Integrand size = 33, antiderivative size = 113 \[ \int \frac {x (1+x) (1+3 x)}{(-1+x) (-1+3 x) \left (-x+x^3\right )^{2/3}} \, dx=-\frac {3 \sqrt [3]{x \left (-1+x^2\right )}}{-1+x}+\frac {1}{2} \left (-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} (-1+x)}{-1+x+2 \sqrt [3]{x \left (-1+x^2\right )}}\right )-2 \log \left (1-x+\sqrt [3]{x \left (-1+x^2\right )}\right )+\log \left (1-2 x+x^2+(-1+x) \sqrt [3]{x \left (-1+x^2\right )}+\left (x \left (-1+x^2\right )\right )^{2/3}\right )\right ) \]
[Out] -3*(x*(x^2-1))^(1/3)/(-1+x)-3^(1/2)*arctan(3^(1/2)*(-1+x)/(-1+x+2*(x*( x^2-1))^(1/3)))-ln(1-x+(x*(x^2-1))^(1/3))+1/2*ln(1-2*x+x^2+(-1+x)*(x*( x^2-1))^(1/3)+(x*(x^2-1))^(2/3))
\[ \int \frac {x (1+x) (1+3 x)}{(-1+x) (-1+3 x) \left (-x+x^3\right )^{2/3}} \, dx=\int \frac {x (1+x) (1+3 x)}{(-1+x) (-1+3 x) \left (-x+x^3\right )^{2/3}} \, dx \]
[In] Int[(x*(1 + x)*(1 + 3*x))/((-1 + x)*(-1 + 3*x)*(-x + x^3)^(2/3)),x]
[Out] (-3*x*(1 + x))/(2*(-x + x^3)^(2/3)) - (3*(1 - x)^(2/3)*x^2*(1 + x)^(2/ 3)*AppellF1[4/3, 5/3, -1/3, 7/3, x, -x])/(4*(-x + x^3)^(2/3)) - (5*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 + 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])/(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][Defe r[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 {\sqrt [3]{x} (1+x) (1+3 x)}{(-1+x) (-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 {x^3 \left (1+x^3\right ) \left (1+3 x^3\right )}{\left (-1+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 {x^3 \sqrt [3]{1+x^3} \left (1+3 x^3\right )}{\left (-1+x^3\right )^{5/3} \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 {2 \sqrt [3]{1+x^3}}{3 \left (-1+x^3\right )^{5/3}}+\frac {x^3 \sqrt [3]{1+x^3}}{\left (-1+x^3\right )^{5/3}}+\frac {2 \sqrt [3]{1+x^3}}{3 \left (-1+x^3\right )^{5/3} \left (-1+3 x^3\right )}\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 {\sqrt [3]{1+x^3}}{\left (-1+x^3\right )^{5/3}} \, 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 {\sqrt [3]{1+x^3}}{\left (-1+x^3\right )^{5/3} \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 \frac {x^3 \sqrt [3]{1+x^3}}{\left (-1+x^3\right )^{5/3}} \, dx,x,\sqrt [3]{x}\right ) \\ = -\frac {x (1+x)}{\left (-x+x^3\right )^{2/3}}-\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 {x^3 \sqrt [3]{1+x^3}}{\left (1-x^3\right )^{5/3}} \, dx,x,\sqrt [3]{x}\right )+\frac {2 (-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+x^3\right )^{5/3} \left (1+x^3\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )-\frac {(-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 {8 (-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+x^3\right )^{5/3} \left (1+x^3\right )^{2/3} \left (-1+3 x^3\right )} \, dx,x,\sqrt [3]{x}\right ) \\ = -\frac {7 x (1+x)}{6 \left (-x+x^3\right )^{2/3}}-\frac {3 (1-x)^{2/3} x^2 (1+x)^{2/3} F_1\left (\frac {4}{3};\frac {5}{3},-\frac {1}{3};\frac {7}{3};x,-x\right )}{4 \left (-x+x^3\right )^{2/3}}-\frac {(-1+x)^{2/3} x^{2/3} (1+x)^{2/3}}{2 \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}}{3 \left (-x+x^3\right )^{2/3}} \text {Subst}\left (\int \frac {1}{\left (-1+x^3\right )^{5/3} \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 {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 x (1+x)}{2 \left (-x+x^3\right )^{2/3}}-\frac {3 (1-x)^{2/3} x^2 (1+x)^{2/3} F_1\left (\frac {4}{3};\frac {5}{3},-\frac {1}{3};\frac {7}{3};x,-x\right )}{4 \left (-x+x^3\right )^{2/3}}-\frac {(-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 \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 {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 {x^{2/3} \left (-1+x^2\right )^{2/3}}{2 \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 x (1+x)}{2 \left (-x+x^3\right )^{2/3}}-\frac {3 (1-x)^{2/3} x^2 (1+x)^{2/3} F_1\left (\frac {4}{3};\frac {5}{3},-\frac {1}{3};\frac {7}{3};x,-x\right )}{4 \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 )}{2 \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 )-\frac {x^{2/3} \sqrt [6]{-1+x^2}}{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 {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 x (1+x)}{2 \left (-x+x^3\right )^{2/3}}-\frac {3 (1-x)^{2/3} x^2 (1+x)^{2/3} F_1\left (\frac {4}{3};\frac {5}{3},-\frac {1}{3};\frac {7}{3};x,-x\right )}{4 \left (-x+x^3\right )^{2/3}}-\frac {3^{3/4} 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 )}{4 \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 )-\frac {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 ) \\ = \text {Too large to display} \\ \end{align*}
Time = 31.58 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00 \[ \int \frac {x (1+x) (1+3 x)}{(-1+x) (-1+3 x) \left (-x+x^3\right )^{2/3}} \, dx=-\frac {3 \sqrt [3]{x \left (-1+x^2\right )}}{-1+x}+\frac {1}{2} \left (-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} (-1+x)}{-1+x+2 \sqrt [3]{x \left (-1+x^2\right )}}\right )-2 \log \left (1-x+\sqrt [3]{x \left (-1+x^2\right )}\right )+\log \left (1-2 x+x^2+(-1+x) \sqrt [3]{x \left (-1+x^2\right )}+\left (x \left (-1+x^2\right )\right )^{2/3}\right )\right ) \]
[In] Integrate[(x*(1 + x)*(1 + 3*x))/((-1 + x)*(-1 + 3*x)*(-x + x^3)^(2/3)) ,x]
[Out] (-3*(x*(-1 + x^2))^(1/3))/(-1 + x) + (-2*Sqrt[3]*ArcTan[(Sqrt[3]*(-1 + x))/(-1 + x + 2*(x*(-1 + x^2))^(1/3))] - 2*Log[1 - x + (x*(-1 + x^2)) ^(1/3)] + Log[1 - 2*x + x^2 + (-1 + x)*(x*(-1 + x^2))^(1/3) + (x*(-1 + x^2))^(2/3)])/2
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.35 (sec) , antiderivative size = 458, normalized size of antiderivative = 4.05
| method | result | size |
| trager | \(-\frac {3 \left (x^{3}-x \right )^{\frac {1}{3}}}{-1+x}-\ln \left (-\frac {-8730 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}+9702 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}-6381 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}} x +22698 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x -411 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}-2127 \left (x^{3}-x \right )^{\frac {2}{3}}+6381 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}}-1107 \left (x^{3}-x \right )^{\frac {1}{3}} x -10476 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}-13947 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x +2264 x^{2}+1107 \left (x^{3}-x \right )^{\frac {1}{3}}+4512 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+1415 x +283}{-1+3 x}\right )+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \ln \left (\frac {4005 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}+9702 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}-3321 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}} x -10413 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x -7716 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}-1107 \left (x^{3}-x \right )^{\frac {2}{3}}+3321 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}}-2127 \left (x^{3}-x \right )^{\frac {1}{3}} x +4806 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}+13173 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x +3679 x^{2}+2127 \left (x^{3}-x \right )^{\frac {1}{3}}-6963 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-2264 x +1981}{-1+3 x}\right )\) | \(458\) |
| risch | \(\text {Expression too large to display}\) | \(912\) |
