ \int \frac {1}{\left ((2-x) x \left (4-2 x+x^2\right )\right )^{5/2}} \, dx=\text {too large to display} \] Input:

int(1/((2-x)*x*(x^2-2*x+4))^(5/2),x)
 

Output:

(sqrt( - x**3 + 4*x**2 - 8*x + 8)*x**6 - 6*sqrt( - x**3 + 4*x**2 - 8*x + 8 
)*x**5 + 18*sqrt( - x**3 + 4*x**2 - 8*x + 8)*x**4 - 32*sqrt( - x**3 + 4*x* 
*2 - 8*x + 8)*x**3 + 30*sqrt( - x**3 + 4*x**2 - 8*x + 8)*x**2 - 12*sqrt( - 
 x**3 + 4*x**2 - 8*x + 8)*x - 2*sqrt( - x**3 + 4*x**2 - 8*x + 8) - 36*sqrt 
(x)*int(sqrt( - x**3 + 4*x**2 - 8*x + 8)/(sqrt(x)*x**9 - 12*sqrt(x)*x**8 + 
 72*sqrt(x)*x**7 - 280*sqrt(x)*x**6 + 768*sqrt(x)*x**5 - 1536*sqrt(x)*x**4 
 + 2240*sqrt(x)*x**3 - 2304*sqrt(x)*x**2 + 1536*sqrt(x)*x - 512*sqrt(x)),x 
)*x**7 + 288*sqrt(x)*int(sqrt( - x**3 + 4*x**2 - 8*x + 8)/(sqrt(x)*x**9 - 
12*sqrt(x)*x**8 + 72*sqrt(x)*x**7 - 280*sqrt(x)*x**6 + 768*sqrt(x)*x**5 - 
1536*sqrt(x)*x**4 + 2240*sqrt(x)*x**3 - 2304*sqrt(x)*x**2 + 1536*sqrt(x)*x 
 - 512*sqrt(x)),x)*x**6 - 1152*sqrt(x)*int(sqrt( - x**3 + 4*x**2 - 8*x + 8 
)/(sqrt(x)*x**9 - 12*sqrt(x)*x**8 + 72*sqrt(x)*x**7 - 280*sqrt(x)*x**6 + 7 
68*sqrt(x)*x**5 - 1536*sqrt(x)*x**4 + 2240*sqrt(x)*x**3 - 2304*sqrt(x)*x** 
2 + 1536*sqrt(x)*x - 512*sqrt(x)),x)*x**5 + 2880*sqrt(x)*int(sqrt( - x**3 
+ 4*x**2 - 8*x + 8)/(sqrt(x)*x**9 - 12*sqrt(x)*x**8 + 72*sqrt(x)*x**7 - 28 
0*sqrt(x)*x**6 + 768*sqrt(x)*x**5 - 1536*sqrt(x)*x**4 + 2240*sqrt(x)*x**3 
- 2304*sqrt(x)*x**2 + 1536*sqrt(x)*x - 512*sqrt(x)),x)*x**4 - 4608*sqrt(x) 
*int(sqrt( - x**3 + 4*x**2 - 8*x + 8)/(sqrt(x)*x**9 - 12*sqrt(x)*x**8 + 72 
*sqrt(x)*x**7 - 280*sqrt(x)*x**6 + 768*sqrt(x)*x**5 - 1536*sqrt(x)*x**4 + 
2240*sqrt(x)*x**3 - 2304*sqrt(x)*x**2 + 1536*sqrt(x)*x - 512*sqrt(x)),x...