\(\int \frac {(-1+x) (1+3 x)}{(-1+3 x) (-x+x^3)^{2/3}} \, dx\) [11]

Optimal result

Integrand size = 27, antiderivative size = 125 \[ \int \frac {(-1+x) (1+3 x)}{(-1+3 x) \left (-x+x^3\right )^{2/3}} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} \left (124616800+1235276981 x+1609127381 x^2\right )[phone] \sqrt {3} (-1+x) \sqrt [3]{-x+x^3}[phone] \sqrt {3} \left (-x+x^3\right )^{2/3}}{-39304000+3108349623 x+2990437623 x^2}\right )-\frac {1}{2} \log \left (\frac {-1+3 x+3 (-1+x) \sqrt [3]{-x+x^3}-3 \left (-x+x^3\right )^{2/3}}{-1+3 x}\right ) \]

[Out] -3^(1/2)*arctan((612314840*3^(1/2)*(x^3-x)^(1/3)*(-1+x)+3^(1/2)*(16091 
27381*x^2+1235276981*x[phone])[phone]*3^(1/2)*(x^3-x)^(2/3))/(2 
990437623*x^2+3108349623*x-39304000))-1/2*ln((3*(-1+x)*(x^3-x)^(1/3)+3 
*x-3*(x^3-x)^(2/3)-1)/(-1+3*x))
 

Rubi [F]

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

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

[Out] (-3*(1 - x)^(2/3)*x*(1 + x)^(2/3)*AppellF1[1/3, -1/3, 2/3, 4/3, x, -x] 
)/(-x + x^3)^(2/3) + (x*(1 - x^2)*(1 - x^(2/3)/(-1 + x^2)^(1/3))*Sqrt[ 
(1 + x^(4/3)/(-1 + x^2)^(2/3) + x^(2/3)/(-1 + x^2)^(1/3))/(1 - ((1 + S 
qrt[3])*x^(2/3))/(-1 + x^2)^(1/3))^2]*EllipticF[ArcCos[(1 - ((1 - Sqrt 
[3])*x^(2/3))/(-1 + x^2)^(1/3))/(1 - ((1 + Sqrt[3])*x^(2/3))/(-1 + x^2 
)^(1/3))], (2 + Sqrt[3])/4])/(3^(1/4)*(-x + x^3)^(2/3)*Sqrt[-((x^(2/3) 
*(1 - x^(2/3)/(-1 + x^2)^(1/3)))/((-1 + x^2)^(1/3)*(1 - ((1 + Sqrt[3]) 
*x^(2/3))/(-1 + x^2)^(1/3))^2))]) + (4*(-1 + x)^(2/3)*x^(2/3)*(1 + x)^ 
(2/3)*Defer[Subst][Defer[Int][1/((1 + (-3)^(1/3)*x)*(-1 + x^3)^(2/3)*( 
1 + x^3)^(2/3)), x], x, x^(1/3)])/(3*(-x + x^3)^(2/3)) + (4*(-1 + x)^( 
2/3)*x^(2/3)*(1 + x)^(2/3)*Defer[Subst][Defer[Int][1/((1 - 3^(1/3)*x)* 
(-1 + x^3)^(2/3)*(1 + x^3)^(2/3)), x], x, x^(1/3)])/(3*(-x + x^3)^(2/3 
)) + (4*(-1 + x)^(2/3)*x^(2/3)*(1 + x)^(2/3)*Defer[Subst][Defer[Int][1 
/((1 - (-1)^(2/3)*3^(1/3)*x)*(-1 + x^3)^(2/3)*(1 + x^3)^(2/3)), x], x, 
 x^(1/3)])/(3*(-x + x^3)^(2/3))
 

