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

Optimal. Leaf size=727 \[ -\frac{27 \sqrt [6]{2} 3^{3/4} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt{3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1}\right ),-7-4 \sqrt{3}\right )}{7 (3-2 x) \sqrt{(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt{\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}}}+\frac{3}{10} (1-x)^{2/3} (2-x)^{2/3} x^2-\frac{81 \sqrt{(3-2 x)^2} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2}}{7 \sqrt [3]{2} (3-2 x) \sqrt [3]{1-x} \sqrt [3]{2-x} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )}+\frac{81 \sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt{3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{14 \sqrt [3]{2} (3-2 x) \sqrt{(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt{\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}}}+\frac{9}{70} (1-x)^{2/3} (2-x)^{2/3} (8 x+23) \]

[Out]

(3*(1 - x)^(2/3)*(2 - x)^(2/3)*x^2)/10 + (9*(1 - x)^(2/3)*(2 - x)^(2/3)*(23 + 8*x))/70 - (81*Sqrt[(3 - 2*x)^2]
*Sqrt[(-3 + 2*x)^2]*(2 - 3*x + x^2)^(1/3))/(7*2^(1/3)*(3 - 2*x)*(1 - x)^(1/3)*(2 - x)^(1/3)*(1 + Sqrt[3] + 2^(
2/3)*(2 - 3*x + x^2)^(1/3))) + (81*3^(1/4)*Sqrt[2 - Sqrt[3]]*Sqrt[(-3 + 2*x)^2]*(2 - 3*x + x^2)^(1/3)*(1 + 2^(
2/3)*(2 - 3*x + x^2)^(1/3))*Sqrt[(1 - 2^(2/3)*(2 - 3*x + x^2)^(1/3) + 2*2^(1/3)*(2 - 3*x + x^2)^(2/3))/(1 + Sq
rt[3] + 2^(2/3)*(2 - 3*x + x^2)^(1/3))^2]*EllipticE[ArcSin[(1 - Sqrt[3] + 2^(2/3)*(2 - 3*x + x^2)^(1/3))/(1 +
Sqrt[3] + 2^(2/3)*(2 - 3*x + x^2)^(1/3))], -7 - 4*Sqrt[3]])/(14*2^(1/3)*(3 - 2*x)*Sqrt[(3 - 2*x)^2]*(1 - x)^(1
/3)*(2 - x)^(1/3)*Sqrt[(1 + 2^(2/3)*(2 - 3*x + x^2)^(1/3))/(1 + Sqrt[3] + 2^(2/3)*(2 - 3*x + x^2)^(1/3))^2]) -
(27*2^(1/6)*3^(3/4)*Sqrt[(-3 + 2*x)^2]*(2 - 3*x + x^2)^(1/3)*(1 + 2^(2/3)*(2 - 3*x + x^2)^(1/3))*Sqrt[(1 - 2^
(2/3)*(2 - 3*x + x^2)^(1/3) + 2*2^(1/3)*(2 - 3*x + x^2)^(2/3))/(1 + Sqrt[3] + 2^(2/3)*(2 - 3*x + x^2)^(1/3))^2
]*EllipticF[ArcSin[(1 - Sqrt[3] + 2^(2/3)*(2 - 3*x + x^2)^(1/3))/(1 + Sqrt[3] + 2^(2/3)*(2 - 3*x + x^2)^(1/3))
], -7 - 4*Sqrt[3]])/(7*(3 - 2*x)*Sqrt[(3 - 2*x)^2]*(1 - x)^(1/3)*(2 - x)^(1/3)*Sqrt[(1 + 2^(2/3)*(2 - 3*x + x^
2)^(1/3))/(1 + Sqrt[3] + 2^(2/3)*(2 - 3*x + x^2)^(1/3))^2])

