\(\int \frac {(-1+x)^2 (1+3 x)}{x (1+x) (-1+3 x) \sqrt [3]{-x+x^3}} \, dx\) [12]

Optimal result

Integrand size = 37, antiderivative size = 166 \[ \int \frac {(-1+x)^2 (1+3 x)}{x (1+x) (-1+3 x) \sqrt [3]{-x+x^3}} \, dx=\frac {-3 \left (-x+x^3\right )^{2/3}+2 \sqrt {3} \left (1-2 x+x^2\right ) \text {Arctan}\left (\frac {\sqrt {3} \left (124616800+1235276981 x+1609127381 x^2\right )[phone] \sqrt {3} (-1+x) \sqrt [3]{-x+x^3}[phone] \sqrt {3} \left (-x+x^3\right )^{2/3}}{-39304000+3108349623 x+2990437623 x^2}\right )-\left (1-2 x+x^2\right ) \log \left (\frac {-1+3 x+3 (-1+x) \sqrt [3]{-x+x^3}-3 \left (-x+x^3\right )^{2/3}}{-1+3 x}\right )}{2 \left (1-2 x+x^2\right )} \]

[Out]

1/2*(2*3^(1/2)*(x^2+x)*arctan((612314840*3^(1/2)*(x^3-x)^(1/3)*(-1+x)+3^(1/2)*(1609127381*x^2+1235276981*x+124
616800)[phone]*3^(1/2)*(x^3-x)^(2/3))/(2990437623*x^2+3108349623*x-39304000))-(x^2+x)*ln((3*(-1+x)*(x^3-x)
^(1/3)+3*x-3*(x^3-x)^(2/3)-1)/(-1+3*x))-6*(x^3-x)^(2/3))/(x^2+x)

Rubi [F]

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

[In]

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

[Out]

3/(-x + x^3)^(1/3) + (3*(1 + Sqrt[3])*x^(2/3))/((-1 + x^2)^(1/3)*(-x + x^3)^(1/3)*(1 - ((1 + Sqrt[3])*x^(2/3))
/(-1 + x^2)^(1/3))) - (18*x^2*(1 - x^2)^(1/3)*AppellF1[5/6, 1/3, 1, 11/6, x^2, 9*x^2])/(5*(-x + x^3)^(1/3)) +
(Sqrt[3]*x^(1/3)*(-1 + x^2)^(1/3)*ArcTan[(1 - (4*x^(2/3))/(-1 + x^2)^(1/3))/Sqrt[3]])/(2*(-x + x^3)^(1/3)) + (
Sqrt[3]*x^(1/3)*(-1 + x^2)^(1/3)*ArcTan[(1 + (2*x^(2/3))/(-1 + x^2)^(1/3))/Sqrt[3]])/(2*(-x + x^3)^(1/3)) - (3
*3^(1/4)*x^(2/3)*(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 + Sqrt[3])*x^(2/3))/(-1 + x^2)^(1/3))^2]*EllipticE[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])/((-1 + x^2)^(1/3)*(-x + x^3)
^(1/3)*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))]) - (3^(3/4)*(1 - Sqrt[3])*x^(2/3)*(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 + Sqrt[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])/(2*(-1 + x^2)^(1/3)*(-x + x^3)^(1/3)*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))]) + (x^(1/3)*(-1 + x^2)^(1/3)*Log[1 - 9*x^2])/(4*(-x + x^3)^(1
/3)) - (3*x^(1/3)*(-1 + x^2)^(1/3)*Log[x^(2/3) - (-1 + x^2)^(1/3)])/(4*(-x + x^3)^(1/3)) - (3*x^(1/3)*(-1 + x^
2)^(1/3)*Log[2*x^(2/3) + (-1 + x^2)^(1/3)])/(4*(-x + x^3)^(1/3)) + (2*x^(1/3)*(-1 + x^2)^(1/3)*Defer[Subst][De
fer[Int][1/((1 + x)*(-1 + x^6)^(1/3)), x], x, x^(1/3)])/(-x + x^3)^(1/3) - (2*(1 - I*Sqrt[3])*x^(1/3)*(-1 + x^
2)^(1/3)*Defer[Subst][Defer[Int][1/((-1 - I*Sqrt[3] + 2*x)*(-1 + x^6)^(1/3)), x], x, x^(1/3)])/(-x + x^3)^(1/3
) - (2*(1 + I*Sqrt[3])*x^(1/3)*(-1 + x^2)^(1/3)*Defer[Subst][Defer[Int][1/((-1 + I*Sqrt[3] + 2*x)*(-1 + x^6)^(
1/3)), x], x, x^(1/3)])/(-x + x^3)^(1/3)