Rubi steps \begin{align*} \text {integral}= \frac {x^{2/3} \left (-1+x^2\right )^{2/3}}{\left (-x+x^3\right )^{2/3}} \int \frac {(-1+x) (1+3 x)}{x^{2/3} (-1+3 x) \left (-1+x^2\right )^{2/3}} \, dx \\ = \frac {3 x^{2/3} \left (-1+x^2\right )^{2/3}}{\left (-x+x^3\right )^{2/3}} \text {Subst}\left (\int \frac {\left (-1+x^3\right ) \left (1+3 x^3\right )}{\left (-1+3 x^3\right ) \left (-1+x^6\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right ) \\ = \frac {3 (-1+x)^{2/3} x^{2/3} (1+x)^{2/3}}{\left (-x+x^3\right )^{2/3}} \text {Subst}\left (\int \frac {\sqrt [3]{-1+x^3} \left (1+3 x^3\right )}{\left (1+x^3\right )^{2/3} \left (-1+3 x^3\right )} \, dx,x,\sqrt [3]{x}\right ) \\ = \frac {3 (-1+x)^{2/3} x^{2/3} (1+x)^{2/3}}{\left (-x+x^3\right )^{2/3}} \text {Subst}\left (\int \left (\frac {\sqrt [3]{-1+x^3}}{\left (1+x^3\right )^{2/3}}+\frac {2 \sqrt [3]{-1+x^3}}{\left (1+x^3\right )^{2/3} \left (-1+3 x^3\right )}\right ) \, dx,x,\sqrt [3]{x}\right ) \\ = \frac {3 (-1+x)^{2/3} x^{2/3} (1+x)^{2/3}}{\left (-x+x^3\right )^{2/3}} \text {Subst}\left (\int \frac {\sqrt [3]{-1+x^3}}{\left (1+x^3\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )+\frac {6 (-1+x)^{2/3} x^{2/3} (1+x)^{2/3}}{\left (-x+x^3\right )^{2/3}} \text {Subst}\left (\int \frac {\sqrt [3]{-1+x^3}}{\left (1+x^3\right )^{2/3} \left (-1+3 x^3\right )} \, dx,x,\sqrt [3]{x}\right ) \\ = \frac {2 (-1+x)^{2/3} x^{2/3} (1+x)^{2/3}}{\left (-x+x^3\right )^{2/3}} \text {Subst}\left (\int \frac {1}{\left (-1+x^3\right )^{2/3} \left (1+x^3\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )-\frac {4 (-1+x)^{2/3} x^{2/3} (1+x)^{2/3}}{\left (-x+x^3\right )^{2/3}} \text {Subst}\left (\int \frac {1}{\left (-1+x^3\right )^{2/3} \left (1+x^3\right )^{2/3} \left (-1+3 x^3\right )} \, dx,x,\sqrt [3]{x}\right )+\frac {3 (-1+x) x^{2/3} (1+x)^{2/3}}{\sqrt [3]{1-x} \left (-x+x^3\right )^{2/3}} \text {Subst}\left (\int \frac {\sqrt [3]{1-x^3}}{\left (1+x^3\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right ) \\ = \frac {3 (-1+x) x (1+x)^{2/3} F_1\left (\frac {1}{3};-\frac {1}{3},\frac {2}{3};\frac {4}{3};x,-x\right )}{\sqrt [3]{1-x} \left (-x+x^3\right )^{2/3}}-\frac {4 (-1+x)^{2/3} x^{2/3} (1+x)^{2/3}}{\left (-x+x^3\right )^{2/3}} \text {Subst}\left (\int \left (-\frac {1}{3 \left (1+\sqrt [3]{-3} x\right ) \left (-1+x^3\right )^{2/3} \left (1+x^3\right )^{2/3}}-\frac {1}{3 \left (1-\sqrt [3]{3} x\right ) \left (-1+x^3\right )^{2/3} \left (1+x^3\right )^{2/3}}-\frac {1}{3 \left (1-(-1)^{2/3} \sqrt [3]{3} x\right ) \left (-1+x^3\right )^{2/3} \left (1+x^3\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )+\frac {2 x^{2/3} \left (-1+x^2\right )^{2/3}}{\left (-x+x^3\right )^{2/3}} \text {Subst}\left (\int \frac {1}{\left (-1+x^6\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right ) \\ = \frac {3 (-1+x) x (1+x)^{2/3} F_1\left (\frac {1}{3};-\frac {1}{3},\frac {2}{3};\frac {4}{3};x,-x\right )}{\sqrt [3]{1-x} \left (-x+x^3\right )^{2/3}}+\frac {4 (-1+x)^{2/3} x^{2/3} (1+x)^{2/3}}{3 \left (-x+x^3\right )^{2/3}} \text {Subst}\left (\int \frac {1}{\left (1+\sqrt [3]{-3} x\right ) \left (-1+x^3\right )^{2/3} \left (1+x^3\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )+\frac {4 (-1+x)^{2/3} x^{2/3} (1+x)^{2/3}}{3 \left (-x+x^3\right )^{2/3}} \text {Subst}\left (\int \frac {1}{\left (1-\sqrt [3]{3} x\right ) \left (-1+x^3\right )^{2/3} \left (1+x^3\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )+\frac {4 (-1+x)^{2/3} x^{2/3} (1+x)^{2/3}}{3 \left (-x+x^3\right )^{2/3}} \text {Subst}\left (\int \frac {1}{\left (1-(-1)^{2/3} \sqrt [3]{3} x\right ) \left (-1+x^3\right )^{2/3} \left (1+x^3\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )+\frac {2 x^{2/3} \sqrt [6]{-1+x^2}}{\sqrt {\frac {1}{1-x^2}} \left (-x+x^3\right )^{2/3}} \text {Subst}\left (\int \frac {1}{\sqrt {1-x^6}} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [6]{-1+x^2}}\right ) \\ = \frac {3 (-1+x) x (1+x)^{2/3} F_1\left (\frac {1}{3};-\frac {1}{3},\frac {2}{3};\frac {4}{3};x,-x\right )}{\sqrt [3]{1-x} \left (-x+x^3\right )^{2/3}}+\frac {x \left (1-x^2\right ) \left (1-\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right ) \sqrt {\frac {1+\frac {x^{4/3}}{\left (-1+x^2\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}}{\left (1-\frac {\left (1+\sqrt {3}\right ) x^{2/3}}{\sqrt [3]{-1+x^2}}\right )^2}} F\left (\cos ^{-1}\left (\frac {1-\frac {\left (1-\sqrt {3}\right ) x^{2/3}}{\sqrt [3]{-1+x^2}}}{1-\frac {\left (1+\sqrt {3}\right ) x^{2/3}}{\sqrt [3]{-1+x^2}}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt [4]{3} \left (-x+x^3\right )^{2/3} \sqrt {-\frac {x^{2/3} \left (1-\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{\sqrt [3]{-1+x^2} \left (1-\frac {\left (1+\sqrt {3}\right ) x^{2/3}}{\sqrt [3]{-1+x^2}}\right )^2}}}+\frac {4 (-1+x)^{2/3} x^{2/3} (1+x)^{2/3}}{3 \left (-x+x^3\right )^{2/3}} \text {Subst}\left (\int \frac {1}{\left (1+\sqrt [3]{-3} x\right ) \left (-1+x^3\right )^{2/3} \left (1+x^3\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )+\frac {4 (-1+x)^{2/3} x^{2/3} (1+x)^{2/3}}{3 \left (-x+x^3\right )^{2/3}} \text {Subst}\left (\int \frac {1}{\left (1-\sqrt [3]{3} x\right ) \left (-1+x^3\right )^{2/3} \left (1+x^3\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )+\frac {4 (-1+x)^{2/3} x^{2/3} (1+x)^{2/3}}{3 \left (-x+x^3\right )^{2/3}} \text {Subst}\left (\int \frac {1}{\left (1-(-1)^{2/3} \sqrt [3]{3} x\right ) \left (-1+x^3\right )^{2/3} \left (1+x^3\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right ) \\ \end{align*}

