Integrand size = 33, antiderivative size = 113 \[ \int \frac {x (1+x) (1+3 x)}{(-1+x) (-1+3 x) \left (-x+x^3\right )^{2/3}} \, dx=-\frac {3 \sqrt [3]{x \left (-1+x^2\right )}}{-1+x}+\frac {1}{2} \left (-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} (-1+x)}{-1+x+2 \sqrt [3]{x \left (-1+x^2\right )}}\right )-2 \log \left (1-x+\sqrt [3]{x \left (-1+x^2\right )}\right )+\log \left (1-2 x+x^2+(-1+x) \sqrt [3]{x \left (-1+x^2\right )}+\left (x \left (-1+x^2\right )\right )^{2/3}\right )\right ) \] Output:
-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))
Time = 31.74 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00 \[ \int \frac {x (1+x) (1+3 x)}{(-1+x) (-1+3 x) \left (-x+x^3\right )^{2/3}} \, dx=-\frac {3 \sqrt [3]{x \left (-1+x^2\right )}}{-1+x}+\frac {1}{2} \left (-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} (-1+x)}{-1+x+2 \sqrt [3]{x \left (-1+x^2\right )}}\right )-2 \log \left (1-x+\sqrt [3]{x \left (-1+x^2\right )}\right )+\log \left (1-2 x+x^2+(-1+x) \sqrt [3]{x \left (-1+x^2\right )}+\left (x \left (-1+x^2\right )\right )^{2/3}\right )\right ) \] Input:
Integrate[(x*(1 + x)*(1 + 3*x))/((-1 + x)*(-1 + 3*x)*(-x + x^3)^(2/3)),x]
Output:
(-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
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.
| \(\displaystyle \int \frac {x (x+1) (3 x+1)}{(x-1) (3 x-1) \left (x^3-x\right )^{2/3}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{2/3} \left (x^2-1\right )^{2/3} \int \frac {\sqrt [3]{x} (x+1) (3 x+1)}{(1-3 x) (1-x) \left (x^2-1\right )^{2/3}}dx}{\left (x^3-x\right )^{2/3}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {3 x^{2/3} \left (x^2-1\right )^{2/3} \int \frac {x (x+1) (3 x+1)}{(1-3 x) (1-x) \left (x^2-1\right )^{2/3}}d\sqrt [3]{x}}{\left (x^3-x\right )^{2/3}}\) |
| \(\Big \downarrow \) 1396 |
| \(\displaystyle \frac {3 (-x-1)^{2/3} (1-x)^{2/3} x^{2/3} \int \frac {x (x+1) (3 x+1)}{(1-3 x) (-x-1)^{2/3} (1-x)^{5/3}}d\sqrt [3]{x}}{\left (x^3-x\right )^{2/3}}\) |
| \(\Big \downarrow \) 281 |
| \(\displaystyle -\frac {3 (-x-1)^{2/3} (1-x)^{2/3} x^{2/3} \int \frac {\sqrt [3]{-x-1} x (3 x+1)}{(1-3 x) (1-x)^{5/3}}d\sqrt [3]{x}}{\left (x^3-x\right )^{2/3}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {3 (-x-1)^{2/3} (1-x)^{2/3} x^{2/3} \int \left (-\frac {\sqrt [3]{-x-1} x}{(1-x)^{5/3}}+\frac {2 \sqrt [3]{-x-1}}{3 (1-3 x) (1-x)^{5/3}}-\frac {2 \sqrt [3]{-x-1}}{3 (1-x)^{5/3}}\right )d\sqrt [3]{x}}{\left (x^3-x\right )^{2/3}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 (-x-1)^{2/3} (1-x)^{2/3} x^{2/3} \left (-\frac {4}{9} \int \frac {1}{\left (\sqrt [3]{-3} \sqrt [3]{x}+1\right ) (-x-1)^{2/3} (1-x)^{2/3}}d\sqrt [3]{x}-\frac {4}{9} \int \frac {1}{\left (1-\sqrt [3]{3} \sqrt [3]{x}\right ) (-x-1)^{2/3} (1-x)^{2/3}}d\sqrt [3]{x}-\frac {4}{9} \int \frac {1}{\left (1-(-1)^{2/3} \sqrt [3]{3} \sqrt [3]{x}\right ) (-x-1)^{2/3} (1-x)^{2/3}}d\sqrt [3]{x}-\frac {\sqrt [3]{-x-1} x^{4/3} \operatorname {AppellF1}\left (\frac {4}{3},\frac {5}{3},-\frac {1}{3},\frac {7}{3},x,-x\right )}{4 \sqrt [3]{x+1}}+\frac {5 \left (1-x^2\right ) \left (1-\frac {x^{2/3}}{\sqrt [3]{x^2-1}}\right ) \sqrt {\frac {\frac {x^{4/3}}{\left (x^2-1\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{x^2-1}}+1}{\left (1-\frac {\left (1+\sqrt {3}\right ) x^{2/3}}{\sqrt [3]{x^2-1}}\right )^2}} \sqrt [3]{x} \operatorname {EllipticF}\left (\arccos \left (\frac {1-\frac {\left (1-\sqrt {3}\right ) x^{2/3}}{\sqrt [3]{x^2-1}}}{1-\frac {\left (1+\sqrt {3}\right ) x^{2/3}}{\sqrt [3]{x^2-1}}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{12 \sqrt [4]{3} (-x-1)^{2/3} (1-x)^{2/3} \sqrt {-\frac {x^{2/3} \left (1-\frac {x^{2/3}}{\sqrt [3]{x^2-1}}\right )}{\sqrt [3]{x^2-1} \left (1-\frac {\left (1+\sqrt {3}\right ) x^{2/3}}{\sqrt [3]{x^2-1}}\right )^2}}}-\frac {\sqrt [3]{-x-1} \sqrt [3]{x}}{2 (1-x)^{2/3}}\right )}{\left (x^3-x\right )^{2/3}}\) |
Input:
Int[(x*(1 + x)*(1 + 3*x))/((-1 + x)*(-1 + 3*x)*(-x + x^3)^(2/3)),x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.53 (sec) , antiderivative size = 671, normalized size of antiderivative = 5.94
| method | result | size |
| trager | \(-\frac {3 \left (x^{3}-x \right )^{\frac {1}{3}}}{-1+x}-3 \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 -5046 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}-2127 \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}}+3234 \left (x^{3}-x \right )^{\frac {1}{3}} x -4806 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}+6231 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -1552 x^{2}-3234 \left (x^{3}-x \right )^{\frac {1}{3}}-3759 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-970 x -194}{-1+3 x}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-\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 -6231 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}-1107 \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}}+3234 \left (x^{3}-x \right )^{\frac {1}{3}} x +10476 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}+1185 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -1157 x^{2}-3234 \left (x^{3}-x \right )^{\frac {1}{3}}-2472 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+712 x -623}{-1+3 x}\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 -5046 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}-2127 \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}}+3234 \left (x^{3}-x \right )^{\frac {1}{3}} x -4806 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}+6231 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -1552 x^{2}-3234 \left (x^{3}-x \right )^{\frac {1}{3}}-3759 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-970 x -194}{-1+3 x}\right )\) | \(671\) |
