Integrand size = 33, antiderivative size = 74 \[ \int \frac {\sqrt [4]{-1+3 x-3 x^2+x^3}}{-1-2 x+x^2+3 x^3} \, dx=\frac {\sqrt [4]{(-1+x)^3} \text {RootSum}\left [1+9 \text {$\#$1}^4+10 \text {$\#$1}^8+3 \text {$\#$1}^{12}\&,\frac {\log \left (\sqrt [4]{-1+x}-\text {$\#$1}\right ) \text {$\#$1}^3}{9+20 \text {$\#$1}^4+9 \text {$\#$1}^8}\&\right ]}{(-1+x)^{3/4}} \]
Time = 0.07 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [4]{-1+3 x-3 x^2+x^3}}{-1-2 x+x^2+3 x^3} \, dx=\frac {\sqrt [4]{(-1+x)^3} \text {RootSum}\left [1+9 \text {$\#$1}^4+10 \text {$\#$1}^8+3 \text {$\#$1}^{12}\&,\frac {\log \left (\sqrt [4]{-1+x}-\text {$\#$1}\right ) \text {$\#$1}^3}{9+20 \text {$\#$1}^4+9 \text {$\#$1}^8}\&\right ]}{(-1+x)^{3/4}} \]
(((-1 + x)^3)^(1/4)*RootSum[1 + 9*#1^4 + 10*#1^8 + 3*#1^12 & , (Log[(-1 + x)^(1/4) - #1]*#1^3)/(9 + 20*#1^4 + 9*#1^8) & ])/(-1 + x)^(3/4)
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.
| \(\displaystyle \int \frac {\sqrt [4]{x^3-3 x^2+3 x-1}}{3 x^3+x^2-2 x-1} \, dx\) |
| \(\Big \downarrow \) 2008 |
| \(\displaystyle \frac {\sqrt [4]{(x-1)^3} \int -\frac {(x-1)^{3/4}}{-3 x^3-x^2+2 x+1}dx}{(x-1)^{3/4}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{(x-1)^3} \int \frac {(x-1)^{3/4}}{-3 x^3-x^2+2 x+1}dx}{(x-1)^{3/4}}\) |
| \(\Big \downarrow \) 2490 |
| \(\displaystyle -\frac {\sqrt [4]{(x-1)^3} \int \frac {(x-1)^{3/4}}{-3 \left (x+\frac {1}{9}\right )^3+\frac {19}{9} \left (x+\frac {1}{9}\right )+\frac {187}{243}}d\left (x+\frac {1}{9}\right )}{(x-1)^{3/4}}\) |
| \(\Big \downarrow \) 2485 |
| \(\displaystyle -\frac {9 \sqrt [4]{(x-1)^3} \int -\frac {12 \sqrt {3} \left (9 \left (x+\frac {1}{9}\right )-10\right )^{3/4}}{\left (\sqrt [3]{2} \left (\frac {38}{\sqrt [3]{187+9 \sqrt {93}}}+\sqrt [3]{374+18 \sqrt {93}}\right )-18 \left (x+\frac {1}{9}\right )\right ) \left (-162 \left (x+\frac {1}{9}\right )^2-9 \sqrt [3]{2} \left (\frac {38}{\sqrt [3]{187+9 \sqrt {93}}}+\sqrt [3]{374+18 \sqrt {93}}\right ) \left (x+\frac {1}{9}\right )-\sqrt [3]{2} \left (187+9 \sqrt {93}\right )^{2/3}-722 \left (\frac {2}{187+9 \sqrt {93}}\right )^{2/3}+38\right )}d\left (x+\frac {1}{9}\right )}{(x-1)^{3/4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {108 \sqrt {3} \sqrt [4]{(x-1)^3} \int \frac {\left (9 \left (x+\frac {1}{9}\right )-10\right )^{3/4}}{\left (\sqrt [3]{2} \left (\frac {38}{\sqrt [3]{187+9 \sqrt {93}}}+\sqrt [3]{374+18 \sqrt {93}}\right )-18 \left (x+\frac {1}{9}\right )\right ) \left (-162 \left (x+\frac {1}{9}\right )^2-9 \sqrt [3]{2} \left (\frac {38}{\sqrt [3]{187+9 \sqrt {93}}}+\sqrt [3]{374+18 \sqrt {93}}\right ) \left (x+\frac {1}{9}\right )-\sqrt [3]{2} \left (187+9 \sqrt {93}\right )^{2/3}-722 \left (\frac {2}{187+9 \sqrt {93}}\right )^{2/3}+38\right )}d\left (x+\frac {1}{9}\right )}{(x-1)^{3/4}}\) |
| \(\Big \downarrow \) 1199 |
