3.24.57 \(\int \frac {1}{\sqrt [4]{-1+3 x-3 x^2+x^3} (-1-2 x+x^2+3 x^3)^4} \, dx\) [2357]

3.24.57.1 Optimal result

Integrand size = 33, antiderivative size = 187 \[ \int \frac {1}{\sqrt [4]{-1+3 x-3 x^2+x^3} \left (-1-2 x+x^2+3 x^3\right )^4} \, dx=\frac {\left ((-1+x)^3\right )^{3/4} \left (\frac {\sqrt [4]{-1+x} \left (307788101+1357068302 x+596630756 x^2-4979849490 x^3-5857310139 x^4+5802065412 x^5+9762576651 x^6-2006712954 x^7-5094769914 x^8\right )}{2859936 \left (-1-2 x+x^2+3 x^3\right )^3}-\frac {\text {RootSum}\left [1+9 \text {$\#$1}^4+10 \text {$\#$1}^8+3 \text {$\#$1}^{12}\&,\frac {-41317673 \log \left (\sqrt [4]{-1+x}-\text {$\#$1}\right )-234521814 \log \left (\sqrt [4]{-1+x}-\text {$\#$1}\right ) \text {$\#$1}^4+566085546 \log \left (\sqrt [4]{-1+x}-\text {$\#$1}\right ) \text {$\#$1}^8}{9 \text {$\#$1}^3+20 \text {$\#$1}^7+9 \text {$\#$1}^{11}}\&\right ]}{3813248}\right )}{(-1+x)^{9/4}} \]

output

Unintegrable
 

3.24.57.2 Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.97 \[ \int \frac {1}{\sqrt [4]{-1+3 x-3 x^2+x^3} \left (-1-2 x+x^2+3 x^3\right )^4} \, dx=\frac {-\frac {4 (-1+x) \left (-307788101-1357068302 x-596630756 x^2+4979849490 x^3+5857310139 x^4-5802065412 x^5-9762576651 x^6+2006712954 x^7+5094769914 x^8\right )}{\left (-1-2 x+x^2+3 x^3\right )^3}-3 (-1+x)^{3/4} \text {RootSum}\left [1+9 \text {$\#$1}^4+10 \text {$\#$1}^8+3 \text {$\#$1}^{12}\&,\frac {-41317673 \log \left (\sqrt [4]{-1+x}-\text {$\#$1}\right )-234521814 \log \left (\sqrt [4]{-1+x}-\text {$\#$1}\right ) \text {$\#$1}^4+566085546 \log \left (\sqrt [4]{-1+x}-\text {$\#$1}\right ) \text {$\#$1}^8}{9 \text {$\#$1}^3+20 \text {$\#$1}^7+9 \text {$\#$1}^{11}}\&\right ]}{11439744 \sqrt [4]{(-1+x)^3}} \]

input

Integrate[1/((-1 + 3*x - 3*x^2 + x^3)^(1/4)*(-1 - 2*x + x^2 + 3*x^3)^4),x]
 

output

((-4*(-1 + x)*(-307788101 - 1357068302*x - 596630756*x^2 + 4979849490*x^3 
+ 5857310139*x^4 - 5802065412*x^5 - 9762576651*x^6 + 2006712954*x^7 + 5094 
769914*x^8))/(-1 - 2*x + x^2 + 3*x^3)^3 - 3*(-1 + x)^(3/4)*RootSum[1 + 9*# 
1^4 + 10*#1^8 + 3*#1^12 & , (-41317673*Log[(-1 + x)^(1/4) - #1] - 23452181 
4*Log[(-1 + x)^(1/4) - #1]*#1^4 + 566085546*Log[(-1 + x)^(1/4) - #1]*#1^8) 
/(9*#1^3 + 20*#1^7 + 9*#1^11) & ])/(11439744*((-1 + x)^3)^(1/4))
 

3.24.57.3 Rubi [F]

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.

