Integrand size = 37, antiderivative size = 166 \[ \int \frac {(-1+x)^2 (1+3 x)}{x (1+x) (-1+3 x) \sqrt [3]{-x+x^3}} \, dx=\frac {-3 \left (-x+x^3\right )^{2/3}+2 \sqrt {3} \left (1-2 x+x^2\right ) \text {Arctan}\left (\frac {\sqrt {3} \left (124616800+1235276981 x+1609127381 x^2\right )+612314840 \sqrt {3} (-1+x) \sqrt [3]{-x+x^3}+2605939922 \sqrt {3} \left (-x+x^3\right )^{2/3}}{-39304000+3108349623 x+2990437623 x^2}\right )-\left (1-2 x+x^2\right ) \log \left (\frac {-1+3 x+3 (-1+x) \sqrt [3]{-x+x^3}-3 \left (-x+x^3\right )^{2/3}}{-1+3 x}\right )}{2 \left (1-2 x+x^2\right )} \]
[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+124616800)+2605939922*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)
\[ \int \frac {(-1+x)^2 (1+3 x)}{x (1+x) (-1+3 x) \sqrt [3]{-x+x^3}} \, dx=\int \frac {(-1+x)^2 (1+3 x)}{x (1+x) (-1+3 x) \sqrt [3]{-x+x^3}} \, dx \]
[In] Int[((-1 + x)^2*(1 + 3*x))/(x*(1 + x)*(-1 + 3*x)*(-x + x^3)^(1/3)),x]
[Out] (3*(1 - x)^(1/3)*(1 + x)^(1/3)*AppellF1[-1/3, -5/3, 4/3, 2/3, x, -x])/ (-x + x^3)^(1/3) + (18*(-1 + x)^(1/3)*x^(1/3)*(1 + x)^(1/3)*Defer[Subs t][Defer[Int][(x*(-1 + x^3)^(5/3))/((1 + x^3)^(4/3)*(-1 + 3*x^3)), x], x, x^(1/3)])/(-x + x^3)^(1/3)
Rubi steps \begin{align*} \text {integral}= \frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{\sqrt [3]{-x+x^3}} \int \frac {(-1+x)^2 (1+3 x)}{x^{4/3} (1+x) (-1+3 x) \sqrt [3]{-1+x^2}} \, dx \\ = \frac {3 \sqrt [3]{x} \sqrt [3]{-1+x^2}}{\sqrt [3]{-x+x^3}} \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 ) \\ = \frac {3 \sqrt [3]{-1+x} \sqrt [3]{x} \sqrt [3]{1+x}}{\sqrt [3]{-x+x^3}} \text {Subst}\left (\int \frac {\left (-1+x^3\right )^{5/3} \left (1+3 x^3\right )}{x^2 \left (1+x^3\right )^{4/3} \left (-1+3 x^3\right )} \, dx,x,\sqrt [3]{x}\right ) \\ = \frac {3 \sqrt [3]{-1+x} \sqrt [3]{x} \sqrt [3]{1+x}}{\sqrt [3]{-x+x^3}} \text {Subst}\left (\int \left (-\frac {\left (-1+x^3\right )^{5/3}}{x^2 \left (1+x^3\right )^{4/3}}+\frac {6 x \left (-1+x^3\right )^{5/3}}{\left (1+x^3\right )^{4/3} \left (-1+3 x^3\right )}\right ) \, dx,x,\sqrt [3]{x}\right ) \\ = -\left (\frac {3 \sqrt [3]{-1+x} \sqrt [3]{x} \sqrt [3]{1+x}}{\sqrt [3]{-x+x^3}} \text {Subst}\left (\int \frac {\left (-1+x^3\right )^{5/3}}{x^2 \left (1+x^3\right )^{4/3}} \, dx,x,\sqrt [3]{x}\right )\right )+\frac {18 \sqrt [3]{-1+x} \sqrt [3]{x} \sqrt [3]{1+x}}{\sqrt [3]{-x+x^3}} \text {Subst}\left (\int \frac {x \left (-1+x^3\right )^{5/3}}{\left (1+x^3\right )^{4/3} \left (-1+3 x^3\right )} \, dx,x,\sqrt [3]{x}\right ) \\ = \frac {18 \sqrt [3]{-1+x} \sqrt [3]{x} \sqrt [3]{1+x}}{\sqrt [3]{-x+x^3}} \text {Subst}\left (\int \frac {x \left (-1+x^3\right )^{5/3}}{\left (1+x^3\right )^{4/3} \left (-1+3 x^3\right )} \, dx,x,\sqrt [3]{x}\right )+\frac {3 (-1+x) \sqrt [3]{x} \sqrt [3]{1+x}}{(1-x)^{2/3} \sqrt [3]{-x+x^3}} \text {Subst}\left (\int \frac {\left (1-x^3\right )^{5/3}}{x^2 \left (1+x^3\right )^{4/3}} \, dx,x,\sqrt [3]{x}\right ) \\ = -\frac {3 (-1+x) \sqrt [3]{1+x} F_1\left (-\frac {1}{3};-\frac {5}{3},\frac {4}{3};\frac {2}{3};x,-x\right )}{(1-x)^{2/3} \sqrt [3]{-x+x^3}}+\frac {18 \sqrt [3]{-1+x} \sqrt [3]{x} \sqrt [3]{1+x}}{\sqrt [3]{-x+x^3}} \text {Subst}\left (\int \frac {x \left (-1+x^3\right )^{5/3}}{\left (1+x^3\right )^{4/3} \left (-1+3 x^3\right )} \, dx,x,\sqrt [3]{x}\right ) \\ \end{align*}
\[ \int \frac {(-1+x)^2 (1+3 x)}{x (1+x) (-1+3 x) \sqrt [3]{-x+x^3}} \, dx=\int \frac {(-1+x)^2 (1+3 x)}{x (1+x) (-1+3 x) \sqrt [3]{-x+x^3}} \, dx \]
[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]
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 +13576896 \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 +13495248 \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=_RETURNVE RBOSE)
[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*Ro otOf(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*RootOf(11664*_Z ^2-108*_Z+1)^2*x^2+349272*RootOf(11664*_Z^2-108*_Z+1)*(x^3-x)^(2/3)-11 9556*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))
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 + 1235276981*x + 124616800) + 260593992 2*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)
\[ \int \frac {(-1+x)^2 (1+3 x)}{x (1+x) (-1+3 x) \sqrt [3]{-x+x^3}} \, dx=\int \frac {\left (x - 1\right )^{2} \cdot \left (3 x + 1\right )}{x \sqrt [3]{x \left (x - 1\right ) \left (x + 1\right )} \left (x + 1\right ) \left (3 x - 1\right )}\, dx \]
[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)
\[ \int \frac {(-1+x)^2 (1+3 x)}{x (1+x) (-1+3 x) \sqrt [3]{-x+x^3}} \, dx=\int { \frac {{\left (3 \, x + 1\right )} {\left (x - 1\right )}^{2}}{{\left (x^{3} - x\right )}^{\frac {1}{3}} {\left (3 \, x - 1\right )} {\left (x + 1\right )} x} \,d x } \]
[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 )
\[ \int \frac {(-1+x)^2 (1+3 x)}{x (1+x) (-1+3 x) \sqrt [3]{-x+x^3}} \, dx=\int { \frac {{\left (3 \, x + 1\right )} {\left (x - 1\right )}^{2}}{{\left (x^{3} - x\right )}^{\frac {1}{3}} {\left (3 \, x - 1\right )} {\left (x + 1\right )} x} \,d x } \]
[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 )
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)