\(\int \frac {1-x^2+x^3}{(-1-x^2+x^3) \sqrt [3]{x^2+x^3}} \, dx\) [2531]

Optimal result

Integrand size = 34, antiderivative size = 212 \[ \int \frac {1-x^2+x^3}{\left (-1-x^2+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x^2+x^3}}\right )-\log \left (-x+\sqrt [3]{x^2+x^3}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{x^2+x^3}+\left (x^2+x^3\right )^{2/3}\right )+2 \text {RootSum}\left [-3+4 \text {$\#$1}^3-3 \text {$\#$1}^6+\text {$\#$1}^9\&,\frac {-\log (x)+\log \left (\sqrt [3]{x^2+x^3}-x \text {$\#$1}\right )+2 \log (x) \text {$\#$1}^3-2 \log \left (\sqrt [3]{x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3-\log (x) \text {$\#$1}^6+\log \left (\sqrt [3]{x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^6}{4 \text {$\#$1}-6 \text {$\#$1}^4+3 \text {$\#$1}^7}\&\right ] \]

[Out]

Unintegrable

Rubi [F]

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

[In]

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

[Out]

(Sqrt[3]*x^(2/3)*(1 + x)^(1/3)*ArcTan[(1 + (2*x^(1/3))/(1 + x)^(1/3))/Sqrt[3]])/(x^2 + x^3)^(1/3) - (3*x^(2/3)
*(1 + x)^(1/3)*Log[x^(1/3) - (1 + x)^(1/3)])/(2*(x^2 + x^3)^(1/3)) + (6*x^(2/3)*(1 + x)^(1/3)*Defer[Subst][Def
er[Int][1/((1 + x^3)^(1/3)*(-1 - x^6 + x^9)), x], x, x^(1/3)])/(x^2 + x^3)^(1/3)

Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1-x^2+x^3}{x^{2/3} \sqrt [3]{1+x} \left (-1-x^2+x^3\right )} \, dx}{\sqrt [3]{x^2+x^3}} \\ & = \frac {\left (3 x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {1-x^6+x^9}{\sqrt [3]{1+x^3} \left (-1-x^6+x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^3}} \\ & = \frac {\left (3 x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \left (\frac {1}{\sqrt [3]{1+x^3}}+\frac {2}{\sqrt [3]{1+x^3} \left (-1-x^6+x^9\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^3}} \\ & = \frac {\left (3 x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^3}}+\frac {\left (6 x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^3} \left (-1-x^6+x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^3}} \\ & = \frac {\sqrt {3} x^{2/3} \sqrt [3]{1+x} \arctan \left (\frac {1+\frac {2 \sqrt [3]{x}}{\sqrt [3]{1+x}}}{\sqrt {3}}\right )}{\sqrt [3]{x^2+x^3}}-\frac {3 x^{2/3} \sqrt [3]{1+x} \log \left (\sqrt [3]{x}-\sqrt [3]{1+x}\right )}{2 \sqrt [3]{x^2+x^3}}+\frac {\left (6 x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^3} \left (-1-x^6+x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.13 \[ \int \frac {1-x^2+x^3}{\left (-1-x^2+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\frac {x^{2/3} \sqrt [3]{1+x} \left (6 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}+2 \sqrt [3]{1+x}}\right )-6 \log \left (-\sqrt [3]{x}+\sqrt [3]{1+x}\right )+3 \log \left (x^{2/3}+\sqrt [3]{x} \sqrt [3]{1+x}+(1+x)^{2/3}\right )-4 \text {RootSum}\left [-3+4 \text {$\#$1}^3-3 \text {$\#$1}^6+\text {$\#$1}^9\&,\frac {\log (x)-3 \log \left (\sqrt [3]{1+x}-\sqrt [3]{x} \text {$\#$1}\right )-2 \log (x) \text {$\#$1}^3+6 \log \left (\sqrt [3]{1+x}-\sqrt [3]{x} \text {$\#$1}\right ) \text {$\#$1}^3+\log (x) \text {$\#$1}^6-3 \log \left (\sqrt [3]{1+x}-\sqrt [3]{x} \text {$\#$1}\right ) \text {$\#$1}^6}{4 \text {$\#$1}-6 \text {$\#$1}^4+3 \text {$\#$1}^7}\&\right ]\right )}{6 \sqrt [3]{x^2 (1+x)}} \]