| \(\displaystyle \frac {48 \sqrt {3} \sqrt [4]{(x-1)^3} \int \left (\frac {\sqrt {9 \left (x+\frac {1}{9}\right )-10} \left (\left (20-38 \sqrt [3]{\frac {2}{187+9 \sqrt {93}}}-2^{2/3} \sqrt [3]{187+9 \sqrt {93}}\right ) \left (9 \left (x+\frac {1}{9}\right )-10\right )+\sqrt [3]{2} \left (187+9 \sqrt {93}\right )^{2/3}+10\ 2^{2/3} \sqrt [3]{187+9 \sqrt {93}}+722 \left (\frac {2}{187+9 \sqrt {93}}\right )^{2/3}+380 \sqrt [3]{\frac {2}{187+9 \sqrt {93}}}+162\right )}{6 \left (38+722 \left (\frac {2}{187+9 \sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (187+9 \sqrt {93}\right )^{2/3}\right ) \left (2 \left (9 \left (x+\frac {1}{9}\right )-10\right )^2+\left (40+38 \sqrt [3]{\frac {2}{187+9 \sqrt {93}}}+2^{2/3} \sqrt [3]{187+9 \sqrt {93}}\right ) \left (9 \left (x+\frac {1}{9}\right )-10\right )+\sqrt [3]{2} \left (187+9 \sqrt {93}\right )^{2/3}+10\ 2^{2/3} \sqrt [3]{187+9 \sqrt {93}}+722 \left (\frac {2}{187+9 \sqrt {93}}\right )^{2/3}+380 \sqrt [3]{\frac {2}{187+9 \sqrt {93}}}+162\right )}-\frac {\left (20-2^{2/3} \left (\frac {19\ 2^{2/3}}{\sqrt [3]{187+9 \sqrt {93}}}+\sqrt [3]{187+9 \sqrt {93}}\right )\right ) \sqrt {9 \left (x+\frac {1}{9}\right )-10}}{6 \left (38+722 \left (\frac {2}{187+9 \sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (187+9 \sqrt {93}\right )^{2/3}\right ) \left (2 \left (9 \left (x+\frac {1}{9}\right )-10\right )-2^{2/3} \sqrt [3]{187+9 \sqrt {93}}-38 \sqrt [3]{\frac {2}{187+9 \sqrt {93}}}+20\right )}\right )d\sqrt [4]{9 \left (x+\frac {1}{9}\right )-10}}{(x-1)^{3/4}}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {48 \sqrt {3} \sqrt [4]{(x-1)^3} \int \left (\frac {\sqrt {9 \left (x+\frac {1}{9}\right )-10} \left (\left (20-38 \sqrt [3]{\frac {2}{187+9 \sqrt {93}}}-2^{2/3} \sqrt [3]{187+9 \sqrt {93}}\right ) \left (9 \left (x+\frac {1}{9}\right )-10\right )+\sqrt [3]{2} \left (187+9 \sqrt {93}\right )^{2/3}+10\ 2^{2/3} \sqrt [3]{187+9 \sqrt {93}}+722 \left (\frac {2}{187+9 \sqrt {93}}\right )^{2/3}+380 \sqrt [3]{\frac {2}{187+9 \sqrt {93}}}+162\right )}{6 \left (38+722 \left (\frac {2}{187+9 \sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (187+9 \sqrt {93}\right )^{2/3}\right ) \left (2 \left (9 \left (x+\frac {1}{9}\right )-10\right )^2+\left (40+38 \sqrt [3]{\frac {2}{187+9 \sqrt {93}}}+2^{2/3} \sqrt [3]{187+9 \sqrt {93}}\right ) \left (9 \left (x+\frac {1}{9}\right )-10\right )+\sqrt [3]{2} \left (187+9 \sqrt {93}\right )^{2/3}+10\ 2^{2/3} \sqrt [3]{187+9 \sqrt {93}}+722 \left (\frac {2}{187+9 \sqrt {93}}\right )^{2/3}+380 \sqrt [3]{\frac {2}{187+9 \sqrt {93}}}+162\right )}-\frac {\left (20-\frac {38 \sqrt [3]{2}+\left (374+18 \sqrt {93}\right )^{2/3}}{\sqrt [3]{187+9 \sqrt {93}}}\right ) \sqrt {9 \left (x+\frac {1}{9}\right )-10}}{6 \left (38+722 \left (\frac {2}{187+9 \sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (187+9 \sqrt {93}\right )^{2/3}\right ) \left (2 \left (9 \left (x+\frac {1}{9}\right )-10\right )-2^{2/3} \sqrt [3]{187+9 \sqrt {93}}-38 \sqrt [3]{\frac {2}{187+9 \sqrt {93}}}+20\right )}\right )d\sqrt [4]{9 \left (x+\frac {1}{9}\right )-10}}{(x-1)^{3/4}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {48 \sqrt {3} \sqrt [4]{(x-1)^3} \int \left (\frac {\sqrt {9 \left (x+\frac {1}{9}\right )-10} \left (\left (20-38 \sqrt [3]{\frac {2}{187+9 \sqrt {93}}}-2^{2/3} \sqrt [3]{187+9 \sqrt {93}}\right ) \left (9 \left (x+\frac {1}{9}\right )-10\right )+\sqrt [3]{2} \left (187+9 \sqrt {93}\right )^{2/3}+10\ 2^{2/3} \sqrt [3]{187+9 \sqrt {93}}+722 \left (\frac {2}{187+9 \sqrt {93}}\right )^{2/3}+380 \sqrt [3]{\frac {2}{187+9 \sqrt {93}}}+162\right )}{6 \left (38+722 \left (\frac {2}{187+9 \sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (187+9 \sqrt {93}\right )^{2/3}\right ) \left (2 \left (9 \left (x+\frac {1}{9}\right )-10\right )^2+\left (40+38 \sqrt [3]{\frac {2}{187+9 \sqrt {93}}}+2^{2/3} \sqrt [3]{187+9 \sqrt {93}}\right ) \left (9 \left (x+\frac {1}{9}\right )-10\right )+\sqrt [3]{2} \left (187+9 \sqrt {93}\right )^{2/3}+10\ 2^{2/3} \sqrt [3]{187+9 \sqrt {93}}+722 \left (\frac {2}{187+9 \sqrt {93}}\right )^{2/3}+380 \sqrt [3]{\frac {2}{187+9 \sqrt {93}}}+162\right )}+\frac {\left (38 \sqrt [3]{2}-20 \sqrt [3]{187+9 \sqrt {93}}+\left (374+18 \sqrt {93}\right )^{2/3}\right ) \sqrt {9 \left (x+\frac {1}{9}\right )-10}}{6 \sqrt [3]{187+9 \sqrt {93}} \left (38+722 \left (\frac {2}{187+9 \sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (187+9 \sqrt {93}\right )^{2/3}\right ) \left (2 \left (9 \left (x+\frac {1}{9}\right )-10\right )-2^{2/3} \sqrt [3]{187+9 \sqrt {93}}-38 \sqrt [3]{\frac {2}{187+9 \sqrt {93}}}+20\right )}\right )d\sqrt [4]{9 \left (x+\frac {1}{9}\right )-10}}{(x-1)^{3/4}}\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \frac {48 \sqrt {3} \sqrt [4]{(x-1)^3} \int \left (\frac {\sqrt {9 \left (x+\frac {1}{9}\right )-10} \left (\left (20-38 \sqrt [3]{\frac {2}{187+9 \sqrt {93}}}-2^{2/3} \sqrt [3]{187+9 \sqrt {93}}\right ) \left (9 \left (x+\frac {1}{9}\right )-10\right )+\sqrt [3]{2} \left (187+9 \sqrt {93}\right )^{2/3}+10\ 2^{2/3} \sqrt [3]{187+9 \sqrt {93}}+722 \left (\frac {2}{187+9 \sqrt {93}}\right )^{2/3}+380 \sqrt [3]{\frac {2}{187+9 \sqrt {93}}}+162\right )}{6 \left (38+722 \left (\frac {2}{187+9 \sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (187+9 \sqrt {93}\right )^{2/3}\right ) \left (2 \left (9 \left (x+\frac {1}{9}\right )-10\right )^2+\left (40+38 \sqrt [3]{\frac {2}{187+9 \sqrt {93}}}+2^{2/3} \sqrt [3]{187+9 \sqrt {93}}\right ) \left (9 \left (x+\frac {1}{9}\right )-10\right )+\sqrt [3]{2} \left (187+9 \sqrt {93}\right )^{2/3}+10\ 2^{2/3} \sqrt [3]{187+9 \sqrt {93}}+722 \left (\frac {2}{187+9 \sqrt {93}}\right )^{2/3}+380 \sqrt [3]{\frac {2}{187+9 \sqrt {93}}}+162\right )}+\frac {\left (38 \sqrt [3]{2}-20 \sqrt [3]{187+9 \sqrt {93}}+\left (374+18 \sqrt {93}\right )^{2/3}\right ) \sqrt {9 \left (x+\frac {1}{9}\right )-10}}{6 \sqrt [3]{187+9 \sqrt {93}} \left (38+722 \left (\frac {2}{187+9 \sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (187+9 \sqrt {93}\right )^{2/3}\right ) \left (2 \left (9 \left (x+\frac {1}{9}\right )-10\right )-2^{2/3} \sqrt [3]{187+9 \sqrt {93}}-38 \sqrt [3]{\frac {2}{187+9 \sqrt {93}}}+20\right )}\right )d\sqrt [4]{9 \left (x+\frac {1}{9}\right )-10}}{(x-1)^{3/4}}\) |
3.10.72.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
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]
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]
Timed out.
