Integrand size = 37, antiderivative size = 166 \[ \int \frac {(-1+x)^2 (1+3 x)}{x (1+x) (-1+3 x) \sqrt [3]{-x+x^3}} \, dx=\frac {-3 \left (-x+x^3\right )^{2/3}+2 \sqrt {3} \left (1-2 x+x^2\right ) \text {Arctan}\left (\frac {\sqrt {3} \left (124616800+1235276981 x+1609127381 x^2\right )+612314840 \sqrt {3} (-1+x) \sqrt [3]{-x+x^3}+2605939922 \sqrt {3} \left (-x+x^3\right )^{2/3}}{-39304000+3108349623 x+2990437623 x^2}\right )-\left (1-2 x+x^2\right ) \log \left (\frac {-1+3 x+3 (-1+x) \sqrt [3]{-x+x^3}-3 \left (-x+x^3\right )^{2/3}}{-1+3 x}\right )}{2 \left (1-2 x+x^2\right )} \] Output:
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)
\[ \int \frac {(-1+x)^2 (1+3 x)}{x (1+x) (-1+3 x) \sqrt [3]{-x+x^3}} \, dx=\int \frac {(-1+x)^2 (1+3 x)}{x (1+x) (-1+3 x) \sqrt [3]{-x+x^3}} \, dx \] Input:
Integrate[((-1 + x)^2*(1 + 3*x))/(x*(1 + x)*(-1 + 3*x)*(-x + x^3)^(1/3)),x ]
Output:
Integrate[((-1 + x)^2*(1 + 3*x))/(x*(1 + x)*(-1 + 3*x)*(-x + x^3)^(1/3)), x]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(x-1)^2 (3 x+1)}{x (x+1) (3 x-1) \sqrt [3]{x^3-x}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [3]{x} \sqrt [3]{x^2-1} \int -\frac {(1-x)^2 (3 x+1)}{(1-3 x) x^{4/3} (x+1) \sqrt [3]{x^2-1}}dx}{\sqrt [3]{x^3-x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [3]{x} \sqrt [3]{x^2-1} \int \frac {(1-x)^2 (3 x+1)}{(1-3 x) x^{4/3} (x+1) \sqrt [3]{x^2-1}}dx}{\sqrt [3]{x^3-x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{x^2-1} \int \frac {(1-x)^2 (3 x+1)}{(1-3 x) x^{2/3} (x+1) \sqrt [3]{x^2-1}}d\sqrt [3]{x}}{\sqrt [3]{x^3-x}}\) |
| \(\Big \downarrow \) 1396 |
| \(\displaystyle -\frac {3 \sqrt [3]{-x-1} \sqrt [3]{1-x} \sqrt [3]{x} \int \frac {(1-x)^{5/3} (3 x+1)}{(1-3 x) \sqrt [3]{-x-1} x^{2/3} (x+1)}d\sqrt [3]{x}}{\sqrt [3]{x^3-x}}\) |
| \(\Big \downarrow \) 281 |
| \(\displaystyle \frac {3 \sqrt [3]{-x-1} \sqrt [3]{1-x} \sqrt [3]{x} \int \frac {(1-x)^{5/3} (3 x+1)}{(1-3 x) (-x-1)^{4/3} x^{2/3}}d\sqrt [3]{x}}{\sqrt [3]{x^3-x}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {3 \sqrt [3]{-x-1} \sqrt [3]{1-x} \sqrt [3]{x} \int \left (\frac {(1-x)^{5/3}}{(-x-1)^{4/3} x^{2/3}}-\frac {6 (1-x)^{5/3} \sqrt [3]{x}}{(-x-1)^{4/3} (3 x-1)}\right )d\sqrt [3]{x}}{\sqrt [3]{x^3-x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 \sqrt [3]{-x-1} \sqrt [3]{1-x} \sqrt [3]{x} \left (\frac {\sqrt [3]{x+1} \operatorname {AppellF1}\left (-\frac {1}{3},-\frac {5}{3},\frac {4}{3},\frac {2}{3},x,-x\right )}{\sqrt [3]{-x-1} \sqrt [3]{x}}-6 \int \frac {(1-x)^{5/3} \sqrt [3]{x}}{(-x-1)^{4/3} (3 x-1)}d\sqrt [3]{x}\right )}{\sqrt [3]{x^3-x}}\) |
