3.10.85 \(\int \frac {1}{\sqrt [4]{-1+3 x-3 x^2+x^3} (-1-2 x+x^2+3 x^3)} \, dx\) [985]

3.10.85.1 Optimal result

Integrand size = 33, antiderivative size = 75 \[ \int \frac {1}{\sqrt [4]{-1+3 x-3 x^2+x^3} \left (-1-2 x+x^2+3 x^3\right )} \, dx=\frac {\left ((-1+x)^3\right )^{3/4} \text {RootSum}\left [1+9 \text {$\#$1}^4+10 \text {$\#$1}^8+3 \text {$\#$1}^{12}\&,\frac {\log \left (\sqrt [4]{-1+x}-\text {$\#$1}\right )}{9 \text {$\#$1}^3+20 \text {$\#$1}^7+9 \text {$\#$1}^{11}}\&\right ]}{(-1+x)^{9/4}} \]

output

Unintegrable
 

3.10.85.2 Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt [4]{-1+3 x-3 x^2+x^3} \left (-1-2 x+x^2+3 x^3\right )} \, dx=\frac {(-1+x)^{3/4} \text {RootSum}\left [1+9 \text {$\#$1}^4+10 \text {$\#$1}^8+3 \text {$\#$1}^{12}\&,\frac {\log \left (\sqrt [4]{-1+x}-\text {$\#$1}\right )}{9 \text {$\#$1}^3+20 \text {$\#$1}^7+9 \text {$\#$1}^{11}}\&\right ]}{\sqrt [4]{(-1+x)^3}} \]

input

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

output

((-1 + x)^(3/4)*RootSum[1 + 9*#1^4 + 10*#1^8 + 3*#1^12 & , Log[(-1 + x)^(1 
/4) - #1]/(9*#1^3 + 20*#1^7 + 9*#1^11) & ])/((-1 + x)^3)^(1/4)
 

3.10.85.3 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.

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

\(\Big \downarrow \) 2008

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2490

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

\(\Big \downarrow \) 2485

\(\displaystyle -\frac {9 (x-1)^{3/4} \int -\frac {324 \sqrt {3}}{\left (\sqrt [3]{2} \left (\frac {38}{\sqrt [3]{187+9 \sqrt {93}}}+\sqrt [3]{374+18 \sqrt {93}}\right )-18 \left (x+\frac {1}{9}\right )\right ) \left (9 \left (x+\frac {1}{9}\right )-10\right )^{3/4} \left (-162 \left (x+\frac {1}{9}\right )^2-9 \sqrt [3]{2} \left (\frac {38}{\sqrt [3]{187+9 \sqrt {93}}}+\sqrt [3]{374+18 \sqrt {93}}\right ) \left (x+\frac {1}{9}\right )-\sqrt [3]{2} \left (187+9 \sqrt {93}\right )^{2/3}-722 \left (\frac {2}{187+9 \sqrt {93}}\right )^{2/3}+38\right )}d\left (x+\frac {1}{9}\right )}{\sqrt [4]{(x-1)^3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2916 \sqrt {3} (x-1)^{3/4} \int \frac {1}{\left (\sqrt [3]{2} \left (\frac {38}{\sqrt [3]{187+9 \sqrt {93}}}+\sqrt [3]{374+18 \sqrt {93}}\right )-18 \left (x+\frac {1}{9}\right )\right ) \left (9 \left (x+\frac {1}{9}\right )-10\right )^{3/4} \left (-162 \left (x+\frac {1}{9}\right )^2-9 \sqrt [3]{2} \left (\frac {38}{\sqrt [3]{187+9 \sqrt {93}}}+\sqrt [3]{374+18 \sqrt {93}}\right ) \left (x+\frac {1}{9}\right )-\sqrt [3]{2} \left (187+9 \sqrt {93}\right )^{2/3}-722 \left (\frac {2}{187+9 \sqrt {93}}\right )^{2/3}+38\right )}d\left (x+\frac {1}{9}\right )}{\sqrt [4]{(x-1)^3}}\)

