Integrand size = 20, antiderivative size = 43 \[ \int \frac {\left (1-x^3\right )^{2/3}}{\left (1+x+x^2\right )^2} \, dx=\frac {1}{\sqrt [3]{1-x^3}}+\frac {x}{\sqrt [3]{1-x^3}}-x^2 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {4}{3},\frac {5}{3},x^3\right ) \] Output:
1/(-x^3+1)^(1/3)+x/(-x^3+1)^(1/3)-x^2*hypergeom([2/3, 4/3],[5/3],x^3)
Time = 10.17 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1-x^3\right )^{2/3}}{\left (1+x+x^2\right )^2} \, dx=\frac {(1+2 x) \left (1-x^3\right )^{2/3}}{1+x+x^2}+x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right ) \] Input:
Integrate[(1 - x^3)^(2/3)/(1 + x + x^2)^2,x]
Output:
((1 + 2*x)*(1 - x^3)^(2/3))/(1 + x + x^2) + x^2*Hypergeometric2F1[1/3, 2/3 , 5/3, x^3]
Time = 0.20 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2583, 2009}
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.
\(\displaystyle \int \frac {\left (1-x^3\right )^{2/3}}{\left (x^2+x+1\right )^2} \, dx\) |
\(\Big \downarrow \) 2583 |
\(\displaystyle \int \left (-\frac {2 x}{\left (1-x^3\right )^{4/3}}+\frac {1}{\left (1-x^3\right )^{4/3}}+\frac {x^2}{\left (1-x^3\right )^{4/3}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x^2 \left (-\operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {4}{3},\frac {5}{3},x^3\right )\right )+\frac {x}{\sqrt [3]{1-x^3}}+\frac {1}{\sqrt [3]{1-x^3}}\) |
Input:
Int[(1 - x^3)^(2/3)/(1 + x + x^2)^2,x]
Output:
(1 - x^3)^(-1/3) + x/(1 - x^3)^(1/3) - x^2*Hypergeometric2F1[2/3, 4/3, 5/3 , x^3]
Int[(Px_.)*((c_) + (d_.)*(x_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^3)^(p _.), x_Symbol] :> Simp[1/c^q Int[ExpandIntegrand[(c^3 - d^3*x^3)^q*(a + b *x^3)^p, Px/(c - d*x)^q, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && Poly Q[Px, x] && EqQ[d^2 - c*e, 0] && ILtQ[q, 0] && RationalQ[p] && EqQ[Denomina tor[p], 3]
Time = 0.41 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.79
method | result | size |
risch | \(-\frac {\left (1+2 x \right ) \left (-1+x \right )}{\left (-x^{3}+1\right )^{\frac {1}{3}}}+x^{2} \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{3}\right )\) | \(34\) |
Input:
int((-x^3+1)^(2/3)/(x^2+x+1)^2,x,method=_RETURNVERBOSE)
Output:
-(1+2*x)*(-1+x)/(-x^3+1)^(1/3)+x^2*hypergeom([1/3,2/3],[5/3],x^3)
\[ \int \frac {\left (1-x^3\right )^{2/3}}{\left (1+x+x^2\right )^2} \, dx=\int { \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{2} + x + 1\right )}^{2}} \,d x } \] Input:
integrate((-x^3+1)^(2/3)/(x^2+x+1)^2,x, algorithm="fricas")
Output:
integral((-x^3 + 1)^(2/3)/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1), x)
\[ \int \frac {\left (1-x^3\right )^{2/3}}{\left (1+x+x^2\right )^2} \, dx=\int \frac {\left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}}}{\left (x^{2} + x + 1\right )^{2}}\, dx \] Input:
integrate((-x**3+1)**(2/3)/(x**2+x+1)**2,x)
Output:
Integral((-(x - 1)*(x**2 + x + 1))**(2/3)/(x**2 + x + 1)**2, x)
\[ \int \frac {\left (1-x^3\right )^{2/3}}{\left (1+x+x^2\right )^2} \, dx=\int { \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{2} + x + 1\right )}^{2}} \,d x } \] Input:
integrate((-x^3+1)^(2/3)/(x^2+x+1)^2,x, algorithm="maxima")
Output:
integrate((-x^3 + 1)^(2/3)/(x^2 + x + 1)^2, x)
\[ \int \frac {\left (1-x^3\right )^{2/3}}{\left (1+x+x^2\right )^2} \, dx=\int { \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{2} + x + 1\right )}^{2}} \,d x } \] Input:
integrate((-x^3+1)^(2/3)/(x^2+x+1)^2,x, algorithm="giac")
Output:
integrate((-x^3 + 1)^(2/3)/(x^2 + x + 1)^2, x)
Timed out. \[ \int \frac {\left (1-x^3\right )^{2/3}}{\left (1+x+x^2\right )^2} \, dx=\int \frac {{\left (1-x^3\right )}^{2/3}}{{\left (x^2+x+1\right )}^2} \,d x \] Input:
int((1 - x^3)^(2/3)/(x + x^2 + 1)^2,x)
Output:
int((1 - x^3)^(2/3)/(x + x^2 + 1)^2, x)
\[ \int \frac {\left (1-x^3\right )^{2/3}}{\left (1+x+x^2\right )^2} \, dx=\frac {-\left (-x^{3}+1\right )^{\frac {2}{3}}+2 \left (\int \frac {\left (-x^{3}+1\right )^{\frac {2}{3}} x}{x^{5}+x^{4}+x^{3}-x^{2}-x -1}d x \right ) x^{2}+2 \left (\int \frac {\left (-x^{3}+1\right )^{\frac {2}{3}} x}{x^{5}+x^{4}+x^{3}-x^{2}-x -1}d x \right ) x +2 \left (\int \frac {\left (-x^{3}+1\right )^{\frac {2}{3}} x}{x^{5}+x^{4}+x^{3}-x^{2}-x -1}d x \right )}{x^{2}+x +1} \] Input:
int((-x^3+1)^(2/3)/(x^2+x+1)^2,x)
Output:
( - ( - x**3 + 1)**(2/3) + 2*int((( - x**3 + 1)**(2/3)*x)/(x**5 + x**4 + x **3 - x**2 - x - 1),x)*x**2 + 2*int((( - x**3 + 1)**(2/3)*x)/(x**5 + x**4 + x**3 - x**2 - x - 1),x)*x + 2*int((( - x**3 + 1)**(2/3)*x)/(x**5 + x**4 + x**3 - x**2 - x - 1),x))/(x**2 + x + 1)