Integrand size = 20, antiderivative size = 49 \[ \int \frac {x}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx=\frac {3}{4} (1-x)^{2/3} (2-x)^{2/3}-\frac {9}{4} (1-x)^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},-1+x\right ) \] Output:
3/4*(1-x)^(2/3)*(2-x)^(2/3)-9/4*(1-x)^(2/3)*hypergeom([1/3, 2/3],[5/3],-1+ x)
Time = 10.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.78 \[ \int \frac {x}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx=\frac {3}{4} (1-x)^{2/3} \left ((2-x)^{2/3}-3 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},-1+x\right )\right ) \] Input:
Integrate[x/((1 - x)^(1/3)*(2 - x)^(1/3)),x]
Output:
(3*(1 - x)^(2/3)*((2 - x)^(2/3) - 3*Hypergeometric2F1[1/3, 2/3, 5/3, -1 + x]))/4
Time = 0.15 (sec) , antiderivative size = 49, 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 = {90, 79}
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.
\(\displaystyle \int \frac {x}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx\) |
\(\Big \downarrow \) 90 |
\(\displaystyle \frac {3}{2} \int \frac {1}{\sqrt [3]{1-x} \sqrt [3]{2-x}}dx+\frac {3}{4} (1-x)^{2/3} (2-x)^{2/3}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle \frac {3}{4} (1-x)^{2/3} (2-x)^{2/3}-\frac {9}{4} (1-x)^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x-1\right )\) |
Input:
Int[x/((1 - x)^(1/3)*(2 - x)^(1/3)),x]
Output:
(3*(1 - x)^(2/3)*(2 - x)^(2/3))/4 - (9*(1 - x)^(2/3)*Hypergeometric2F1[1/3 , 2/3, 5/3, -1 + x])/4
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 , m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] || !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
\[\int \frac {x}{\left (1-x \right )^{\frac {1}{3}} \left (-x +2\right )^{\frac {1}{3}}}d x\]
Input:
int(x/(1-x)^(1/3)/(-x+2)^(1/3),x)
Output:
int(x/(1-x)^(1/3)/(-x+2)^(1/3),x)
\[ \int \frac {x}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx=\int { \frac {x}{{\left (-x + 2\right )}^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {1}{3}}} \,d x } \] Input:
integrate(x/(1-x)^(1/3)/(2-x)^(1/3),x, algorithm="fricas")
Output:
integral(x*(-x + 2)^(2/3)*(-x + 1)^(2/3)/(x^2 - 3*x + 2), x)
\[ \int \frac {x}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx=\int \frac {x}{\sqrt [3]{1 - x} \sqrt [3]{2 - x}}\, dx \] Input:
integrate(x/(1-x)**(1/3)/(2-x)**(1/3),x)
Output:
Integral(x/((1 - x)**(1/3)*(2 - x)**(1/3)), x)
\[ \int \frac {x}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx=\int { \frac {x}{{\left (-x + 2\right )}^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {1}{3}}} \,d x } \] Input:
integrate(x/(1-x)^(1/3)/(2-x)^(1/3),x, algorithm="maxima")
Output:
integrate(x/((-x + 2)^(1/3)*(-x + 1)^(1/3)), x)
\[ \int \frac {x}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx=\int { \frac {x}{{\left (-x + 2\right )}^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {1}{3}}} \,d x } \] Input:
integrate(x/(1-x)^(1/3)/(2-x)^(1/3),x, algorithm="giac")
Output:
integrate(x/((-x + 2)^(1/3)*(-x + 1)^(1/3)), x)
Timed out. \[ \int \frac {x}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx=\int \frac {x}{{\left (1-x\right )}^{1/3}\,{\left (2-x\right )}^{1/3}} \,d x \] Input:
int(x/((1 - x)^(1/3)*(2 - x)^(1/3)),x)
Output:
int(x/((1 - x)^(1/3)*(2 - x)^(1/3)), x)
\[ \int \frac {x}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx=\int \frac {x}{\left (1-x \right )^{\frac {1}{3}} \left (-x +2\right )^{\frac {1}{3}}}d x \] Input:
int(x/(1-x)^(1/3)/(2-x)^(1/3),x)
Output:
int(x/(( - x + 1)**(1/3)*( - x + 2)**(1/3)),x)