3.875 \(\int \frac{x^4}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx\)

Optimal. Leaf size=752 \[ -\frac{594 \sqrt [6]{2} 3^{3/4} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt{3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1}\right ),-7-4 \sqrt{3}\right )}{91 (3-2 x) \sqrt{(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt{\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}}}+\frac{3}{13} (1-x)^{2/3} (2-x)^{2/3} x^3+\frac{99}{130} (1-x)^{2/3} (2-x)^{2/3} x^2-\frac{891\ 2^{2/3} \sqrt{(3-2 x)^2} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2}}{91 (3-2 x) \sqrt [3]{1-x} \sqrt [3]{2-x} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )}+\frac{891 \sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt{3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{91 \sqrt [3]{2} (3-2 x) \sqrt{(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt{\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}}}+\frac{27}{455} (1-x)^{2/3} (2-x)^{2/3} (34 x+89) \]

[Out]

(99*(1 - x)^(2/3)*(2 - x)^(2/3)*x^2)/130 + (3*(1 - x)^(2/3)*(2 - x)^(2/3)*x^3)/13 + (27*(1 - x)^(2/3)*(2 - x)^
(2/3)*(89 + 34*x))/455 - (891*2^(2/3)*Sqrt[(3 - 2*x)^2]*Sqrt[(-3 + 2*x)^2]*(2 - 3*x + x^2)^(1/3))/(91*(3 - 2*x
)*(1 - x)^(1/3)*(2 - x)^(1/3)*(1 + Sqrt[3] + 2^(2/3)*(2 - 3*x + x^2)^(1/3))) + (891*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 + Sqrt[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]])/(
91*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]) - (594*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]])/(91*(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])

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Rubi [A]  time = 0.644883, antiderivative size = 752, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {100, 153, 147, 61, 623, 303, 218, 1877} \[ \frac{3}{13} (1-x)^{2/3} (2-x)^{2/3} x^3+\frac{99}{130} (1-x)^{2/3} (2-x)^{2/3} x^2-\frac{891\ 2^{2/3} \sqrt{(3-2 x)^2} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2}}{91 (3-2 x) \sqrt [3]{1-x} \sqrt [3]{2-x} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )}-\frac{594 \sqrt [6]{2} 3^{3/4} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt{3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{91 (3-2 x) \sqrt{(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt{\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}}}+\frac{891 \sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt{3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{91 \sqrt [3]{2} (3-2 x) \sqrt{(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt{\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}}}+\frac{27}{455} (1-x)^{2/3} (2-x)^{2/3} (34 x+89) \]

Antiderivative was successfully verified.

[In]

Int[x^4/((1 - x)^(1/3)*(2 - x)^(1/3)),x]

[Out]

(99*(1 - x)^(2/3)*(2 - x)^(2/3)*x^2)/130 + (3*(1 - x)^(2/3)*(2 - x)^(2/3)*x^3)/13 + (27*(1 - x)^(2/3)*(2 - x)^
(2/3)*(89 + 34*x))/455 - (891*2^(2/3)*Sqrt[(3 - 2*x)^2]*Sqrt[(-3 + 2*x)^2]*(2 - 3*x + x^2)^(1/3))/(91*(3 - 2*x
)*(1 - x)^(1/3)*(2 - x)^(1/3)*(1 + Sqrt[3] + 2^(2/3)*(2 - 3*x + x^2)^(1/3))) + (891*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 + Sqrt[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]])/(
91*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]) - (594*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]])/(91*(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 153