[In]

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

[Out]

(x^(2/3)*(1 + x)^(1/3)*(6*Sqrt[3]*ArcTan[(Sqrt[3]*x^(1/3))/(x^(1/3) + 2*(1 + x)^(1/3))] - 6*Log[-x^(1/3) + (1
+ x)^(1/3)] + 3*Log[x^(2/3) + x^(1/3)*(1 + x)^(1/3) + (1 + x)^(2/3)] - 4*RootSum[-3 + 4*#1^3 - 3*#1^6 + #1^9 &
, (Log[x] - 3*Log[(1 + x)^(1/3) - x^(1/3)*#1] - 2*Log[x]*#1^3 + 6*Log[(1 + x)^(1/3) - x^(1/3)*#1]*#1^3 + Log[
x]*#1^6 - 3*Log[(1 + x)^(1/3) - x^(1/3)*#1]*#1^6)/(4*#1 - 6*#1^4 + 3*#1^7) & ]))/(6*(x^2*(1 + x))^(1/3))

Maple [N/A] (verified)

Time = 184.56 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.69

method	result	size
pseudoelliptic	\(2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{9}-3 \textit {\_Z}^{6}+4 \textit {\_Z}^{3}-3\right )}{\sum }\frac {\left (\textit {\_R}^{6}-2 \textit {\_R}^{3}+1\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}}{x}\right )}{3 \textit {\_R}^{7}-6 \textit {\_R}^{4}+4 \textit {\_R}}\right )-\ln \left (\frac {-x +\left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}}{x}\right )+\frac {\ln \left (\frac {\left (x^{2} \left (1+x \right )\right )^{\frac {2}{3}}+\left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )\)	\(147\)
trager	\(\text {Expression too large to display}\)	\(206332\)

[In]

int((x^3-x^2+1)/(x^3-x^2-1)/(x^3+x^2)^(1/3),x,method=_RETURNVERBOSE)

[Out]

2*sum((_R^6-2*_R^3+1)*ln((-_R*x+(x^2*(1+x))^(1/3))/x)/(3*_R^7-6*_R^4+4*_R),_R=RootOf(_Z^9-3*_Z^6+4*_Z^3-3))-ln
((-x+(x^2*(1+x))^(1/3))/x)+1/2*ln(((x^2*(1+x))^(2/3)+(x^2*(1+x))^(1/3)*x+x^2)/x^2)-3^(1/2)*arctan(1/3*(2*(x^2*
(1+x))^(1/3)+x)*3^(1/2)/x)

Fricas [C] (verification not implemented)