________________________________________________________________________________________

Rubi [A]  time = 0.454973, antiderivative size = 727, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {100, 147, 61, 623, 303, 218, 1877} \[ \frac{3}{10} (1-x)^{2/3} (2-x)^{2/3} x^2-\frac{81 \sqrt{(3-2 x)^2} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2}}{7 \sqrt [3]{2} (3-2 x) \sqrt [3]{1-x} \sqrt [3]{2-x} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )}-\frac{27 \sqrt [6]{2} 3^{3/4} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt{3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{7 (3-2 x) \sqrt{(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt{\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}}}+\frac{81 \sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt{3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{14 \sqrt [3]{2} (3-2 x) \sqrt{(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt{\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}}}+\frac{9}{70} (1-x)^{2/3} (2-x)^{2/3} (8 x+23) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(1 - x)^(2/3)*(2 - x)^(2/3)*x^2)/10 + (9*(1 - x)^(2/3)*(2 - x)^(2/3)*(23 + 8*x))/70 - (81*Sqrt[(3 - 2*x)^2]
*Sqrt[(-3 + 2*x)^2]*(2 - 3*x + x^2)^(1/3))/(7*2^(1/3)*(3 - 2*x)*(1 - x)^(1/3)*(2 - x)^(1/3)*(1 + Sqrt[3] + 2^(
2/3)*(2 - 3*x + x^2)^(1/3))) + (81*3^(1/4)*Sqrt[2 - Sqrt[3]]*Sqrt[(-3 + 2*x)^2]*(2 - 3*x + x^2)^(1/3)*(1 + 2^(
2/3)*(2 - 3*x + x^2)^(1/3))*Sqrt[(1 - 2^(2/3)*(2 - 3*x + x^2)^(1/3) + 2*2^(1/3)*(2 - 3*x + x^2)^(2/3))/(1 + Sq
rt[3] + 2^(2/3)*(2 - 3*x + x^2)^(1/3))^2]*EllipticE[ArcSin[(1 - Sqrt[3] + 2^(2/3)*(2 - 3*x + x^2)^(1/3))/(1 +
Sqrt[3] + 2^(2/3)*(2 - 3*x + x^2)^(1/3))], -7 - 4*Sqrt[3]])/(14*2^(1/3)*(3 - 2*x)*Sqrt[(3 - 2*x)^2]*(1 - x)^(1
/3)*(2 - x)^(1/3)*Sqrt[(1 + 2^(2/3)*(2 - 3*x + x^2)^(1/3))/(1 + Sqrt[3] + 2^(2/3)*(2 - 3*x + x^2)^(1/3))^2]) -
(27*2^(1/6)*3^(3/4)*Sqrt[(-3 + 2*x)^2]*(2 - 3*x + x^2)^(1/3)*(1 + 2^(2/3)*(2 - 3*x + x^2)^(1/3))*Sqrt[(1 - 2^
(2/3)*(2 - 3*x + x^2)^(1/3) + 2*2^(1/3)*(2 - 3*x + x^2)^(2/3))/(1 + Sqrt[3] + 2^(2/3)*(2 - 3*x + x^2)^(1/3))^2
]*EllipticF[ArcSin[(1 - Sqrt[3] + 2^(2/3)*(2 - 3*x + x^2)^(1/3))/(1 + Sqrt[3] + 2^(2/3)*(2 - 3*x + x^2)^(1/3))
], -7 - 4*Sqrt[3]])/(7*(3 - 2*x)*Sqrt[(3 - 2*x)^2]*(1 - x)^(1/3)*(2 - x)^(1/3)*Sqrt[(1 + 2^(2/3)*(2 - 3*x + x^
2)^(1/3))/(1 + Sqrt[3] + 2^(2/3)*(2 - 3*x + x^2)^(1/3))^2])

Rule 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m - 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 1)), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(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 147

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> -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] + Dist[(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]

Rule 61

Int[((a_.) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Dist[((a + b*x)^m*(c + d*x)^m)/(a*c + (b*
c + a*d)*x + b*d*x^2)^m, Int[(a*c + (b*c + a*d)*x + b*d*x^2)^m, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c -
a*d, 0] && LtQ[-1, m, 0] && LeQ[3, Denominator[m], 4] && AtomQ[b*c + a*d]