[In] int(x*(1+x)*(1+3*x)/(-1+x)/(-1+3*x)/(x^3-x)^(2/3),x,method=_RETURNVERB OSE)
[Out] -3/(-1+x)*(x^3-x)^(1/3)-ln(-(-8730*RootOf(9*_Z^2-3*_Z+1)^2*x^2+9702*Ro otOf(9*_Z^2-3*_Z+1)*(x^3-x)^(2/3)-6381*RootOf(9*_Z^2-3*_Z+1)*(x^3-x)^( 1/3)*x+22698*RootOf(9*_Z^2-3*_Z+1)^2*x-411*RootOf(9*_Z^2-3*_Z+1)*x^2-2 127*(x^3-x)^(2/3)+6381*RootOf(9*_Z^2-3*_Z+1)*(x^3-x)^(1/3)-1107*(x^3-x )^(1/3)*x-10476*RootOf(9*_Z^2-3*_Z+1)^2-13947*RootOf(9*_Z^2-3*_Z+1)*x+ 2264*x^2+1107*(x^3-x)^(1/3)+4512*RootOf(9*_Z^2-3*_Z+1)+1415*x+283)/(-1 +3*x))+3*RootOf(9*_Z^2-3*_Z+1)*ln((4005*RootOf(9*_Z^2-3*_Z+1)^2*x^2+97 02*RootOf(9*_Z^2-3*_Z+1)*(x^3-x)^(2/3)-3321*RootOf(9*_Z^2-3*_Z+1)*(x^3 -x)^(1/3)*x-10413*RootOf(9*_Z^2-3*_Z+1)^2*x-7716*RootOf(9*_Z^2-3*_Z+1) *x^2-1107*(x^3-x)^(2/3)+3321*RootOf(9*_Z^2-3*_Z+1)*(x^3-x)^(1/3)-2127* (x^3-x)^(1/3)*x+4806*RootOf(9*_Z^2-3*_Z+1)^2+13173*RootOf(9*_Z^2-3*_Z+ 1)*x+3679*x^2+2127*(x^3-x)^(1/3)-6963*RootOf(9*_Z^2-3*_Z+1)-2264*x+198 1)/(-1+3*x))
none
Time = 0.52 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.15 \[ \int \frac {x (1+x) (1+3 x)}{(-1+x) (-1+3 x) \left (-x+x^3\right )^{2/3}} \, dx=-\frac {2 \, \sqrt {3} {\left (x - 1\right )} \arctan \left (\frac {612314840 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {1}{3}} {\left (x - 1\right )} + \sqrt {3} {\left (1609127381 \, x^{2} + 1235276981 \, x + 124616800\right )} + 2605939922 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {2}{3}}}{2990437623 \, x^{2} + 3108349623 \, x - 39304000}\right ) + {\left (x - 1\right )} \log \left (\frac {3 \, {\left (x^{3} - x\right )}^{\frac {1}{3}} {\left (x - 1\right )} + 3 \, x - 3 \, {\left (x^{3} - x\right )}^{\frac {2}{3}} - 1}{3 \, x - 1}\right ) + 6 \, {\left (x^{3} - x\right )}^{\frac {1}{3}}}{2 \, {\left (x - 1\right )}} \]
[In] integrate(x*(1+x)*(1+3*x)/(-1+x)/(-1+3*x)/(x^3-x)^(2/3),x, algorithm=" fricas")
[Out] -1/2*(2*sqrt(3)*(x - 1)*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)) + (x - 1)*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)^(1/3))/(x - 1)
\[ \int \frac {x (1+x) (1+3 x)}{(-1+x) (-1+3 x) \left (-x+x^3\right )^{2/3}} \, dx=\int \frac {x \left (x + 1\right ) \left (3 x + 1\right )}{\left (x \left (x - 1\right ) \left (x + 1\right )\right )^{\frac {2}{3}} \left (x - 1\right ) \left (3 x - 1\right )}\, dx \]
[In] integrate(x*(1+x)*(1+3*x)/(-1+x)/(-1+3*x)/(x**3-x)**(2/3),x)
[Out] Integral(x*(x + 1)*(3*x + 1)/((x*(x - 1)*(x + 1))**(2/3)*(x - 1)*(3*x - 1)), x)
\[ \int \frac {x (1+x) (1+3 x)}{(-1+x) (-1+3 x) \left (-x+x^3\right )^{2/3}} \, dx=\int { \frac {{\left (3 \, x + 1\right )} {\left (x + 1\right )} x}{{\left (x^{3} - x\right )}^{\frac {2}{3}} {\left (3 \, x - 1\right )} {\left (x - 1\right )}} \,d x } \]
[In] integrate(x*(1+x)*(1+3*x)/(-1+x)/(-1+3*x)/(x^3-x)^(2/3),x, algorithm=" maxima")
[Out] integrate((3*x + 1)*(x + 1)*x/((x^3 - x)^(2/3)*(3*x - 1)*(x - 1)), x)
\[ \int \frac {x (1+x) (1+3 x)}{(-1+x) (-1+3 x) \left (-x+x^3\right )^{2/3}} \, dx=\int { \frac {{\left (3 \, x + 1\right )} {\left (x + 1\right )} x}{{\left (x^{3} - x\right )}^{\frac {2}{3}} {\left (3 \, x - 1\right )} {\left (x - 1\right )}} \,d x } \]
[In] integrate(x*(1+x)*(1+3*x)/(-1+x)/(-1+3*x)/(x^3-x)^(2/3),x, algorithm=" giac")
[Out] integrate((3*x + 1)*(x + 1)*x/((x^3 - x)^(2/3)*(3*x - 1)*(x - 1)), x)
Timed out. \[ \int \frac {x (1+x) (1+3 x)}{(-1+x) (-1+3 x) \left (-x+x^3\right )^{2/3}} \, dx=\int \frac {x\,\left (3\,x+1\right )\,\left (x+1\right )}{{\left (x^3-x\right )}^{2/3}\,\left (3\,x-1\right )\,\left (x-1\right )} \,d x \]
[In] int((x*(3*x + 1)*(x + 1))/((x^3 - x)^(2/3)*(3*x - 1)*(x - 1)),x)
[Out] int((x*(3*x + 1)*(x + 1))/((x^3 - x)^(2/3)*(3*x - 1)*(x - 1)), x)