Rubi steps \begin{align*} \text {integral}= \frac {\left (\sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \int \frac {(-1+x)^2 (1+3 x)}{x^{4/3} (1+x) (-1+3 x) \sqrt [3]{-1+x^2}} \, dx}{\sqrt [3]{-x+x^3}} \\ = \frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {\left (-1+x^3\right )^2 \left (1+3 x^3\right )}{x^2 \left (1+x^3\right ) \left (-1+3 x^3\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^3}} \\ = \frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \left (-\frac {1}{x^2 \sqrt [3]{-1+x^6}}+\frac {x}{\sqrt [3]{-1+x^6}}+\frac {2}{3 (1+x) \sqrt [3]{-1+x^6}}-\frac {2 (1+x)}{3 \left (1-x+x^2\right ) \sqrt [3]{-1+x^6}}+\frac {2 x}{\left (-1+3 x^3\right ) \sqrt [3]{-1+x^6}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^3}} \\ = \frac {\left (2 \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^3}}-\frac {\left (2 \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1+x}{\left (1-x+x^2\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^3}}-\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{x^2 \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^3}}+\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^3}}+\frac {\left (6 \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {x}{\left (-1+3 x^3\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^3}} \\ = -\frac {\left (3 \sqrt [3]{x}\right ) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1-x^6}} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [6]{-1+x^2}}\right )}{\sqrt {-\frac {1}{-1+x^2}} \sqrt [6]{-1+x^2} \sqrt [3]{-x+x^3}}+\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^3}} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^3}}+\frac {\left (2 \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^3}}-\frac {\left (2 \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \left (\frac {1-i \sqrt {3}}{\left (-1-i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^6}}+\frac {1+i \sqrt {3}}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^6}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^3}}+\frac {\left (6 \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \left (\frac {x}{\sqrt [3]{-1+x^6} \left (-1+9 x^6\right )}+\frac {3 x^4}{\sqrt [3]{-1+x^6} \left (-1+9 x^6\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^3}} \\ = \frac {3}{\sqrt [3]{-x+x^3}}+\frac {\sqrt {3} \sqrt [3]{x} \sqrt [3]{-1+x^2} \arctan \left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{2 \sqrt [3]{-x+x^3}}-\frac {3 \sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (x^{2/3}-\sqrt [3]{-1+x^2}\right )}{4 \sqrt [3]{-x+x^3}}+\frac {\left (6 \sqrt [3]{x}\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {1-x^6}} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [6]{-1+x^2}}\right )}{\sqrt {-\frac {1}{-1+x^2}} \sqrt [6]{-1+x^2} \sqrt [3]{-x+x^3}}+\frac {\left (2 \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^3}}+\frac {\left (6 \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt [3]{-1+x^6} \left (-1+9 x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^3}}+\frac {\left (18 \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt [3]{-1+x^6} \left (-1+9 x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^3}}-\frac {\left (2 \left (1-i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-1-i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^3}}-\frac {\left (2 \left (1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^3}} \\ = \frac {3}{\sqrt [3]{-x+x^3}}+\frac {\sqrt {3} \sqrt [3]{x} \sqrt [3]{-1+x^2} \arctan \left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{2 \sqrt [3]{-x+x^3}}-\frac {3 \sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (x^{2/3}-\sqrt [3]{-1+x^2}\right )}{4 \sqrt [3]{-x+x^3}}+\frac {\left (18 \sqrt [3]{x} \sqrt [3]{1-x^2}\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt [3]{1-x^6} \left (-1+9 x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^3}}-\frac {\left (3 \sqrt [3]{x}\right ) \text {Subst}\left (\int \frac {-1+\sqrt {3}-2 x^4}{\sqrt {1-x^6}} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [6]{-1+x^2}}\right )}{\sqrt {-\frac {1}{-1+x^2}} \sqrt [6]{-1+x^2} \sqrt [3]{-x+x^3}}+\frac {\left (3 \left (-1+\sqrt {3}\right ) \sqrt [3]{x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^6}} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt [6]{-1+x^2}}\right )}{\sqrt {-\frac {1}{-1+x^2}} \sqrt [6]{-1+x^2} \sqrt [3]{-x+x^3}}+\frac {\left (2 \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^3}}+\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^3} \left (-1+9 x^3\right )} \, dx,x,x^{2/3}\right )}{\sqrt [3]{-x+x^3}}-\frac {\left (2 \left (1-i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-1-i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^3}}-\frac {\left (2 \left (1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^3}} \\ = \frac {3}{\sqrt [3]{-x+x^3}}+\frac {3 \left (1+\sqrt {3}\right ) x^{2/3}}{\sqrt [3]{-1+x^2} \sqrt [3]{-x+x^3} \left (1-\frac {\left (1+\sqrt {3}\right ) x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}-\frac {18 x^2 \sqrt [3]{1-x^2} \operatorname {AppellF1}\left (\frac {5}{6},\frac {1}{3},1,\frac {11}{6},x^2,9 x^2\right )}{5 \sqrt [3]{-x+x^3}}+\frac {\sqrt {3} \sqrt [3]{x} \sqrt [3]{-1+x^2} \arctan \left (\frac {1-\frac {4 x^{2/3}}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{2 \sqrt [3]{-x+x^3}}+\frac {\sqrt {3} \sqrt [3]{x} \sqrt [3]{-1+x^2} \arctan \left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{2 \sqrt [3]{-x+x^3}}-\frac {3 \sqrt [4]{3} x^{2/3} \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}} E\left (\arccos \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 [3]{-1+x^2} \sqrt [3]{-x+x^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 {3^{3/4} \left (1-\sqrt {3}\right ) x^{2/3} \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}} \operatorname {EllipticF}\left (\arccos \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 )}{2 \sqrt [3]{-1+x^2} \sqrt [3]{-x+x^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 {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (1-9 x^2\right )}{4 \sqrt [3]{-x+x^3}}-\frac {3 \sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (x^{2/3}-\sqrt [3]{-1+x^2}\right )}{4 \sqrt [3]{-x+x^3}}-\frac {3 \sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (2 x^{2/3}+\sqrt [3]{-1+x^2}\right )}{4 \sqrt [3]{-x+x^3}}+\frac {\left (2 \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^3}}-\frac {\left (2 \left (1-i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-1-i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^3}}-\frac {\left (2 \left (1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^3}} \\ \end{align*}

