\(\int \frac {1+x}{(-1-x+x^3) \sqrt [3]{-x^2+x^3}} \, dx\) [1344]
Optimal result
Integrand size = 27, antiderivative size = 97 \[
\int \frac {1+x}{\left (-1-x+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=-\text {RootSum}\left [-1+5 \text {$\#$1}^3-4 \text {$\#$1}^6+\text {$\#$1}^9\&,\frac {-2 \log (x)+2 \log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-5 \text {$\#$1}+3 \text {$\#$1}^4}\&\right ]
\]
[Out]
Unintegrable
Rubi [F]
\[
\int \frac {1+x}{\left (-1-x+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\int \frac {1+x}{\left (-1-x+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx
\]
[In]
Int[(1 + x)/((-1 - x + x^3)*(-x^2 + x^3)^(1/3)),x]
[Out]
(3*(-1 + x)^(1/3)*x^(2/3)*Defer[Subst][Defer[Int][1/((-1 + x^3)^(1/3)*(-1 - x^3 + x^9)), x], x, x^(1/3)])/(-x^
2 + x^3)^(1/3) + (3*(-1 + x)^(1/3)*x^(2/3)*Defer[Subst][Defer[Int][x^3/((-1 + x^3)^(1/3)*(-1 - x^3 + x^9)), x]
, x, x^(1/3)])/(-x^2 + x^3)^(1/3)
Rubi steps \begin{align*}
\text {integral}& = \frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1+x}{\sqrt [3]{-1+x} x^{2/3} \left (-1-x+x^3\right )} \, dx}{\sqrt [3]{-x^2+x^3}} \\ & = \frac {\left (3 \sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {1+x^3}{\sqrt [3]{-1+x^3} \left (-1-x^3+x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^3}} \\ & = \frac {\left (3 \sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \left (\frac {1}{\sqrt [3]{-1+x^3} \left (-1-x^3+x^9\right )}+\frac {x^3}{\sqrt [3]{-1+x^3} \left (-1-x^3+x^9\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^3}} \\ & = \frac {\left (3 \sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^3} \left (-1-x^3+x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^3}}+\frac {\left (3 \sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt [3]{-1+x^3} \left (-1-x^3+x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^3}} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.27 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.22
\[
\int \frac {1+x}{\left (-1-x+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=-\frac {\sqrt [3]{-1+x} x^{2/3} \text {RootSum}\left [-1+5 \text {$\#$1}^3-4 \text {$\#$1}^6+\text {$\#$1}^9\&,\frac {-2 \log (x)+6 \log \left (\sqrt [3]{-1+x}-\sqrt [3]{x} \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-3 \log \left (\sqrt [3]{-1+x}-\sqrt [3]{x} \text {$\#$1}\right ) \text {$\#$1}^3}{-5 \text {$\#$1}+3 \text {$\#$1}^4}\&\right ]}{3 \sqrt [3]{(-1+x) x^2}}
\]
[In]
Integrate[(1 + x)/((-1 - x + x^3)*(-x^2 + x^3)^(1/3)),x]
[Out]
-1/3*((-1 + x)^(1/3)*x^(2/3)*RootSum[-1 + 5*#1^3 - 4*#1^6 + #1^9 & , (-2*Log[x] + 6*Log[(-1 + x)^(1/3) - x^(1/
3)*#1] + Log[x]*#1^3 - 3*Log[(-1 + x)^(1/3) - x^(1/3)*#1]*#1^3)/(-5*#1 + 3*#1^4) & ])/((-1 + x)*x^2)^(1/3)
Maple [N/A] (verified)
Time = 189.62 (sec) , antiderivative size = 56, normalized size of antiderivative =
0.58
| | |
| method | result | size |
| | |
| pseudoelliptic |
\(\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{9}-4 \textit {\_Z}^{6}+5 \textit {\_Z}^{3}-1\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (\left (x -1\right ) x^{2}\right )^{\frac {1}{3}}}{x}\right ) \left (\textit {\_R}^{3}-2\right )}{3 \textit {\_R}^{4}-5 \textit {\_R}}\) |
\(56\) |
| trager |
\(\text {Expression too large to display}\) |
\(190624\) |
| | |
|
|
|
|
[In]
int((1+x)/(x^3-x-1)/(x^3-x^2)^(1/3),x,method=_RETURNVERBOSE)
[Out]
sum(ln((-_R*x+((x-1)*x^2)^(1/3))/x)*(_R^3-2)/(3*_R^4-5*_R),_R=RootOf(_Z^9-4*_Z^6+5*_Z^3-1))
Fricas [C] (verification not implemented)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.94 (sec) , antiderivative size = 3219, normalized size of antiderivative = 33.19
\[
\int \frac {1+x}{\left (-1-x+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\text {Too large to display}
\]
[In]
integrate((1+x)/(x^3-x-1)/(x^3-x^2)^(1/3),x, algorithm="fricas")
[Out]
1/2116*1058^(2/3)*(sqrt(23)*sqrt(-23/3*((4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 5221*(4/529)^(2/3)/(1041
*sqrt(69) + 33925)^(1/3) + 7)^2 + 322/3*(4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 1681162/3*(4/529)^(2/3)/
(1041*sqrt(69) + 33925)^(1/3) + 4759/3) - 23/3*(4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) - 120083/3*(4/529)^
(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 322/3)^(1/3)*(sqrt(-3) - 1)*log(-1/207*((2921*1058^(1/3)*(sqrt(-3)*x + x