| risch | \(\text {Expression too large to display}\) | \(912\) |
Input:
int(x*(1+x)*(1+3*x)/(-1+x)/(-1+3*x)/(x^3-x)^(2/3),x,method=_RETURNVERBOSE)
Output:
-3/(-1+x)*(x^3-x)^(1/3)-3*ln(-(-4005*RootOf(9*_Z^2-3*_Z+1)^2*x^2+9702*Root Of(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-5046*RootOf(9*_Z^2-3*_Z+1)*x^2-2127*(x^3- x)^(2/3)+3321*RootOf(9*_Z^2-3*_Z+1)*(x^3-x)^(1/3)+3234*(x^3-x)^(1/3)*x-480 6*RootOf(9*_Z^2-3*_Z+1)^2+6231*RootOf(9*_Z^2-3*_Z+1)*x-1552*x^2-3234*(x^3- x)^(1/3)-3759*RootOf(9*_Z^2-3*_Z+1)-970*x-194)/(-1+3*x))*RootOf(9*_Z^2-3*_ Z+1)-ln((8730*RootOf(9*_Z^2-3*_Z+1)^2*x^2+9702*RootOf(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-6231*RootOf(9*_Z^2-3*_Z+1)*x^2-1107*(x^3-x)^(2/3)+6381*RootOf(9* _Z^2-3*_Z+1)*(x^3-x)^(1/3)+3234*(x^3-x)^(1/3)*x+10476*RootOf(9*_Z^2-3*_Z+1 )^2+1185*RootOf(9*_Z^2-3*_Z+1)*x-1157*x^2-3234*(x^3-x)^(1/3)-2472*RootOf(9 *_Z^2-3*_Z+1)+712*x-623)/(-1+3*x))+ln(-(-4005*RootOf(9*_Z^2-3*_Z+1)^2*x^2+ 9702*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-5046*RootOf(9*_Z^2-3*_Z+1)*x^2-2 127*(x^3-x)^(2/3)+3321*RootOf(9*_Z^2-3*_Z+1)*(x^3-x)^(1/3)+3234*(x^3-x)^(1 /3)*x-4806*RootOf(9*_Z^2-3*_Z+1)^2+6231*RootOf(9*_Z^2-3*_Z+1)*x-1552*x^2-3 234*(x^3-x)^(1/3)-3759*RootOf(9*_Z^2-3*_Z+1)-970*x-194)/(-1+3*x))
Time = 0.44 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.15 \[ \int \frac {x (1+x) (1+3 x)}{(-1+x) (-1+3 x) \left (-x+x^3\right )^{2/3}} \, dx=-\frac {2 \, \sqrt {3} {\left (x - 1\right )} \arctan \left (\frac {612314840 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {1}{3}} {\left (x - 1\right )} + \sqrt {3} {\left (1609127381 \, x^{2} + 1235276981 \, x + 124616800\right )} + 2605939922 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {2}{3}}}{2990437623 \, x^{2} + 3108349623 \, x - 39304000}\right ) + {\left (x - 1\right )} \log \left (\frac {3 \, {\left (x^{3} - x\right )}^{\frac {1}{3}} {\left (x - 1\right )} + 3 \, x - 3 \, {\left (x^{3} - x\right )}^{\frac {2}{3}} - 1}{3 \, x - 1}\right ) + 6 \, {\left (x^{3} - x\right )}^{\frac {1}{3}}}{2 \, {\left (x - 1\right )}} \] Input:
integrate(x*(1+x)*(1+3*x)/(-1+x)/(-1+3*x)/(x^3-x)^(2/3),x, algorithm="fric as")
Output:
-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)*lo g((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)
