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

Optimal. Leaf size=720 \[ -\frac{33\ 3^{3/4} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt{3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1}\right ),-7-4 \sqrt{3}\right )}{7\ 2^{5/6} (3-2 x) \sqrt{(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt{\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}}}-\frac{99 \sqrt{(3-2 x)^2} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2}}{14 \sqrt [3]{2} (3-2 x) \sqrt [3]{1-x} \sqrt [3]{2-x} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )}+\frac{99 \sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt{3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{28 \sqrt [3]{2} (3-2 x) \sqrt{(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt{\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}}}+\frac{3}{7} (1-x)^{2/3} (2-x)^{2/3} x+\frac{45}{28} (1-x)^{2/3} (2-x)^{2/3} \]

[Out]

(45*(1 - x)^(2/3)*(2 - x)^(2/3))/28 + (3*(1 - x)^(2/3)*(2 - x)^(2/3)*x)/7 - (99*Sqrt[(3 - 2*x)^2]*Sqrt[(-3 + 2
*x)^2]*(2 - 3*x + x^2)^(1/3))/(14*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))) + (99*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]])/(28*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]) - (33*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*2^(5/6)*(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 = 0.452305, antiderivative size = 720, 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 = {90, 80, 61, 623, 303, 218, 1877} \[ -\frac{99 \sqrt{(3-2 x)^2} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2}}{14 \sqrt [3]{2} (3-2 x) \sqrt [3]{1-x} \sqrt [3]{2-x} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )}-\frac{33\ 3^{3/4} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt{3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{7\ 2^{5/6} (3-2 x) \sqrt{(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt{\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}}}+\frac{99 \sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt{3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{28 \sqrt [3]{2} (3-2 x) \sqrt{(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt{\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}}}+\frac{3}{7} (1-x)^{2/3} (2-x)^{2/3} x+\frac{45}{28} (1-x)^{2/3} (2-x)^{2/3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(45*(1 - x)^(2/3)*(2 - x)^(2/3))/28 + (3*(1 - x)^(2/3)*(2 - x)^(2/3)*x)/7 - (99*Sqrt[(3 - 2*x)^2]*Sqrt[(-3 + 2
*x)^2]*(2 - 3*x + x^2)^(1/3))/(14*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))) + (99*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]])/(28*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]) - (33*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*2^(5/6)*(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 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(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]

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^2}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx &=\frac{3}{7} (1-x)^{2/3} (2-x)^{2/3} x+\frac{3}{7} \int \frac{-2+5 x}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx\\ &=\frac{45}{28} (1-x)^{2/3} (2-x)^{2/3}+\frac{3}{7} (1-x)^{2/3} (2-x)^{2/3} x+\frac{33}{14} \int \frac{1}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx\\ &=\frac{45}{28} (1-x)^{2/3} (2-x)^{2/3}+\frac{3}{7} (1-x)^{2/3} (2-x)^{2/3} x+\frac{\left (33 \sqrt [3]{2-3 x+x^2}\right ) \int \frac{1}{\sqrt [3]{2-3 x+x^2}} \, dx}{14 \sqrt [3]{1-x} \sqrt [3]{2-x}}\\ &=\frac{45}{28} (1-x)^{2/3} (2-x)^{2/3}+\frac{3}{7} (1-x)^{2/3} (2-x)^{2/3} x+\frac{\left (99 \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 )}{14 \sqrt [3]{1-x} \sqrt [3]{2-x} (-3+2 x)}\\ &=\frac{45}{28} (1-x)^{2/3} (2-x)^{2/3}+\frac{3}{7} (1-x)^{2/3} (2-x)^{2/3} x+\frac{\left (99 \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 )}{14\ 2^{2/3} \sqrt [3]{1-x} \sqrt [3]{2-x} (-3+2 x)}+\frac{\left (99 \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 )}{14 \sqrt [6]{2} \sqrt{2+\sqrt{3}} \sqrt [3]{1-x} \sqrt [3]{2-x} (-3+2 x)}\\ &=\frac{45}{28} (1-x)^{2/3} (2-x)^{2/3}+\frac{3}{7} (1-x)^{2/3} (2-x)^{2/3} x-\frac{99 \sqrt{(3-2 x)^2} \sqrt{(-3+2 x)^2} \sqrt [3]{2-3 x+x^2}}{14 \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{99 \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 )}{28 \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{33\ 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\ 2^{5/6} (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 = 0.0165523, size = 44, normalized size = 0.06 \[ \frac{3}{28} (1-x)^{2/3} \left ((2-x)^{2/3} (4 x+15)-33 \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};x-1\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(1 - x)^(2/3)*((2 - x)^(2/3)*(15 + 4*x) - 33*Hypergeometric2F1[1/3, 2/3, 5/3, -1 + x]))/28

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Maple [F]  time = 0.071, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2}{\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^2/(1-x)^(1/3)/(2-x)^(1/3),x)

[Out]

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

[Out]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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