Mathematica [F]

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

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

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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.60 (sec) , antiderivative size = 606, normalized size of antiderivative = 4.85

method	result	size
trager	\(\ln \left (-\frac {1415 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+3234 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}+3234 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}} x -3679 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +1819 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-2127 \left (x^{3}-x \right )^{\frac {2}{3}}-3234 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}}-2127 \left (x^{3}-x \right )^{\frac {1}{3}} x +1698 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}+2572 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -712 x^{2}+2127 \left (x^{3}-x \right )^{\frac {1}{3}}-251 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-445 x -89}{-1+3 x}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\frac {-1415 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+3234 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}+3234 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}} x +3679 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +4649 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-1107 \left (x^{3}-x \right )^{\frac {2}{3}}-3234 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}}-1107 \left (x^{3}-x \right )^{\frac {1}{3}} x -1698 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-4786 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -2522 x^{2}+1107 \left (x^{3}-x \right )^{\frac {1}{3}}+3145 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+1552 x -1358}{-1+3 x}\right )+\ln \left (\frac {-1415 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+3234 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}+3234 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}} x +3679 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +4649 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-1107 \left (x^{3}-x \right )^{\frac {2}{3}}-3234 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}}-1107 \left (x^{3}-x \right )^{\frac {1}{3}} x -1698 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-4786 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -2522 x^{2}+1107 \left (x^{3}-x \right )^{\frac {1}{3}}+3145 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+1552 x -1358}{-1+3 x}\right )\)	\(606\)