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

Time = 0.99 (sec) , antiderivative size = 3326, normalized size of antiderivative = 15.69 \[ \int \frac {1-x^2+x^3}{\left (-1-x^2+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/11532*5766^(2/3)*(sqrt(31)*sqrt(-496/3*(11*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) - 5797*(4/961)^(2/3)/(81*s
qrt(93) + 31)^(1/3) + 2)^2 + 21824/3*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) - 11501248/3*(4/961)^(2/3)/(81*sqr
t(93) + 31)^(1/3) - 524608/3) - 1364/3*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) + 718828/3*(4/961)^(2/3)/(81*sqr
t(93) + 31)^(1/3) + 496/3)^(1/3)*(sqrt(-3) - 1)*log(-4/837*((25420*5766^(1/3)*(sqrt(-3)*x + x)*(11*(4/961)^(1/
3)*(81*sqrt(93) + 31)^(1/3) - 5797*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) + 2)^2 - 221712*5766^(1/3)*(sqrt(-3)
*x + x)*(11*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) - 5797*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) + 2) + 3*(205
*5766^(1/3)*sqrt(31)*(sqrt(-3)*x + x)*(11*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) - 5797*(4/961)^(2/3)/(81*sqrt
(93) + 31)^(1/3) + 2) + 558*5766^(1/3)*sqrt(31)*(sqrt(-3)*x + x))*sqrt(-496/3*(11*(4/961)^(1/3)*(81*sqrt(93) +
31)^(1/3) - 5797*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) + 2)^2 + 21824/3*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/
3) - 11501248/3*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) - 524608/3) - 1218672*5766^(1/3)*(sqrt(-3)*x + x))*(sqr
t(31)*sqrt(-496/3*(11*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) - 5797*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) + 2
)^2 + 21824/3*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) - 11501248/3*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) - 524
608/3) - 1364/3*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) + 718828/3*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) + 496
/3)^(2/3) - 43401650688*(x^3 + x^2)^(1/3))/x) - 1/11532*5766^(2/3)*(sqrt(31)*sqrt(-496/3*(11*(4/961)^(1/3)*(81
*sqrt(93) + 31)^(1/3) - 5797*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) + 2)^2 + 21824/3*(4/961)^(1/3)*(81*sqrt(93
) + 31)^(1/3) - 11501248/3*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) - 524608/3) - 1364/3*(4/961)^(1/3)*(81*sqrt(
93) + 31)^(1/3) + 718828/3*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) + 496/3)^(1/3)*(sqrt(-3) + 1)*log(4/837*((25
420*5766^(1/3)*(sqrt(-3)*x - x)*(11*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) - 5797*(4/961)^(2/3)/(81*sqrt(93) +
31)^(1/3) + 2)^2 - 221712*5766^(1/3)*(sqrt(-3)*x - x)*(11*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) - 5797*(4/96
1)^(2/3)/(81*sqrt(93) + 31)^(1/3) + 2) + 3*(205*5766^(1/3)*sqrt(31)*(sqrt(-3)*x - x)*(11*(4/961)^(1/3)*(81*sqr
t(93) + 31)^(1/3) - 5797*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) + 2) + 558*5766^(1/3)*sqrt(31)*(sqrt(-3)*x - x
))*sqrt(-496/3*(11*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) - 5797*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) + 2)^2
+ 21824/3*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) - 11501248/3*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) - 524608
/3) - 1218672*5766^(1/3)*(sqrt(-3)*x - x))*(sqrt(31)*sqrt(-496/3*(11*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) -
5797*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) + 2)^2 + 21824/3*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) - 11501248
/3*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) - 524608/3) - 1364/3*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) + 718828
/3*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) + 496/3)^(2/3) + 43401650688*(x^3 + x^2)^(1/3))/x) + 1/11532*5766^(2
/3)*(-sqrt(31)*sqrt(-496/3*(11*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) - 5797*(4/961)^(2/3)/(81*sqrt(93) + 31)^
(1/3) + 2)^2 + 21824/3*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) - 11501248/3*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1
/3) - 524608/3) - 1364/3*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) + 718828/3*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1
/3) + 496/3)^(1/3)*(sqrt(-3) - 1)*log(-4/837*((25420*5766^(1/3)*(sqrt(-3)*x + x)*(11*(4/961)^(1/3)*(81*sqrt(93
) + 31)^(1/3) - 5797*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) + 2)^2 - 221712*5766^(1/3)*(sqrt(-3)*x + x)*(11*(4
/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) - 5797*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) + 2) - 3*(205*5766^(1/3)*sq