)*((4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 5221*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 7)^2 - 645
15*1058^(1/3)*(sqrt(-3)*x + x)*((4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 5221*(4/529)^(2/3)/(1041*sqrt(69
) + 33925)^(1/3) + 7) + 3*(127*1058^(1/3)*sqrt(23)*(sqrt(-3)*x + x)*((4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/
3) + 5221*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 7) + 138*1058^(1/3)*sqrt(23)*(sqrt(-3)*x + x))*sqrt(-2
3/3*((4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 5221*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 7)^2 + 3
22/3*(4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 1681162/3*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 475
9/3) - 67896*1058^(1/3)*(sqrt(-3)*x + x))*(sqrt(23)*sqrt(-23/3*((4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) +
5221*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 7)^2 + 322/3*(4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) +
1681162/3*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 4759/3) - 23/3*(4/529)^(1/3)*(1041*sqrt(69) + 33925)^(
1/3) - 120083/3*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 322/3)^(2/3) - 26433072*(x^3 - x^2)^(1/3))/x) -
1/2116*1058^(2/3)*(sqrt(23)*sqrt(-23/3*((4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 5221*(4/529)^(2/3)/(1041
*sqrt(69) + 33925)^(1/3) + 7)^2 + 322/3*(4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 1681162/3*(4/529)^(2/3)/
(1041*sqrt(69) + 33925)^(1/3) + 4759/3) - 23/3*(4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) - 120083/3*(4/529)^
(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 322/3)^(1/3)*(sqrt(-3) + 1)*log(1/207*((2921*1058^(1/3)*(sqrt(-3)*x - x)
*((4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 5221*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 7)^2 - 6451
5*1058^(1/3)*(sqrt(-3)*x - x)*((4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 5221*(4/529)^(2/3)/(1041*sqrt(69)
+ 33925)^(1/3) + 7) + 3*(127*1058^(1/3)*sqrt(23)*(sqrt(-3)*x - x)*((4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3
) + 5221*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 7) + 138*1058^(1/3)*sqrt(23)*(sqrt(-3)*x - x))*sqrt(-23
/3*((4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 5221*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 7)^2 + 32
2/3*(4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 1681162/3*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 4759
/3) - 67896*1058^(1/3)*(sqrt(-3)*x - x))*(sqrt(23)*sqrt(-23/3*((4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 5
221*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 7)^2 + 322/3*(4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 1
681162/3*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 4759/3) - 23/3*(4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1
/3) - 120083/3*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 322/3)^(2/3) + 26433072*(x^3 - x^2)^(1/3))/x) + 1
/2116*1058^(2/3)*(-sqrt(23)*sqrt(-23/3*((4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 5221*(4/529)^(2/3)/(1041
*sqrt(69) + 33925)^(1/3) + 7)^2 + 322/3*(4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 1681162/3*(4/529)^(2/3)/
(1041*sqrt(69) + 33925)^(1/3) + 4759/3) - 23/3*(4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) - 120083/3*(4/529)^
(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 322/3)^(1/3)*(sqrt(-3) - 1)*log(-1/207*((2921*1058^(1/3)*(sqrt(-3)*x + x
)*((4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 5221*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 7)^2 - 645
15*1058^(1/3)*(sqrt(-3)*x + x)*((4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 5221*(4/529)^(2/3)/(1041*sqrt(69
) + 33925)^(1/3) + 7) - 3*(127*1058^(1/3)*sqrt(23)*(sqrt(-3)*x + x)*((4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/
3) + 5221*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 7) + 138*1058^(1/3)*sqrt(23)*(sqrt(-3)*x + x))*sqrt(-2
3/3*((4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 5221*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 7)^2 + 3
22/3*(4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 1681162/3*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 475
9/3) - 67896*1058^(1/3)*(sqrt(-3)*x + x))*(-sqrt(23)*sqrt(-23/3*((4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) +