\(\Big \downarrow \) 1199

\(\displaystyle \frac {1296 \sqrt {3} (x-1)^{3/4} \int \left (\frac {1}{3 \left (38+722 \left (\frac {2}{187+9 \sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (187+9 \sqrt {93}\right )^{2/3}\right ) \left (2 \left (9 \left (x+\frac {1}{9}\right )-10\right )-2^{2/3} \sqrt [3]{187+9 \sqrt {93}}-38 \sqrt [3]{\frac {2}{187+9 \sqrt {93}}}+20\right )}-\frac {9 \left (x+\frac {1}{9}\right )+2^{2/3} \sqrt [3]{187+9 \sqrt {93}}+38 \sqrt [3]{\frac {2}{187+9 \sqrt {93}}}}{3 \left (38+722 \left (\frac {2}{187+9 \sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (187+9 \sqrt {93}\right )^{2/3}\right ) \left (2 \left (9 \left (x+\frac {1}{9}\right )-10\right )^2+\left (40+38 \sqrt [3]{\frac {2}{187+9 \sqrt {93}}}+2^{2/3} \sqrt [3]{187+9 \sqrt {93}}\right ) \left (9 \left (x+\frac {1}{9}\right )-10\right )+\sqrt [3]{2} \left (187+9 \sqrt {93}\right )^{2/3}+10\ 2^{2/3} \sqrt [3]{187+9 \sqrt {93}}+722 \left (\frac {2}{187+9 \sqrt {93}}\right )^{2/3}+380 \sqrt [3]{\frac {2}{187+9 \sqrt {93}}}+162\right )}\right )d\sqrt [4]{9 \left (x+\frac {1}{9}\right )-10}}{\sqrt [4]{(x-1)^3}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {1296 \sqrt {3} (x-1)^{3/4} \int \frac {1}{4 \left (\left (9 \left (x+\frac {1}{9}\right )-10\right )^3+30 \left (9 \left (x+\frac {1}{9}\right )-10\right )^2+243 \left (9 \left (x+\frac {1}{9}\right )-10\right )+243\right )}d\sqrt [4]{9 \left (x+\frac {1}{9}\right )-10}}{\sqrt [4]{(x-1)^3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {324 \sqrt {3} (x-1)^{3/4} \int \frac {1}{\left (9 \left (x+\frac {1}{9}\right )-10\right )^3+30 \left (9 \left (x+\frac {1}{9}\right )-10\right )^2+243 \left (9 \left (x+\frac {1}{9}\right )-10\right )+243}d\sqrt [4]{9 \left (x+\frac {1}{9}\right )-10}}{\sqrt [4]{(x-1)^3}}\)

\(\Big \downarrow \) 7299

\(\displaystyle \frac {324 \sqrt {3} (x-1)^{3/4} \int \frac {1}{\left (9 \left (x+\frac {1}{9}\right )-10\right )^3+30 \left (9 \left (x+\frac {1}{9}\right )-10\right )^2+243 \left (9 \left (x+\frac {1}{9}\right )-10\right )+243}d\sqrt [4]{9 \left (x+\frac {1}{9}\right )-10}}{\sqrt [4]{(x-1)^3}}\)

input

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

output

$Aborted
 

3.10.85.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 1199

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_.) + (b_.)*(x 
_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Denominator[m]}, Simp[q/e   Subs 
t[Int[ExpandIntegrand[x^(q*(m + 1) - 1)*(((e*f - d*g)/e + g*(x^q/e))^n/((c* 
d^2 - b*d*e + a*e^2)/e^2 - (2*c*d - b*e)*(x^q/e^2) + c*(x^(2*q)/e^2))), x], 
 x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && Integer 
Q[n] && FractionQ[m]
 

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]
 

rule 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, x]]
 

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

3.10.85.4 Maple [F(-1)]

Timed out.

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

input

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

output

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

3.10.85.5 Fricas [C] (verification not implemented)

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

Time = 0.94 (sec) , antiderivative size = 6285, normalized size of antiderivative = 83.80 \[ \int \frac {1}{\sqrt [4]{-1+3 x-3 x^2+x^3} \left (-1-2 x+x^2+3 x^3\right )} \, dx=\text {Too large to display} \]

input

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

output

Too large to include
 

3.10.85.6 Sympy [N/A]

Not integrable

Time = 8.14 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.32 \[ \int \frac {1}{\sqrt [4]{-1+3 x-3 x^2+x^3} \left (-1-2 x+x^2+3 x^3\right )} \, dx=\int \frac {1}{\left (3 x^{3} + x^{2} - 2 x - 1\right ) \sqrt [4]{\left (x - 1\right )^{3}}}\, dx \]

input

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

output

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

3.10.85.7 Maxima [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.44 \[ \int \frac {1}{\sqrt [4]{-1+3 x-3 x^2+x^3} \left (-1-2 x+x^2+3 x^3\right )} \, dx=\int { \frac {1}{{\left (3 \, x^{3} + x^{2} - 2 \, x - 1\right )} {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}^{\frac {1}{4}}} \,d x } \]

input

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

output

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

3.10.85.8 Giac [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.44 \[ \int \frac {1}{\sqrt [4]{-1+3 x-3 x^2+x^3} \left (-1-2 x+x^2+3 x^3\right )} \, dx=\int { \frac {1}{{\left (3 \, x^{3} + x^{2} - 2 \, x - 1\right )} {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}^{\frac {1}{4}}} \,d x } \]

input

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

output

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

3.10.85.9 Mupad [N/A]

Not integrable

Time = 5.67 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.49 \[ \int \frac {1}{\sqrt [4]{-1+3 x-3 x^2+x^3} \left (-1-2 x+x^2+3 x^3\right )} \, dx=-\int \frac {1}{{\left (x^3-3\,x^2+3\,x-1\right )}^{1/4}\,\left (-3\,x^3-x^2+2\,x+1\right )} \,d x \]

input

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

output

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