Input:
Int[((-1 + x)^2*(1 + 3*x))/(x*(1 + x)*(-1 + 3*x)*(-x + x^3)^(1/3)),x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.78 (sec) , antiderivative size = 621, normalized size of antiderivative = 3.74
| method | result | size |
| 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 )\) | \(621\) |
| trager | \(\text {Expression too large to display}\) | \(685\) |
Input:
int((-1+x)^2*(1+3*x)/x/(1+x)/(-1+3*x)/(x^3-x)^(1/3),x,method=_RETURNVERBOS E)
Output:
-3*(-1+x)/(x*(x^2-1))^(1/3)+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*Root Of(_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*RootOf(_Z^2-_Z+1)*(x^3-x)^(1/3) *x-3679*RootOf(_Z^2-_Z+1)^2*x+1819*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)+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+1819*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*R ootOf(_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))
Time = 0.61 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.83 \[ \int \frac {(-1+x)^2 (1+3 x)}{x (1+x) (-1+3 x) \sqrt [3]{-x+x^3}} \, dx=\frac {2 \, \sqrt {3} {\left (x^{2} + x\right )} \arctan \left (\frac {612314840 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {1}{3}} {\left (x - 1\right )} + \sqrt {3} {\left (1609127381 \, x^{2} + 1235276981 \, x + 124616800\right )} + 2605939922 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {2}{3}}}{2990437623 \, x^{2} + 3108349623 \, x - 39304000}\right ) - {\left (x^{2} + x\right )} \log \left (\frac {3 \, {\left (x^{3} - x\right )}^{\frac {1}{3}} {\left (x - 1\right )} + 3 \, x - 3 \, {\left (x^{3} - x\right )}^{\frac {2}{3}} - 1}{3 \, x - 1}\right ) - 6 \, {\left (x^{3} - x\right )}^{\frac {2}{3}}}{2 \, {\left (x^{2} + x\right )}} \] Input:
integrate((-1+x)^2*(1+3*x)/x/(1+x)/(-1+3*x)/(x^3-x)^(1/3),x, algorithm="fr icas")
Output:
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) + 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)
\[ \int \frac {(-1+x)^2 (1+3 x)}{x (1+x) (-1+3 x) \sqrt [3]{-x+x^3}} \, dx=\int \frac {\left (x - 1\right )^{2} \cdot \left (3 x + 1\right )}{x \sqrt [3]{x \left (x - 1\right ) \left (x + 1\right )} \left (x + 1\right ) \left (3 x - 1\right )}\, dx \] Input:
integrate((-1+x)**2*(1+3*x)/x/(1+x)/(-1+3*x)/(x**3-x)**(1/3),x)
Output:
Integral((x - 1)**2*(3*x + 1)/(x*(x*(x - 1)*(x + 1))**(1/3)*(x + 1)*(3*x - 1)), x)
