\(\int \frac {x^3}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx\) [486]

Optimal result

Integrand size = 22, antiderivative size = 93 \[ \int \frac {x^3}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx=\frac {30}{7} (1-x)^{2/3} (2-x)^{2/3}-\frac {57}{35} (1-x)^{5/3} (2-x)^{2/3}+\frac {3}{10} (1-x)^{8/3} (2-x)^{2/3}-\frac {81}{14} (1-x)^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},-1+x\right ) \] Output:

30/7*(1-x)^(2/3)*(2-x)^(2/3)-57/35*(1-x)^(5/3)*(2-x)^(2/3)+3/10*(1-x)^(8/3 
)*(2-x)^(2/3)-81/14*(1-x)^(2/3)*hypergeom([1/3, 2/3],[5/3],-1+x)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 10.05 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.77 \[ \int \frac {x^3}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx=\frac {3}{10} (1-x)^{2/3} \left ((2-x)^{2/3} x^2-12 \operatorname {Hypergeometric2F1}\left (-\frac {5}{3},\frac {2}{3},\frac {5}{3},-1+x\right )+42 \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {2}{3},\frac {5}{3},-1+x\right )-36 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},-1+x\right )\right ) \] Input:

Integrate[x^3/((1 - x)^(1/3)*(2 - x)^(1/3)),x]
 

Output:

(3*(1 - x)^(2/3)*((2 - x)^(2/3)*x^2 - 12*Hypergeometric2F1[-5/3, 2/3, 5/3, 
 -1 + x] + 42*Hypergeometric2F1[-2/3, 2/3, 5/3, -1 + x] - 36*Hypergeometri 
c2F1[1/3, 2/3, 5/3, -1 + x]))/10
 

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.90, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {111, 27, 164, 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.

\(\displaystyle \int \frac {x^3}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx\)

\(\Big \downarrow \) 111

\(\displaystyle \frac {3}{10} \int -\frac {4 (1-2 x) x}{\sqrt [3]{1-x} \sqrt [3]{2-x}}dx+\frac {3}{10} (1-x)^{2/3} (2-x)^{2/3} x^2\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{10} (1-x)^{2/3} (2-x)^{2/3} x^2-\frac {6}{5} \int \frac {(1-2 x) x}{\sqrt [3]{1-x} \sqrt [3]{2-x}}dx\)

\(\Big \downarrow \) 164

\(\displaystyle \frac {3}{10} (1-x)^{2/3} (2-x)^{2/3} x^2-\frac {6}{5} \left (-\frac {45}{14} \int \frac {1}{\sqrt [3]{1-x} \sqrt [3]{2-x}}dx-\frac {3}{28} (1-x)^{2/3} (2-x)^{2/3} (8 x+23)\right )\)

\(\Big \downarrow \) 79

\(\displaystyle \frac {3}{10} (1-x)^{2/3} (2-x)^{2/3} x^2-\frac {6}{5} \left (\frac {135}{28} (1-x)^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x-1\right )-\frac {3}{28} (1-x)^{2/3} (2-x)^{2/3} (8 x+23)\right )\)

Input:

Int[x^3/((1 - x)^(1/3)*(2 - x)^(1/3)),x]
 

Output:

(3*(1 - x)^(2/3)*(2 - x)^(2/3)*x^2)/10 - (6*((-3*(1 - x)^(2/3)*(2 - x)^(2/ 
3)*(23 + 8*x))/28 + (135*(1 - x)^(2/3)*Hypergeometric2F1[1/3, 2/3, 5/3, -1 
 + x])/28))/5
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 79

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]))
 

rule 111

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1))   Int[(a + b*x) 
^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m 
 - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m 
 + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & 
& GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]
 

rule 164

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ 
))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - 
 b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( 
c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h 
*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 
3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + 
d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3))   Int[( 
a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] 
&& NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]
 

Maple [F]

\[\int \frac {x^{3}}{\left (1-x \right )^{\frac {1}{3}} \left (-x +2\right )^{\frac {1}{3}}}d x\]

Input:

int(x^3/(1-x)^(1/3)/(-x+2)^(1/3),x)
 

Output:

int(x^3/(1-x)^(1/3)/(-x+2)^(1/3),x)
 

Fricas [F]

\[ \int \frac {x^3}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx=\int { \frac {x^{3}}{{\left (-x + 2\right )}^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {1}{3}}} \,d x } \] Input:

integrate(x^3/(1-x)^(1/3)/(2-x)^(1/3),x, algorithm="fricas")
 

Output:

integral(x^3*(-x + 2)^(2/3)*(-x + 1)^(2/3)/(x^2 - 3*x + 2), x)
 

Sympy [F]

\[ \int \frac {x^3}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx=\int \frac {x^{3}}{\sqrt [3]{1 - x} \sqrt [3]{2 - x}}\, dx \] Input:

integrate(x**3/(1-x)**(1/3)/(2-x)**(1/3),x)
 

Output:

Integral(x**3/((1 - x)**(1/3)*(2 - x)**(1/3)), x)
 

Maxima [F]

\[ \int \frac {x^3}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx=\int { \frac {x^{3}}{{\left (-x + 2\right )}^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {1}{3}}} \,d x } \] Input:

integrate(x^3/(1-x)^(1/3)/(2-x)^(1/3),x, algorithm="maxima")
 

Output:

integrate(x^3/((-x + 2)^(1/3)*(-x + 1)^(1/3)), x)
 

Giac [F]

\[ \int \frac {x^3}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx=\int { \frac {x^{3}}{{\left (-x + 2\right )}^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {1}{3}}} \,d x } \] Input:

integrate(x^3/(1-x)^(1/3)/(2-x)^(1/3),x, algorithm="giac")
 

Output:

integrate(x^3/((-x + 2)^(1/3)*(-x + 1)^(1/3)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx=\int \frac {x^3}{{\left (1-x\right )}^{1/3}\,{\left (2-x\right )}^{1/3}} \,d x \] Input:

int(x^3/((1 - x)^(1/3)*(2 - x)^(1/3)),x)
 

Output:

int(x^3/((1 - x)^(1/3)*(2 - x)^(1/3)), x)
 

Reduce [F]

\[ \int \frac {x^3}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx=\int \frac {x^{3}}{\left (1-x \right )^{\frac {1}{3}} \left (-x +2\right )^{\frac {1}{3}}}d x \] Input:

int(x^3/(1-x)^(1/3)/(2-x)^(1/3),x)
 

Output:

int(x**3/(( - x + 1)**(1/3)*( - x + 2)**(1/3)),x)