Rule 623

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{d = Denominator[p]}, Dist[(d*Sqrt[(b + 2*c*x)
^2])/(b + 2*c*x), Subst[Int[x^(d*(p + 1) - 1)/Sqrt[b^2 - 4*a*c + 4*c*x^d], x], x, (a + b*x + c*x^2)^(1/d)], x]
/; 3 <= d <= 4] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && RationalQ[p]

Rule 303

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Dist[(Sq
rt[2]*s)/(Sqrt[2 + Sqrt[3]]*r), Int[1/Sqrt[a + b*x^3], x], x] + Dist[1/r, Int[((1 - Sqrt[3])*s + r*x)/Sqrt[a +
b*x^3], x], x]] /; FreeQ[{a, b}, x] && PosQ[a]

Rule 218

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(2*Sqr
t[2 + Sqrt[3]]*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3
])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]])/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(s*(s + r*x))/((1 + Sqr
t[3])*s + r*x)^2]), x]] /; FreeQ[{a, b}, x] && PosQ[a]

Rule 1877

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Simplify[((1 - Sqrt[3])*d)/c]]
, s = Denom[Simplify[((1 - Sqrt[3])*d)/c]]}, Simp[(2*d*s^3*Sqrt[a + b*x^3])/(a*r^2*((1 + Sqrt[3])*s + r*x)), x
] - Simp[(3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]*Elli
pticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]])/(r^2*Sqrt[a + b*x^3]*Sqrt[(s*(
s + r*x))/((1 + Sqrt[3])*s + r*x)^2]), x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && EqQ[b*c^3 - 2*(5 - 3*Sqrt[3
])*a*d^3, 0]