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

\(\Big \downarrow \) 2008

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

\(\Big \downarrow \) 2490

\(\displaystyle \frac {(x-1)^{3/4} \int \frac {1}{(x-1)^{3/4} \left (-3 \left (x+\frac {1}{9}\right )^3+\frac {19}{9} \left (x+\frac {1}{9}\right )+\frac {187}{243}\right )^4}d\left (x+\frac {1}{9}\right )}{\sqrt [4]{(x-1)^3}}\)

\(\Big \downarrow \) 2485

\(\displaystyle \frac {6561 (x-1)^{3/4} \int \frac {408146688 \sqrt {3}}{\left (\sqrt [3]{2} \left (\frac {38}{\sqrt [3]{187+9 \sqrt {93}}}+\sqrt [3]{374+18 \sqrt {93}}\right )-18 \left (x+\frac {1}{9}\right )\right )^4 \left (9 \left (x+\frac {1}{9}\right )-10\right )^{3/4} \left (-162 \left (x+\frac {1}{9}\right )^2-9 \sqrt [3]{2} \left (\frac {38}{\sqrt [3]{187+9 \sqrt {93}}}+\sqrt [3]{374+18 \sqrt {93}}\right ) \left (x+\frac {1}{9}\right )-\sqrt [3]{2} \left (187+9 \sqrt {93}\right )^{2/3}-722 \left (\frac {2}{187+9 \sqrt {93}}\right )^{2/3}+38\right )^4}d\left (x+\frac {1}{9}\right )}{\sqrt [4]{(x-1)^3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2677850419968 \sqrt {3} (x-1)^{3/4} \int \frac {1}{\left (\sqrt [3]{2} \left (\frac {38}{\sqrt [3]{187+9 \sqrt {93}}}+\sqrt [3]{374+18 \sqrt {93}}\right )-18 \left (x+\frac {1}{9}\right )\right )^4 \left (9 \left (x+\frac {1}{9}\right )-10\right )^{3/4} \left (-162 \left (x+\frac {1}{9}\right )^2-9 \sqrt [3]{2} \left (\frac {38}{\sqrt [3]{187+9 \sqrt {93}}}+\sqrt [3]{374+18 \sqrt {93}}\right ) \left (x+\frac {1}{9}\right )-\sqrt [3]{2} \left (187+9 \sqrt {93}\right )^{2/3}-722 \left (\frac {2}{187+9 \sqrt {93}}\right )^{2/3}+38\right )^4}d\left (x+\frac {1}{9}\right )}{\sqrt [4]{(x-1)^3}}\)

\(\Big \downarrow \) 1292

\(\displaystyle \frac {2677850419968 \sqrt {3} (x-1)^{3/4} \int \frac {1}{\left (\sqrt [3]{2} \left (\frac {38}{\sqrt [3]{187+9 \sqrt {93}}}+\sqrt [3]{374+18 \sqrt {93}}\right )-18 \left (x+\frac {1}{9}\right )\right )^4 \left (9 \left (x+\frac {1}{9}\right )-10\right )^{3/4} \left (-162 \left (x+\frac {1}{9}\right )^2-9 \sqrt [3]{2} \left (\frac {38}{\sqrt [3]{187+9 \sqrt {93}}}+\sqrt [3]{374+18 \sqrt {93}}\right ) \left (x+\frac {1}{9}\right )-\sqrt [3]{2} \left (187+9 \sqrt {93}\right )^{2/3}-722 \left (\frac {2}{187+9 \sqrt {93}}\right )^{2/3}+38\right )^4}d\left (x+\frac {1}{9}\right )}{\sqrt [4]{(x-1)^3}}\)

input

Int[1/((-1 + 3*x - 3*x^2 + x^3)^(1/4)*(-1 - 2*x + x^2 + 3*x^3)^4),x]
 

output

$Aborted
 

3.24.57.3.1 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 1292

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x 
_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Unintegrable[(d + e*x)^m*(f + g*x)^n* 
(a + b*x + c*x^2)^p, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x]
 