[In] int((-1+x)*(1+3*x)/(-1+3*x)/(x^3-x)^(2/3),x,method=_RETURNVERBOSE)
 

[Out] ln(-(1415*RootOf(_Z^2-_Z+1)^2*x^2+3234*RootOf(_Z^2-_Z+1)*(x^3-x)^(2/3) 
+3234*RootOf(_Z^2-_Z+1)*(x^3-x)^(1/3)*x-3679*RootOf(_Z^2-_Z+1)^2*x+181 
9*RootOf(_Z^2-_Z+1)*x^2-2127*(x^3-x)^(2/3)-3234*RootOf(_Z^2-_Z+1)*(x^3 
-x)^(1/3)-2127*(x^3-x)^(1/3)*x+1698*RootOf(_Z^2-_Z+1)^2+2572*RootOf(_Z 
^2-_Z+1)*x-712*x^2+2127*(x^3-x)^(1/3)-251*RootOf(_Z^2-_Z+1)-445*x-89)/ 
(-1+3*x))*RootOf(_Z^2-_Z+1)-RootOf(_Z^2-_Z+1)*ln((-1415*RootOf(_Z^2-_Z 
+1)^2*x^2+3234*RootOf(_Z^2-_Z+1)*(x^3-x)^(2/3)+3234*RootOf(_Z^2-_Z+1)* 
(x^3-x)^(1/3)*x+3679*RootOf(_Z^2-_Z+1)^2*x+4649*RootOf(_Z^2-_Z+1)*x^2- 
1107*(x^3-x)^(2/3)-3234*RootOf(_Z^2-_Z+1)*(x^3-x)^(1/3)-1107*(x^3-x)^( 
1/3)*x-1698*RootOf(_Z^2-_Z+1)^2-4786*RootOf(_Z^2-_Z+1)*x-2522*x^2+1107 
*(x^3-x)^(1/3)+3145*RootOf(_Z^2-_Z+1)+1552*x-1358)/(-1+3*x))+ln((-1415 
*RootOf(_Z^2-_Z+1)^2*x^2+3234*RootOf(_Z^2-_Z+1)*(x^3-x)^(2/3)+3234*Roo 
tOf(_Z^2-_Z+1)*(x^3-x)^(1/3)*x+3679*RootOf(_Z^2-_Z+1)^2*x+4649*RootOf( 
_Z^2-_Z+1)*x^2-1107*(x^3-x)^(2/3)-3234*RootOf(_Z^2-_Z+1)*(x^3-x)^(1/3) 
-1107*(x^3-x)^(1/3)*x-1698*RootOf(_Z^2-_Z+1)^2-4786*RootOf(_Z^2-_Z+1)* 
x-2522*x^2+1107*(x^3-x)^(1/3)+3145*RootOf(_Z^2-_Z+1)+1552*x-1358)/(-1+ 
3*x))
 

Fricas [A] (verification not implemented)

none

Time = 0.50 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.86 \[ \int \frac {(-1+x) (1+3 x)}{(-1+3 x) \left (-x+x^3\right )^{2/3}} \, dx=-\sqrt {3} \arctan \left (\frac {612314840 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {1}{3}} {\left (x - 1\right )} + \sqrt {3} {\left (1609127381 \, x^{2} + 1235276981 \, x + 124616800\right )} + 2605939922 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {2}{3}}}{2990437623 \, x^{2} + 3108349623 \, x - 39304000}\right ) - \frac {1}{2} \, \log \left (\frac {3 \, {\left (x^{3} - x\right )}^{\frac {1}{3}} {\left (x - 1\right )} + 3 \, x - 3 \, {\left (x^{3} - x\right )}^{\frac {2}{3}} - 1}{3 \, x - 1}\right ) \]

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

[Out] -sqrt(3)*arctan((612314840*sqrt(3)*(x^3 - x)^(1/3)*(x - 1) + sqrt(3)*( 
1609127381*x^2 + 1235276981*x + 124616800) + 2605939922*sqrt(3)*(x^3 - 
 x)^(2/3))/(2990437623*x^2 + 3108349623*x - 39304000)) - 1/2*log((3*(x 
^3 - x)^(1/3)*(x - 1) + 3*x - 3*(x^3 - x)^(2/3) - 1)/(3*x - 1))
 

Sympy [F]

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

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

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

Maxima [F]

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

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

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

Giac [F]

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

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

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

Mupad [F(-1)]

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

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

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