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

Optimal result

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

[Out]

Unintegrable

Rubi [F]

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

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \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 \\ \end{align*}

Mathematica [A] (verified)

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

[In]

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

[Out]

((-4*(-1 + x)*(-307788101 - 1357068302*x - 596630756*x^2 + 4979849490*x^3 + 5857310139*x^4 - 5802065412*x^5 -
9762576651*x^6 + 2006712954*x^7 + 5094769914*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] - 234521814*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))

Maple [F(-1)]

Timed out.

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

[In]

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

[Out]

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

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 \[ \int \frac {1}{\sqrt [4]{-1+3 x-3 x^2+x^3} \left (-1-2 x+x^2+3 x^3\right )^4} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

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

[In]

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

[Out]

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

Giac [N/A]

Not integrable

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

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

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

[In]

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

[Out]

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