rule 2008

Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, 
x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Simp[((a + b*x)^Exp 
on[Px, x])^p/(a + b*x)^(Expon[Px, x]*p)   Int[u*(a + b*x)^(Expon[Px, x]*p), 
 x], x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /;  !IntegerQ[p] && PolyQ[Px, x 
] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
 

rule 2485

Int[((e_.) + (f_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_S 
ymbol] :> With[{r = Rt[-9*a*d^2 + Sqrt[3]*d*Sqrt[4*b^3*d + 27*a^2*d^2], 3]} 
, Simp[1/d^(2*p)   Int[(e + f*x)^m*Simp[18^(1/3)*b*(d/(3*r)) - r/18^(1/3) + 
 d*x, x]^p*Simp[b*(d/3) + 12^(1/3)*b^2*(d^2/(3*r^2)) + r^2/(3*12^(1/3)) - d 
*(2^(1/3)*b*(d/(3^(1/3)*r)) - r/18^(1/3))*x + d^2*x^2, x]^p, x], x]] /; Fre 
eQ[{a, b, d, e, f, m}, x] && NeQ[4*b^3 + 27*a^2*d, 0] && ILtQ[p, 0]
 

rule 2490

Int[(P3_)^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> With[{a = Coeff[P3 
, x, 0], b = Coeff[P3, x, 1], c = Coeff[P3, x, 2], d = Coeff[P3, x, 3]}, Su 
bst[Int[((3*d*e - c*f)/(3*d) + f*x)^m*Simp[(2*c^3 - 9*b*c*d + 27*a*d^2)/(27 
*d^2) - (c^2 - 3*b*d)*(x/(3*d)) + d*x^3, x]^p, x], x, x + c/(3*d)] /; NeQ[c 
, 0]] /; FreeQ[{e, f, m, p}, x] && PolyQ[P3, x, 3]
 

3.24.57.4 Maple [F(-1)]

Timed out.

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

input

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

output

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

3.24.57.5 Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 1.12 (sec) , antiderivative size = 5503, normalized size of antiderivative = 29.43 \[ \int \frac {1}{\sqrt [4]{-1+3 x-3 x^2+x^3} \left (-1-2 x+x^2+3 x^3\right )^4} \, dx=\text {Too large to display} \]

input

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

output

Too large to include
 

3.24.57.6 Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt [4]{-1+3 x-3 x^2+x^3} \left (-1-2 x+x^2+3 x^3\right )^4} \, dx=\text {Timed out} \]

input

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

output

Timed out
 

3.24.57.7 Maxima [N/A]

Not integrable

Time = 0.22 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.18 \[ \int \frac {1}{\sqrt [4]{-1+3 x-3 x^2+x^3} \left (-1-2 x+x^2+3 x^3\right )^4} \, dx=\int { \frac {1}{{\left (3 \, x^{3} + x^{2} - 2 \, x - 1\right )}^{4} {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}^{\frac {1}{4}}} \,d x } \]

input

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

output

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

3.24.57.8 Giac [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.18 \[ \int \frac {1}{\sqrt [4]{-1+3 x-3 x^2+x^3} \left (-1-2 x+x^2+3 x^3\right )^4} \, dx=\int { \frac {1}{{\left (3 \, x^{3} + x^{2} - 2 \, x - 1\right )}^{4} {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}^{\frac {1}{4}}} \,d x } \]

input

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

output

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

3.24.57.9 Mupad [N/A]

Not integrable

Time = 6.34 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.19 \[ \int \frac {1}{\sqrt [4]{-1+3 x-3 x^2+x^3} \left (-1-2 x+x^2+3 x^3\right )^4} \, dx=\int \frac {1}{{\left (x^3-3\,x^2+3\,x-1\right )}^{1/4}\,{\left (-3\,x^3-x^2+2\,x+1\right )}^4} \,d x \]

input

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

output

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