rt(31)*(sqrt(-3)*x + x)*(11*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) - 5797*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/
3) + 2) + 558*5766^(1/3)*sqrt(31)*(sqrt(-3)*x + x))*sqrt(-496/3*(11*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) - 5
797*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) + 2)^2 + 21824/3*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) - 11501248/
3*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) - 524608/3) - 1218672*5766^(1/3)*(sqrt(-3)*x + x))*(-sqrt(31)*sqrt(-4
96/3*(11*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) - 5797*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) + 2)^2 + 21824/3
*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) - 11501248/3*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) - 524608/3) - 1364
/3*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) + 718828/3*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) + 496/3)^(2/3) - 4
3401650688*(x^3 + x^2)^(1/3))/x) - 1/11532*5766^(2/3)*(-sqrt(31)*sqrt(-496/3*(11*(4/961)^(1/3)*(81*sqrt(93) +
31)^(1/3) - 5797*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) + 2)^2 + 21824/3*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3
) - 11501248/3*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) - 524608/3) - 1364/3*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1
/3) + 718828/3*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) + 496/3)^(1/3)*(sqrt(-3) + 1)*log(4/837*((25420*5766^(1/
3)*(sqrt(-3)*x - x)*(11*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) - 5797*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) +
2)^2 - 221712*5766^(1/3)*(sqrt(-3)*x - x)*(11*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) - 5797*(4/961)^(2/3)/(81
*sqrt(93) + 31)^(1/3) + 2) - 3*(205*5766^(1/3)*sqrt(31)*(sqrt(-3)*x - x)*(11*(4/961)^(1/3)*(81*sqrt(93) + 31)^
(1/3) - 5797*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) + 2) + 558*5766^(1/3)*sqrt(31)*(sqrt(-3)*x - x))*sqrt(-496
/3*(11*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) - 5797*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) + 2)^2 + 21824/3*(
4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) - 11501248/3*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) - 524608/3) - 121867
2*5766^(1/3)*(sqrt(-3)*x - x))*(-sqrt(31)*sqrt(-496/3*(11*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) - 5797*(4/961
)^(2/3)/(81*sqrt(93) + 31)^(1/3) + 2)^2 + 21824/3*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) - 11501248/3*(4/961)^
(2/3)/(81*sqrt(93) + 31)^(1/3) - 524608/3) - 1364/3*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) + 718828/3*(4/961)^
(2/3)/(81*sqrt(93) + 31)^(1/3) + 496/3)^(2/3) + 43401650688*(x^3 + x^2)^(1/3))/x) + 1/2*(44/279*(4/961)^(1/3)*
(81*sqrt(93) + 31)^(1/3) - 748/9*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) + 8/279)^(1/3)*(sqrt(-3) - 1)*log(16/2
7*((6355*(sqrt(-3)*x + x)*(11*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) - 5797*(4/961)^(2/3)/(81*sqrt(93) + 31)^(
1/3) + 2)^2 - 55428*(sqrt(-3)*x + x)*(11*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) - 5797*(4/961)^(2/3)/(81*sqrt(
93) + 31)^(1/3) + 2) + 5340708*sqrt(-3)*x + 5340708*x)*(44/279*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) - 748/9*
(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) + 8/279)^(2/3) + 940896*(x^3 + x^2)^(1/3))/x) - 1/2*(44/279*(4/961)^(1/
3)*(81*sqrt(93) + 31)^(1/3) - 748/9*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) + 8/279)^(1/3)*(sqrt(-3) + 1)*log(-
16/27*((6355*(sqrt(-3)*x - x)*(11*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) - 5797*(4/961)^(2/3)/(81*sqrt(93) + 3
1)^(1/3) + 2)^2 - 55428*(sqrt(-3)*x - x)*(11*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) - 5797*(4/961)^(2/3)/(81*s
qrt(93) + 31)^(1/3) + 2) + 5340708*sqrt(-3)*x - 5340708*x)*(44/279*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) - 74
8/9*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) + 8/279)^(2/3) - 940896*(x^3 + x^2)^(1/3))/x) + 1/5766*5766^(2/3)*(
sqrt(31)*sqrt(-496/3*(11*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) - 5797*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3)
+ 2)^2 + 21824/3*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) - 11501248/3*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) -
524608/3) - 1364/3*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) + 718828/3*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) +
496/3)^(1/3)*log(4/837*((25420*5766^(1/3)*x*(11*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) - 5797*(4/961)^(2/3)/(8
