3.9.75 \(\int \frac {x^4}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx\) [875]

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

[Out]

99/130*(1-x)^(2/3)*(2-x)^(2/3)*x^2+3/13*(1-x)^(2/3)*(2-x)^(2/3)*x^3+27/455*(1-x)^(2/3)*(2-x)^(2/3)*(89+34*x)-8
91/91*2^(2/3)*(x^2-3*x+2)^(1/3)*((3-2*x)^2)^(1/2)*((-3+2*x)^2)^(1/2)/(3-2*x)/(1-x)^(1/3)/(2-x)^(1/3)/(1+2^(2/3
)*(x^2-3*x+2)^(1/3)+3^(1/2))-594/91*2^(1/6)*3^(3/4)*(x^2-3*x+2)^(1/3)*(1+2^(2/3)*(x^2-3*x+2)^(1/3))*EllipticF(
(1+2^(2/3)*(x^2-3*x+2)^(1/3)-3^(1/2))/(1+2^(2/3)*(x^2-3*x+2)^(1/3)+3^(1/2)),I*3^(1/2)+2*I)*((-3+2*x)^2)^(1/2)*
((1-2^(2/3)*(x^2-3*x+2)^(1/3)+2*2^(1/3)*(x^2-3*x+2)^(2/3))/(1+2^(2/3)*(x^2-3*x+2)^(1/3)+3^(1/2))^2)^(1/2)/(3-2
*x)/(1-x)^(1/3)/(2-x)^(1/3)/((3-2*x)^2)^(1/2)/((1+2^(2/3)*(x^2-3*x+2)^(1/3))/(1+2^(2/3)*(x^2-3*x+2)^(1/3)+3^(1
/2))^2)^(1/2)+891/182*3^(1/4)*(x^2-3*x+2)^(1/3)*(1+2^(2/3)*(x^2-3*x+2)^(1/3))*EllipticE((1+2^(2/3)*(x^2-3*x+2)
^(1/3)-3^(1/2))/(1+2^(2/3)*(x^2-3*x+2)^(1/3)+3^(1/2)),I*3^(1/2)+2*I)*((-3+2*x)^2)^(1/2)*(1/2*6^(1/2)-1/2*2^(1/
2))*((1-2^(2/3)*(x^2-3*x+2)^(1/3)+2*2^(1/3)*(x^2-3*x+2)^(2/3))/(1+2^(2/3)*(x^2-3*x+2)^(1/3)+3^(1/2))^2)^(1/2)*
2^(2/3)/(3-2*x)/(1-x)^(1/3)/(2-x)^(1/3)/((3-2*x)^2)^(1/2)/((1+2^(2/3)*(x^2-3*x+2)^(1/3))/(1+2^(2/3)*(x^2-3*x+2
)^(1/3)+3^(1/2))^2)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.45, antiderivative size = 752, normalized size of antiderivative = 1.00, 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 = {102, 158, 152, 63, 637, 309, 224, 1891} \begin {gather*} -\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 (\text {ArcSin}\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 (\text {ArcSin}\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 {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 {27}{455} (1-x)^{2/3} (2-x)^{2/3} (34 x+89) \end {gather*}

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 63

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 102

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 152

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 158

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 224

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

Rule 309

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Dist[(-(
1 - Sqrt[3]))*(s/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 637

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 1891

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]/(r^2
*Sqrt[a + b*x^3]*Sqrt[s*((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1
+ Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]], 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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.03, size = 72, normalized size = 0.10 \begin {gather*} \frac {3 (1-x)^{2/3} \left ((2-x)^{2/3} \left (-261+1224 x+462 x^2+140 x^3\right )-2475 \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};-1+x\right )-2772 (-1+x) \, _2F_1\left (\frac {1}{3},\frac {5}{3};\frac {8}{3};-1+x\right )\right )}{1820} \end {gather*}

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 = 0.01, size = 0, normalized size = 0.00 \[\int \frac {x^{4}}{\left (1-x \right )^{\frac {1}{3}} \left (2-x \right )^{\frac {1}{3}}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4}}{\sqrt [3]{1 - x} \sqrt [3]{2 - x}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^4}{{\left (1-x\right )}^{1/3}\,{\left (2-x\right )}^{1/3}} \,d x \end {gather*}

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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