\[ \int \frac {(-1+x)^2 (1+3 x)}{x (1+x) (-1+3 x) \sqrt [3]{-x+x^3}} \, dx=\int { \frac {{\left (3 \, x + 1\right )} {\left (x - 1\right )}^{2}}{{\left (x^{3} - x\right )}^{\frac {1}{3}} {\left (3 \, x - 1\right )} {\left (x + 1\right )} x} \,d x } \] Input:
integrate((-1+x)^2*(1+3*x)/x/(1+x)/(-1+3*x)/(x^3-x)^(1/3),x, algorithm="ma xima")
Output:
integrate((3*x + 1)*(x - 1)^2/((x^3 - x)^(1/3)*(3*x - 1)*(x + 1)*x), x)
\[ \int \frac {(-1+x)^2 (1+3 x)}{x (1+x) (-1+3 x) \sqrt [3]{-x+x^3}} \, dx=\int { \frac {{\left (3 \, x + 1\right )} {\left (x - 1\right )}^{2}}{{\left (x^{3} - x\right )}^{\frac {1}{3}} {\left (3 \, x - 1\right )} {\left (x + 1\right )} x} \,d x } \] Input:
integrate((-1+x)^2*(1+3*x)/x/(1+x)/(-1+3*x)/(x^3-x)^(1/3),x, algorithm="gi ac")
Output:
integrate((3*x + 1)*(x - 1)^2/((x^3 - x)^(1/3)*(3*x - 1)*(x + 1)*x), x)
Timed out. \[ \int \frac {(-1+x)^2 (1+3 x)}{x (1+x) (-1+3 x) \sqrt [3]{-x+x^3}} \, dx=\int \frac {\left (3\,x+1\right )\,{\left (x-1\right )}^2}{x\,{\left (x^3-x\right )}^{1/3}\,\left (3\,x-1\right )\,\left (x+1\right )} \,d x \] Input:
int(((3*x + 1)*(x - 1)^2)/(x*(x^3 - x)^(1/3)*(3*x - 1)*(x + 1)),x)
Output:
int(((3*x + 1)*(x - 1)^2)/(x*(x^3 - x)^(1/3)*(3*x - 1)*(x + 1)), x)
\[ \int \frac {(-1+x)^2 (1+3 x)}{x (1+x) (-1+3 x) \sqrt [3]{-x+x^3}} \, dx=3 \left (\int \frac {x^{2}}{3 x^{\frac {7}{3}} \left (x^{2}-1\right )^{\frac {1}{3}}+2 x^{\frac {4}{3}} \left (x^{2}-1\right )^{\frac {1}{3}}-x^{\frac {1}{3}} \left (x^{2}-1\right )^{\frac {1}{3}}}d x \right )-5 \left (\int \frac {x}{3 x^{\frac {7}{3}} \left (x^{2}-1\right )^{\frac {1}{3}}+2 x^{\frac {4}{3}} \left (x^{2}-1\right )^{\frac {1}{3}}-x^{\frac {1}{3}} \left (x^{2}-1\right )^{\frac {1}{3}}}d x \right )+\int \frac {1}{3 x^{\frac {10}{3}} \left (x^{2}-1\right )^{\frac {1}{3}}+2 x^{\frac {7}{3}} \left (x^{2}-1\right )^{\frac {1}{3}}-x^{\frac {4}{3}} \left (x^{2}-1\right )^{\frac {1}{3}}}d x +\int \frac {1}{3 x^{\frac {7}{3}} \left (x^{2}-1\right )^{\frac {1}{3}}+2 x^{\frac {4}{3}} \left (x^{2}-1\right )^{\frac {1}{3}}-x^{\frac {1}{3}} \left (x^{2}-1\right )^{\frac {1}{3}}}d x \] Input:
int((-1+x)^2*(1+3*x)/x/(1+x)/(-1+3*x)/(x^3-x)^(1/3),x)
Output:
3*int(x**2/(3*x**(1/3)*(x**2 - 1)**(1/3)*x**2 + 2*x**(1/3)*(x**2 - 1)**(1/ 3)*x - x**(1/3)*(x**2 - 1)**(1/3)),x) - 5*int(x/(3*x**(1/3)*(x**2 - 1)**(1 /3)*x**2 + 2*x**(1/3)*(x**2 - 1)**(1/3)*x - x**(1/3)*(x**2 - 1)**(1/3)),x) + int(1/(3*x**(1/3)*(x**2 - 1)**(1/3)*x**3 + 2*x**(1/3)*(x**2 - 1)**(1/3) *x**2 - x**(1/3)*(x**2 - 1)**(1/3)*x),x) + int(1/(3*x**(1/3)*(x**2 - 1)**( 1/3)*x**2 + 2*x**(1/3)*(x**2 - 1)**(1/3)*x - x**(1/3)*(x**2 - 1)**(1/3)),x )