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 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^4}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx &=\frac{3}{13} (1-x)^{2/3} (2-x)^{2/3} x^3+\frac{3}{13} \int \frac{x^2 (-6+11 x)}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx\\ &=\frac{99}{130} (1-x)^{2/3} (2-x)^{2/3} x^2+\frac{3}{13} (1-x)^{2/3} (2-x)^{2/3} x^3+\frac{9}{130} \int \frac{x (-44+68 x)}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx\\ &=\frac{99}{130} (1-x)^{2/3} (2-x)^{2/3} x^2+\frac{3}{13} (1-x)^{2/3} (2-x)^{2/3} x^3+\frac{27}{455} (1-x)^{2/3} (2-x)^{2/3} (89+34 x)+\frac{594}{91} \int \frac{1}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx\\ &=\frac{99}{130} (1-x)^{2/3} (2-x)^{2/3} x^2+\frac{3}{13} (1-x)^{2/3} (2-x)^{2/3} x^3+\frac{27}{455} (1-x)^{2/3} (2-x)^{2/3} (89+34 x)+\frac{\left (594 \sqrt [3]{2-3 x+x^2}\right ) \int \frac{1}{\sqrt [3]{2-3 x+x^2}} \, dx}{91 \sqrt [3]{1-x} \sqrt [3]{2-x}}\\ &=\frac{99}{130} (1-x)^{2/3} (2-x)^{2/3} x^2+\frac{3}{13} (1-x)^{2/3} (2-x)^{2/3} x^3+\frac{27}{455} (1-x)^{2/3} (2-x)^{2/3} (89+34 x)+\frac{\left (1782 \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 )}{91 \sqrt [3]{1-x} \sqrt [3]{2-x} (-3+2 x)}\\ &=\frac{99}{130} (1-x)^{2/3} (2-x)^{2/3} x^2+\frac{3}{13} (1-x)^{2/3} (2-x)^{2/3} x^3+\frac{27}{455} (1-x)^{2/3} (2-x)^{2/3} (89+34 x)+\frac{\left (891 \sqrt [3]{2} \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 )}{91 \sqrt [3]{1-x} \sqrt [3]{2-x} (-3+2 x)}+\frac{\left (891\ 2^{5/6} \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 )}{91 \sqrt{2+\sqrt{3}} \sqrt [3]{1-x} \sqrt [3]{2-x} (-3+2 x)}\\ &=\frac{99}{130} (1-x)^{2/3} (2-x)^{2/3} x^2+\frac{3}{13} (1-x)^{2/3} (2-x)^{2/3} x^3+\frac{27}{455} (1-x)^{2/3} (2-x)^{2/3} (89+34 x)-\frac{891\ 2^{2/3} \sqrt{(3-2 x)^2} \sqrt{(-3+2 x)^2} \sqrt [3]{2-3 x+x^2}}{91 (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{891 \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 )}{91 \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{594 \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 )}{91 (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 = 0.0445775, size = 72, normalized size = 0.1 \[ \frac{3 (1-x)^{2/3} \left (-2475 \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};x-1\right )-2772 (x-1) \, _2F_1\left (\frac{1}{3},\frac{5}{3};\frac{8}{3};x-1\right )+(2-x)^{2/3} \left (140 x^3+462 x^2+1224 x-261\right )\right )}{1820} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/((1 - x)^(1/3)*(2 - x)^(1/3)),x]

[Out]

(3*(1 - x)^(2/3)*((2 - x)^(2/3)*(-261 + 1224*x + 462*x^2 + 140*x^3) - 2475*Hypergeometric2F1[1/3, 2/3, 5/3, -1
+ x] - 2772*(-1 + x)*Hypergeometric2F1[1/3, 5/3, 8/3, -1 + x]))/1820

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Maple [F]  time = 0.081, size = 0, normalized size = 0. \begin{align*} \int{{x}^{4}{\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^4/(1-x)^(1/3)/(2-x)^(1/3),x)

[Out]

int(x^4/(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^{4}}{{\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^4/(1-x)^(1/3)/(2-x)^(1/3),x, algorithm="maxima")

[Out]

integrate(x^4/((-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^{4}{\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^4/(1-x)^(1/3)/(2-x)^(1/3),x, algorithm="fricas")

[Out]

integral(x^4*(-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^{4}}{\sqrt [3]{1 - x} \sqrt [3]{2 - x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(1-x)**(1/3)/(2-x)**(1/3),x)

[Out]

Integral(x**4/((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^{4}}{{\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^4/(1-x)^(1/3)/(2-x)^(1/3),x, algorithm="giac")

[Out]

integrate(x^4/((-x + 2)^(1/3)*(-x + 1)^(1/3)), x)