5221*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 7)^2 + 322/3*(4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) +
1681162/3*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 4759/3) - 23/3*(4/529)^(1/3)*(1041*sqrt(69) + 33925)^
(1/3) - 120083/3*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 322/3)^(2/3) - 26433072*(x^3 - x^2)^(1/3))/x) -
1/2116*1058^(2/3)*(-sqrt(23)*sqrt(-23/3*((4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 5221*(4/529)^(2/3)/(10
41*sqrt(69) + 33925)^(1/3) + 7)^2 + 322/3*(4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 1681162/3*(4/529)^(2/3
)/(1041*sqrt(69) + 33925)^(1/3) + 4759/3) - 23/3*(4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) - 120083/3*(4/529
)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 322/3)^(1/3)*(sqrt(-3) + 1)*log(1/207*((2921*1058^(1/3)*(sqrt(-3)*x -
x)*((4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 5221*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 7)^2 - 64
515*1058^(1/3)*(sqrt(-3)*x - x)*((4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 5221*(4/529)^(2/3)/(1041*sqrt(6
9) + 33925)^(1/3) + 7) - 3*(127*1058^(1/3)*sqrt(23)*(sqrt(-3)*x - x)*((4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1
/3) + 5221*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 7) + 138*1058^(1/3)*sqrt(23)*(sqrt(-3)*x - x))*sqrt(-
23/3*((4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 5221*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 7)^2 +
322/3*(4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 1681162/3*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 47
59/3) - 67896*1058^(1/3)*(sqrt(-3)*x - x))*(-sqrt(23)*sqrt(-23/3*((4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3)
+ 5221*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 7)^2 + 322/3*(4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3)
+ 1681162/3*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 4759/3) - 23/3*(4/529)^(1/3)*(1041*sqrt(69) + 33925)
^(1/3) - 120083/3*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 322/3)^(2/3) + 26433072*(x^3 - x^2)^(1/3))/x)
+ 1/2*(1/69*(4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 227/3*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) +
7/69)^(1/3)*(sqrt(-3) - 1)*log(1/9*((2921*(sqrt(-3)*x + x)*((4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 5221
*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 7)^2 - 64515*(sqrt(-3)*x + x)*((4/529)^(1/3)*(1041*sqrt(69) + 3
3925)^(1/3) + 5221*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 7) + 150714*sqrt(-3)*x + 150714*x)*(1/69*(4/5
29)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 227/3*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 7/69)^(2/3) + 12
492*(x^3 - x^2)^(1/3))/x) - 1/2*(1/69*(4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 227/3*(4/529)^(2/3)/(1041*
sqrt(69) + 33925)^(1/3) + 7/69)^(1/3)*(sqrt(-3) + 1)*log(-1/9*((2921*(sqrt(-3)*x - x)*((4/529)^(1/3)*(1041*sqr
t(69) + 33925)^(1/3) + 5221*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 7)^2 - 64515*(sqrt(-3)*x - x)*((4/52
9)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 5221*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 7) + 150714*sqrt(-
3)*x - 150714*x)*(1/69*(4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 227/3*(4/529)^(2/3)/(1041*sqrt(69) + 3392
5)^(1/3) + 7/69)^(2/3) - 12492*(x^3 - x^2)^(1/3))/x) + 1/1058*1058^(2/3)*(sqrt(23)*sqrt(-23/3*((4/529)^(1/3)*(
1041*sqrt(69) + 33925)^(1/3) + 5221*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 7)^2 + 322/3*(4/529)^(1/3)*(
1041*sqrt(69) + 33925)^(1/3) + 1681162/3*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 4759/3) - 23/3*(4/529)^
(1/3)*(1041*sqrt(69) + 33925)^(1/3) - 120083/3*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 322/3)^(1/3)*log(
1/207*((2921*1058^(1/3)*x*((4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 5221*(4/529)^(2/3)/(1041*sqrt(69) + 3
3925)^(1/3) + 7)^2 - 64515*1058^(1/3)*x*((4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 5221*(4/529)^(2/3)/(104
1*sqrt(69) + 33925)^(1/3) + 7) + 3*(127*1058^(1/3)*sqrt(23)*x*((4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 5
221*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 7) + 138*1058^(1/3)*sqrt(23)*x)*sqrt(-23/3*((4/529)^(1/3)*(1