Mathematica [F]

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

[In]

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

[Out]

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

Maple [C] (verified)

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

Time = 0.70 (sec) , antiderivative size = 461, normalized size of antiderivative = 2.78

method	result	size
trager	\(-\frac {3 \left (x^{3}-x \right )^{\frac {2}{3}}}{\left (1+x \right ) x}-\ln \left (\frac {11314080 \operatorname {RootOf}\left (11664 \textit {\_Z}^{2}-108 \textit {\_Z} +1\right )^{2} x^{2}+349272 \operatorname {RootOf}\left (11664 \textit {\_Z}^{2}-108 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}-229716 \operatorname {RootOf}\left (11664 \textit {\_Z}^{2}-108 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}} x -29416608 \operatorname {RootOf}\left (11664 \textit {\_Z}^{2}-108 \textit {\_Z} +1\right )^{2} x -224316 \operatorname {RootOf}\left (11664 \textit {\_Z}^{2}-108 \textit {\_Z} +1\right ) x^{2}-1107 \left (x^{3}-x \right )^{\frac {2}{3}}+229716 \operatorname {RootOf}\left (11664 \textit {\_Z}^{2}-108 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}}+3234 \left (x^{3}-x \right )^{\frac {1}{3}} x [phone] \operatorname {RootOf}\left (11664 \textit {\_Z}^{2}-108 \textit {\_Z} +1\right )^{2}+42660 \operatorname {RootOf}\left (11664 \textit {\_Z}^{2}-108 \textit {\_Z} +1\right ) x -1157 x^{2}-3234 \left (x^{3}-x \right )^{\frac {1}{3}}-88992 \operatorname {RootOf}\left (11664 \textit {\_Z}^{2}-108 \textit {\_Z} +1\right )+712 x -623}{-1+3 x}\right )+108 \operatorname {RootOf}\left (11664 \textit {\_Z}^{2}-108 \textit {\_Z} +1\right ) \ln \left (-\frac {-5190480 \operatorname {RootOf}\left (11664 \textit {\_Z}^{2}-108 \textit {\_Z} +1\right )^{2} x^{2}+349272 \operatorname {RootOf}\left (11664 \textit {\_Z}^{2}-108 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}-119556 \operatorname {RootOf}\left (11664 \textit {\_Z}^{2}-108 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}} x [phone] \operatorname {RootOf}\left (11664 \textit {\_Z}^{2}-108 \textit {\_Z} +1\right )^{2} x -181656 \operatorname {RootOf}\left (11664 \textit {\_Z}^{2}-108 \textit {\_Z} +1\right ) x^{2}-2127 \left (x^{3}-x \right )^{\frac {2}{3}}+119556 \operatorname {RootOf}\left (11664 \textit {\_Z}^{2}-108 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}}+3234 \left (x^{3}-x \right )^{\frac {1}{3}} x -6228576 \operatorname {RootOf}\left (11664 \textit {\_Z}^{2}-108 \textit {\_Z} +1\right )^{2}+224316 \operatorname {RootOf}\left (11664 \textit {\_Z}^{2}-108 \textit {\_Z} +1\right ) x -1552 x^{2}-3234 \left (x^{3}-x \right )^{\frac {1}{3}}-135324 \operatorname {RootOf}\left (11664 \textit {\_Z}^{2}-108 \textit {\_Z} +1\right )-970 x -194}{-1+3 x}\right )\)	\(461\)
risch	\(-\frac {3 \left (-1+x \right )}{{\left (x \left (x^{2}-1\right )\right )}^{\frac {1}{3}}}+\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 +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 )+\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 )\)	\(621\)