1*sqrt(93) + 31)^(1/3) + 2)^2 - 221712*5766^(1/3)*x*(11*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) - 5797*(4/961)^
(2/3)/(81*sqrt(93) + 31)^(1/3) + 2) + 3*(205*5766^(1/3)*sqrt(31)*x*(11*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3)
- 5797*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) + 2) + 558*5766^(1/3)*sqrt(31)*x)*sqrt(-496/3*(11*(4/961)^(1/3)*
(81*sqrt(93) + 31)^(1/3) - 5797*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) + 2)^2 + 21824/3*(4/961)^(1/3)*(81*sqrt
(93) + 31)^(1/3) - 11501248/3*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) - 524608/3) - 1218672*5766^(1/3)*x)*(sqrt
(31)*sqrt(-496/3*(11*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) - 5797*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) + 2)
^2 + 21824/3*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) - 11501248/3*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) - 5246
08/3) - 1364/3*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) + 718828/3*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) + 496/
3)^(2/3) + 21700825344*(x^3 + x^2)^(1/3))/x) + 1/5766*5766^(2/3)*(-sqrt(31)*sqrt(-496/3*(11*(4/961)^(1/3)*(81*
sqrt(93) + 31)^(1/3) - 5797*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) + 2)^2 + 21824/3*(4/961)^(1/3)*(81*sqrt(93)
+ 31)^(1/3) - 11501248/3*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) - 524608/3) - 1364/3*(4/961)^(1/3)*(81*sqrt(9
3) + 31)^(1/3) + 718828/3*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) + 496/3)^(1/3)*log(4/837*((25420*5766^(1/3)*x
*(11*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) - 5797*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) + 2)^2 - 221712*5766
^(1/3)*x*(11*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) - 5797*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) + 2) - 3*(20
5*5766^(1/3)*sqrt(31)*x*(11*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) - 5797*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/
3) + 2) + 558*5766^(1/3)*sqrt(31)*x)*sqrt(-496/3*(11*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) - 5797*(4/961)^(2/
3)/(81*sqrt(93) + 31)^(1/3) + 2)^2 + 21824/3*(4/961)^(1/3)*(81*sqrt(93) + 31)^(1/3) - 11501248/3*(4/961)^(2/3)
/(81*sqrt(93) + 31)^(1/3) - 524608/3) - 1218672*5766^(1/3)*x)*(-sqrt(31)*sqrt(-496/3*(11*(4/961)^(1/3)*(81*sqr
t(93) + 31)^(1/3) - 5797*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) + 2)^2 + 21824/3*(4/961)^(1/3)*(81*sqrt(93) +
31)^(1/3) - 11501248/3*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) - 524608/3) - 1364/3*(4/961)^(1/3)*(81*sqrt(93)
+ 31)^(1/3) + 718828/3*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) + 496/3)^(2/3) + 21700825344*(x^3 + x^2)^(1/3))/
x) - sqrt(3)*arctan(1/3*(sqrt(3)*x + 2*sqrt(3)*(x^3 + x^2)^(1/3))/x) + (44/279*(4/961)^(1/3)*(81*sqrt(93) + 31
)^(1/3) - 748/9*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) + 8/279)^(1/3)*log(-16/27*((6355*x*(11*(4/961)^(1/3)*(8
1*sqrt(93) + 31)^(1/3) - 5797*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) + 2)^2 - 55428*x*(11*(4/961)^(1/3)*(81*sq
rt(93) + 31)^(1/3) - 5797*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) + 2) + 5340708*x)*(44/279*(4/961)^(1/3)*(81*s
qrt(93) + 31)^(1/3) - 748/9*(4/961)^(2/3)/(81*sqrt(93) + 31)^(1/3) + 8/279)^(2/3) - 470448*(x^3 + x^2)^(1/3))/
x) - log(-(x - (x^3 + x^2)^(1/3))/x) + 1/2*log((x^2 + (x^3 + x^2)^(1/3)*x + (x^3 + x^2)^(2/3))/x^2)

Sympy [N/A]

Not integrable

Time = 15.32 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.13 \[ \int \frac {1-x^2+x^3}{\left (-1-x^2+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\int \frac {x^{3} - x^{2} + 1}{\sqrt [3]{x^{2} \left (x + 1\right )} \left (x^{3} - x^{2} - 1\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.16 \[ \int \frac {1-x^2+x^3}{\left (-1-x^2+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\int { \frac {x^{3} - x^{2} + 1}{{\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} - x^{2} - 1\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.16 \[ \int \frac {1-x^2+x^3}{\left (-1-x^2+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\int { \frac {x^{3} - x^{2} + 1}{{\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} - x^{2} - 1\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.17 \[ \int \frac {1-x^2+x^3}{\left (-1-x^2+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\int -\frac {x^3-x^2+1}{{\left (x^3+x^2\right )}^{1/3}\,\left (-x^3+x^2+1\right )} \,d x \]

[In]

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

[Out]

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