041*sqrt(69) + 33925)^(1/3) + 5221*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 7)^2 + 322/3*(4/529)^(1/3)*(1
041*sqrt(69) + 33925)^(1/3) + 1681162/3*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 4759/3) - 67896*1058^(1/
3)*x)*(sqrt(23)*sqrt(-23/3*((4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 5221*(4/529)^(2/3)/(1041*sqrt(69) +
33925)^(1/3) + 7)^2 + 322/3*(4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 1681162/3*(4/529)^(2/3)/(1041*sqrt(6
9) + 33925)^(1/3) + 4759/3) - 23/3*(4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) - 120083/3*(4/529)^(2/3)/(1041*
sqrt(69) + 33925)^(1/3) + 322/3)^(2/3) + 13216536*(x^3 - x^2)^(1/3))/x) + 1/1058*1058^(2/3)*(-sqrt(23)*sqrt(-2
3/3*((4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 5221*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 7)^2 + 3
22/3*(4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 1681162/3*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 475
9/3) - 23/3*(4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) - 120083/3*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3)
+ 322/3)^(1/3)*log(1/207*((2921*1058^(1/3)*x*((4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 5221*(4/529)^(2/3
)/(1041*sqrt(69) + 33925)^(1/3) + 7)^2 - 64515*1058^(1/3)*x*((4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 522
1*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 7) - 3*(127*1058^(1/3)*sqrt(23)*x*((4/529)^(1/3)*(1041*sqrt(69
) + 33925)^(1/3) + 5221*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 7) + 138*1058^(1/3)*sqrt(23)*x)*sqrt(-23
/3*((4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 5221*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 7)^2 + 32
2/3*(4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 1681162/3*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 4759
/3) - 67896*1058^(1/3)*x)*(-sqrt(23)*sqrt(-23/3*((4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 5221*(4/529)^(2
/3)/(1041*sqrt(69) + 33925)^(1/3) + 7)^2 + 322/3*(4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 1681162/3*(4/52
9)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 4759/3) - 23/3*(4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) - 120083/3
*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 322/3)^(2/3) + 13216536*(x^3 - x^2)^(1/3))/x) + (1/69*(4/529)^(
1/3)*(1041*sqrt(69) + 33925)^(1/3) + 227/3*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 7/69)^(1/3)*log(-1/9*
((2921*x*((4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 5221*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) + 7)^
2 - 64515*x*((4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 5221*(4/529)^(2/3)/(1041*sqrt(69) + 33925)^(1/3) +
7) + 150714*x)*(1/69*(4/529)^(1/3)*(1041*sqrt(69) + 33925)^(1/3) + 227/3*(4/529)^(2/3)/(1041*sqrt(69) + 33925)
^(1/3) + 7/69)^(2/3) - 6246*(x^3 - x^2)^(1/3))/x)
Sympy [F(-1)]
Timed out. \[
\int \frac {1+x}{\left (-1-x+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\text {Timed out}
\]
[In]
integrate((1+x)/(x**3-x-1)/(x**3-x**2)**(1/3),x)
[Out]
Timed out
Maxima [N/A]
Not integrable
Time = 0.23 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.28
\[
\int \frac {1+x}{\left (-1-x+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\int { \frac {x + 1}{{\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} - x - 1\right )}} \,d x }
\]
[In]
integrate((1+x)/(x^3-x-1)/(x^3-x^2)^(1/3),x, algorithm="maxima")
[Out]
integrate((x + 1)/((x^3 - x^2)^(1/3)*(x^3 - x - 1)), x)
Giac [N/A]
Not integrable
Time = 0.32 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.28
\[
\int \frac {1+x}{\left (-1-x+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\int { \frac {x + 1}{{\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} - x - 1\right )}} \,d x }
\]
[In]
integrate((1+x)/(x^3-x-1)/(x^3-x^2)^(1/3),x, algorithm="giac")
[Out]
integrate((x + 1)/((x^3 - x^2)^(1/3)*(x^3 - x - 1)), x)
Mupad [N/A]
Not integrable
Time = 6.19 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.29
\[
\int \frac {1+x}{\left (-1-x+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\int -\frac {x+1}{{\left (x^3-x^2\right )}^{1/3}\,\left (-x^3+x+1\right )} \,d x
\]
[In]
int(-(x + 1)/((x^3 - x^2)^(1/3)*(x - x^3 + 1)),x)
[Out]
int(-(x + 1)/((x^3 - x^2)^(1/3)*(x - x^3 + 1)), x)