[In]

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

[Out]

-3/(1+x)/x*(x^3-x)^(2/3)-ln((11314080*RootOf(11664*_Z^2-108*_Z+1)^2*x^2+349272*RootOf(11664*_Z^2-108*_Z+1)*(x^
3-x)^(2/3)-229716*RootOf(11664*_Z^2-108*_Z+1)*(x^3-x)^(1/3)*x-29416608*RootOf(11664*_Z^2-108*_Z+1)^2*x-224316*
RootOf(11664*_Z^2-108*_Z+1)*x^2-1107*(x^3-x)^(2/3)+229716*RootOf(11664*_Z^2-108*_Z+1)*(x^3-x)^(1/3)+3234*(x^3-
x)^(1/3)*x+13576896*RootOf(11664*_Z^2-108*_Z+1)^2+42660*RootOf(11664*_Z^2-108*_Z+1)*x-1157*x^2-3234*(x^3-x)^(1
/3)-88992*RootOf(11664*_Z^2-108*_Z+1)+712*x-623)/(-1+3*x))+108*RootOf(11664*_Z^2-108*_Z+1)*ln(-(-5190480*RootO
f(11664*_Z^2-108*_Z+1)^2*x^2+349272*RootOf(11664*_Z^2-108*_Z+1)*(x^3-x)^(2/3)-119556*RootOf(11664*_Z^2-108*_Z+
1)*(x^3-x)^(1/3)*x+13495248*RootOf(11664*_Z^2-108*_Z+1)^2*x-181656*RootOf(11664*_Z^2-108*_Z+1)*x^2-2127*(x^3-x
)^(2/3)+119556*RootOf(11664*_Z^2-108*_Z+1)*(x^3-x)^(1/3)+3234*(x^3-x)^(1/3)*x-6228576*RootOf(11664*_Z^2-108*_Z
+1)^2+224316*RootOf(11664*_Z^2-108*_Z+1)*x-1552*x^2-3234*(x^3-x)^(1/3)-135324*RootOf(11664*_Z^2-108*_Z+1)-970*
x-194)/(-1+3*x))

Fricas [A] (verification not implemented)

none

Time = 0.65 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.83 \[ \int \frac {(-1+x)^2 (1+3 x)}{x (1+x) (-1+3 x) \sqrt [3]{-x+x^3}} \, dx=\frac {2 \, \sqrt {3} {\left (x^{2} + x\right )} \arctan \left (\frac {612314840 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {1}{3}} {\left (x - 1\right )} + \sqrt {3} {\left (1609127381 \, x^{2} + 1235276981 \, x + 124616800\right )} + 2605939922 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {2}{3}}}{2990437623 \, x^{2} + 3108349623 \, x - 39304000}\right ) - {\left (x^{2} + x\right )} \log \left (\frac {3 \, {\left (x^{3} - x\right )}^{\frac {1}{3}} {\left (x - 1\right )} + 3 \, x - 3 \, {\left (x^{3} - x\right )}^{\frac {2}{3}} - 1}{3 \, x - 1}\right ) - 6 \, {\left (x^{3} - x\right )}^{\frac {2}{3}}}{2 \, {\left (x^{2} + x\right )}} \]

[In]

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

[Out]

1/2*(2*sqrt(3)*(x^2 + x)*arctan((612314840*sqrt(3)*(x^3 - x)^(1/3)*(x - 1) + sqrt(3)*(1609127381*x^2 + 1235276
981*x + 124616800) + 2605939922*sqrt(3)*(x^3 - x)^(2/3))/(2990437623*x^2 + 3108349623*x - 39304000)) - (x^2 +
x)*log((3*(x^3 - x)^(1/3)*(x - 1) + 3*x - 3*(x^3 - x)^(2/3) - 1)/(3*x - 1)) - 6*(x^3 - x)^(2/3))/(x^2 + x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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