\[\int \frac {\left (x^{3}-3 x^{2}+3 x -1\right )^{\frac {1}{4}}}{3 x^{3}+x^{2}-2 x -1}d x\]
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.94 (sec) , antiderivative size = 4191, normalized size of antiderivative = 56.64 \[ \int \frac {\sqrt [4]{-1+3 x-3 x^2+x^3}}{-1-2 x+x^2+3 x^3} \, dx=\text {Too large to display} \]
-1/62*sqrt(31)*sqrt(2)*sqrt(-sqrt(2*sqrt(31)*sqrt(-3/31*((4/9)^(1/3)*(2246 9903*sqrt(93) + 389553327)^(1/3) + 617793*(4/9)^(2/3)/(22469903*sqrt(93) + 389553327)^(1/3) - 1071)^2 - 6426/31*(4/9)^(1/3)*(22469903*sqrt(93) + 389 553327)^(1/3) - 3969937818/31*(4/9)^(2/3)/(22469903*sqrt(93) + 389553327)^ (1/3) + 6736019/31) - 2*(4/9)^(1/3)*(22469903*sqrt(93) + 389553327)^(1/3) - 1235586*(4/9)^(2/3)/(22469903*sqrt(93) + 389553327)^(1/3) - 4284))*log(1 /31*(sqrt(2)*(1527*sqrt(31)*(x - 1)*((4/9)^(1/3)*(22469903*sqrt(93) + 3895 53327)^(1/3) + 617793*(4/9)^(2/3)/(22469903*sqrt(93) + 389553327)^(1/3) - 1071)^2 + 5272704*sqrt(31)*(x - 1)*((4/9)^(1/3)*(22469903*sqrt(93) + 38955 3327)^(1/3) + 617793*(4/9)^(2/3)/(22469903*sqrt(93) + 389553327)^(1/3) - 1 071) + 93*(509*(x - 1)*((4/9)^(1/3)*(22469903*sqrt(93) + 389553327)^(1/3) + 617793*(4/9)^(2/3)/(22469903*sqrt(93) + 389553327)^(1/3) - 1071) - 12215 1*x + 122151)*sqrt(-3/31*((4/9)^(1/3)*(22469903*sqrt(93) + 389553327)^(1/3 ) + 617793*(4/9)^(2/3)/(22469903*sqrt(93) + 389553327)^(1/3) - 1071)^2 - 6 426/31*(4/9)^(1/3)*(22469903*sqrt(93) + 389553327)^(1/3) - 3969937818/31*( 4/9)^(2/3)/(22469903*sqrt(93) + 389553327)^(1/3) + 6736019/31) + 1076537*s qrt(31)*(x - 1))*sqrt(-sqrt(2*sqrt(31)*sqrt(-3/31*((4/9)^(1/3)*(22469903*s qrt(93) + 389553327)^(1/3) + 617793*(4/9)^(2/3)/(22469903*sqrt(93) + 38955 3327)^(1/3) - 1071)^2 - 6426/31*(4/9)^(1/3)*(22469903*sqrt(93) + 389553327 )^(1/3) - 3969937818/31*(4/9)^(2/3)/(22469903*sqrt(93) + 389553327)^(1/...
Not integrable
Time = 3.39 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.30 \[ \int \frac {\sqrt [4]{-1+3 x-3 x^2+x^3}}{-1-2 x+x^2+3 x^3} \, dx=\int \frac {\sqrt [4]{\left (x - 1\right )^{3}}}{3 x^{3} + x^{2} - 2 x - 1}\, dx \]
Not integrable
Time = 0.22 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.45 \[ \int \frac {\sqrt [4]{-1+3 x-3 x^2+x^3}}{-1-2 x+x^2+3 x^3} \, dx=\int { \frac {{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}^{\frac {1}{4}}}{3 \, x^{3} + x^{2} - 2 \, x - 1} \,d x } \]
Not integrable
Time = 0.27 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.45 \[ \int \frac {\sqrt [4]{-1+3 x-3 x^2+x^3}}{-1-2 x+x^2+3 x^3} \, dx=\int { \frac {{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}^{\frac {1}{4}}}{3 \, x^{3} + x^{2} - 2 \, x - 1} \,d x } \]
Not integrable
Time = 6.18 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.50 \[ \int \frac {\sqrt [4]{-1+3 x-3 x^2+x^3}}{-1-2 x+x^2+3 x^3} \, dx=-\int \frac {{\left (x^3-3\,x^2+3\,x-1\right )}^{1/4}}{-3\,x^3-x^2+2\,x+1} \,d x \]