\[ \int \frac {x (1+x) (1+3 x)}{(-1+x) (-1+3 x) \left (-x+x^3\right )^{2/3}} \, dx=\int \frac {x \left (x + 1\right ) \left (3 x + 1\right )}{\left (x \left (x - 1\right ) \left (x + 1\right )\right )^{\frac {2}{3}} \left (x - 1\right ) \left (3 x - 1\right )}\, dx \] Input:
integrate(x*(1+x)*(1+3*x)/(-1+x)/(-1+3*x)/(x**3-x)**(2/3),x)
Output:
Integral(x*(x + 1)*(3*x + 1)/((x*(x - 1)*(x + 1))**(2/3)*(x - 1)*(3*x - 1) ), x)
\[ \int \frac {x (1+x) (1+3 x)}{(-1+x) (-1+3 x) \left (-x+x^3\right )^{2/3}} \, dx=\int { \frac {{\left (3 \, x + 1\right )} {\left (x + 1\right )} x}{{\left (x^{3} - x\right )}^{\frac {2}{3}} {\left (3 \, x - 1\right )} {\left (x - 1\right )}} \,d x } \] Input:
integrate(x*(1+x)*(1+3*x)/(-1+x)/(-1+3*x)/(x^3-x)^(2/3),x, algorithm="maxi ma")
Output:
integrate((3*x + 1)*(x + 1)*x/((x^3 - x)^(2/3)*(3*x - 1)*(x - 1)), x)
\[ \int \frac {x (1+x) (1+3 x)}{(-1+x) (-1+3 x) \left (-x+x^3\right )^{2/3}} \, dx=\int { \frac {{\left (3 \, x + 1\right )} {\left (x + 1\right )} x}{{\left (x^{3} - x\right )}^{\frac {2}{3}} {\left (3 \, x - 1\right )} {\left (x - 1\right )}} \,d x } \] Input:
integrate(x*(1+x)*(1+3*x)/(-1+x)/(-1+3*x)/(x^3-x)^(2/3),x, algorithm="giac ")
Output:
integrate((3*x + 1)*(x + 1)*x/((x^3 - x)^(2/3)*(3*x - 1)*(x - 1)), x)
Timed out. \[ \int \frac {x (1+x) (1+3 x)}{(-1+x) (-1+3 x) \left (-x+x^3\right )^{2/3}} \, dx=\int \frac {x\,\left (3\,x+1\right )\,\left (x+1\right )}{{\left (x^3-x\right )}^{2/3}\,\left (3\,x-1\right )\,\left (x-1\right )} \,d x \] Input:
int((x*(3*x + 1)*(x + 1))/((x^3 - x)^(2/3)*(3*x - 1)*(x - 1)),x)
Output:
int((x*(3*x + 1)*(x + 1))/((x^3 - x)^(2/3)*(3*x - 1)*(x - 1)), x)
\[ \int \frac {x (1+x) (1+3 x)}{(-1+x) (-1+3 x) \left (-x+x^3\right )^{2/3}} \, dx=3 \left (\int \frac {x^{3}}{3 x^{\frac {8}{3}} \left (x^{2}-1\right )^{\frac {2}{3}}-4 x^{\frac {5}{3}} \left (x^{2}-1\right )^{\frac {2}{3}}+x^{\frac {2}{3}} \left (x^{2}-1\right )^{\frac {2}{3}}}d x \right )+4 \left (\int \frac {x^{2}}{3 x^{\frac {8}{3}} \left (x^{2}-1\right )^{\frac {2}{3}}-4 x^{\frac {5}{3}} \left (x^{2}-1\right )^{\frac {2}{3}}+x^{\frac {2}{3}} \left (x^{2}-1\right )^{\frac {2}{3}}}d x \right )+\int \frac {x}{3 x^{\frac {8}{3}} \left (x^{2}-1\right )^{\frac {2}{3}}-4 x^{\frac {5}{3}} \left (x^{2}-1\right )^{\frac {2}{3}}+x^{\frac {2}{3}} \left (x^{2}-1\right )^{\frac {2}{3}}}d x \] Input:
int(x*(1+x)*(1+3*x)/(-1+x)/(-1+3*x)/(x^3-x)^(2/3),x)
Output:
3*int(x**3/(3*x**(2/3)*(x**2 - 1)**(2/3)*x**2 - 4*x**(2/3)*(x**2 - 1)**(2/ 3)*x + x**(2/3)*(x**2 - 1)**(2/3)),x) + 4*int(x**2/(3*x**(2/3)*(x**2 - 1)* *(2/3)*x**2 - 4*x**(2/3)*(x**2 - 1)**(2/3)*x + x**(2/3)*(x**2 - 1)**(2/3)) ,x) + int(x/(3*x**(2/3)*(x**2 - 1)**(2/3)*x**2 - 4*x**(2/3)*(x**2 - 1)**(2 /3)*x + x**(2/3)*(x**2 - 1)**(2/3)),x)