Rubi steps

\begin{align*} \int \frac{x^3}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx &=\frac{3}{10} (1-x)^{2/3} (2-x)^{2/3} x^2+\frac{3}{10} \int \frac{x (-4+8 x)}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx\\ &=\frac{3}{10} (1-x)^{2/3} (2-x)^{2/3} x^2+\frac{9}{70} (1-x)^{2/3} (2-x)^{2/3} (23+8 x)+\frac{27}{7} \int \frac{1}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx\\ &=\frac{3}{10} (1-x)^{2/3} (2-x)^{2/3} x^2+\frac{9}{70} (1-x)^{2/3} (2-x)^{2/3} (23+8 x)+\frac{\left (27 \sqrt [3]{2-3 x+x^2}\right ) \int \frac{1}{\sqrt [3]{2-3 x+x^2}} \, dx}{7 \sqrt [3]{1-x} \sqrt [3]{2-x}}\\ &=\frac{3}{10} (1-x)^{2/3} (2-x)^{2/3} x^2+\frac{9}{70} (1-x)^{2/3} (2-x)^{2/3} (23+8 x)+\frac{\left (81 \sqrt{(-3+2 x)^2} \sqrt [3]{2-3 x+x^2}\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{1+4 x^3}} \, dx,x,\sqrt [3]{2-3 x+x^2}\right )}{7 \sqrt [3]{1-x} \sqrt [3]{2-x} (-3+2 x)}\\ &=\frac{3}{10} (1-x)^{2/3} (2-x)^{2/3} x^2+\frac{9}{70} (1-x)^{2/3} (2-x)^{2/3} (23+8 x)+\frac{\left (81 \sqrt{(-3+2 x)^2} \sqrt [3]{2-3 x+x^2}\right ) \operatorname{Subst}\left (\int \frac{1-\sqrt{3}+2^{2/3} x}{\sqrt{1+4 x^3}} \, dx,x,\sqrt [3]{2-3 x+x^2}\right )}{7\ 2^{2/3} \sqrt [3]{1-x} \sqrt [3]{2-x} (-3+2 x)}+\frac{\left (81 \sqrt{(-3+2 x)^2} \sqrt [3]{2-3 x+x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+4 x^3}} \, dx,x,\sqrt [3]{2-3 x+x^2}\right )}{7 \sqrt [6]{2} \sqrt{2+\sqrt{3}} \sqrt [3]{1-x} \sqrt [3]{2-x} (-3+2 x)}\\ &=\frac{3}{10} (1-x)^{2/3} (2-x)^{2/3} x^2+\frac{9}{70} (1-x)^{2/3} (2-x)^{2/3} (23+8 x)-\frac{81 \sqrt{(3-2 x)^2} \sqrt{(-3+2 x)^2} \sqrt [3]{2-3 x+x^2}}{7 \sqrt [3]{2} (3-2 x) \sqrt [3]{1-x} \sqrt [3]{2-x} \left (1+\sqrt{3}+2^{2/3} \sqrt [3]{2-3 x+x^2}\right )}+\frac{81 \sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt{(-3+2 x)^2} \sqrt [3]{2-3 x+x^2} \left (1+2^{2/3} \sqrt [3]{2-3 x+x^2}\right ) \sqrt{\frac{1-2^{2/3} \sqrt [3]{2-3 x+x^2}+2 \sqrt [3]{2} \left (2-3 x+x^2\right )^{2/3}}{\left (1+\sqrt{3}+2^{2/3} \sqrt [3]{2-3 x+x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{1-\sqrt{3}+2^{2/3} \sqrt [3]{2-3 x+x^2}}{1+\sqrt{3}+2^{2/3} \sqrt [3]{2-3 x+x^2}}\right )|-7-4 \sqrt{3}\right )}{14 \sqrt [3]{2} (3-2 x) \sqrt{(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt{\frac{1+2^{2/3} \sqrt [3]{2-3 x+x^2}}{\left (1+\sqrt{3}+2^{2/3} \sqrt [3]{2-3 x+x^2}\right )^2}}}-\frac{27 \sqrt [6]{2} 3^{3/4} \sqrt{(-3+2 x)^2} \sqrt [3]{2-3 x+x^2} \left (1+2^{2/3} \sqrt [3]{2-3 x+x^2}\right ) \sqrt{\frac{1-2^{2/3} \sqrt [3]{2-3 x+x^2}+2 \sqrt [3]{2} \left (2-3 x+x^2\right )^{2/3}}{\left (1+\sqrt{3}+2^{2/3} \sqrt [3]{2-3 x+x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{1-\sqrt{3}+2^{2/3} \sqrt [3]{2-3 x+x^2}}{1+\sqrt{3}+2^{2/3} \sqrt [3]{2-3 x+x^2}}\right )|-7-4 \sqrt{3}\right )}{7 (3-2 x) \sqrt{(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt{\frac{1+2^{2/3} \sqrt [3]{2-3 x+x^2}}{\left (1+\sqrt{3}+2^{2/3} \sqrt [3]{2-3 x+x^2}\right )^2}}}\\ \end{align*}

Mathematica [C]  time = 0.0564724, size = 72, normalized size = 0.1 \[ \frac{3}{10} (1-x)^{2/3} \left (-12 \, _2F_1\left (-\frac{5}{3},\frac{2}{3};\frac{5}{3};x-1\right )+42 \, _2F_1\left (-\frac{2}{3},\frac{2}{3};\frac{5}{3};x-1\right )-36 \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};x-1\right )+(2-x)^{2/3} x^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(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*Hypergeometric2F1[1/3, 2/3, 5/3, -1 + x]))/10

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Maple [F]  time = 0.072, size = 0, normalized size = 0. \begin{align*} \int{{x}^{3}{\frac{1}{\sqrt [3]{1-x}}}{\frac{1}{\sqrt [3]{2-x}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{{\left (-x + 2\right )}^{\frac{1}{3}}{\left (-x + 1\right )}^{\frac{1}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{3}{\left (-x + 2\right )}^{\frac{2}{3}}{\left (-x + 1\right )}^{\frac{2}{3}}}{x^{2} - 3 \, x + 2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\sqrt [3]{1 - x} \sqrt [3]{2 - x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{{\left (-x + 2\right )}^{\frac{1}{3}}{\left (-x + 1\right )}^{\frac{1}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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