3.9.78 \(\int \frac {x}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx\) [878]

Optimal. Leaf size=695 \[ \frac {3}{4} (1-x)^{2/3} (2-x)^{2/3}-\frac {9 \sqrt {(3-2 x)^2} \sqrt {(-3+2 x)^2} \sqrt [3]{2-3 x+x^2}}{2 \sqrt [3]{2} (3-2 x) \sqrt [3]{1-x} \sqrt [3]{2-x} \left (1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}\right )}+\frac {9 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt {(-3+2 x)^2} \sqrt [3]{2-3 x+x^2} \left (1+2^{2/3} \sqrt [3]{2-3 x+x^2}\right ) \sqrt {\frac {1-2^{2/3} \sqrt [3]{2-3 x+x^2}+2 \sqrt [3]{2} \left (2-3 x+x^2\right )^{2/3}}{\left (1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}}{1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}}\right )|-7-4 \sqrt {3}\right )}{4 \sqrt [3]{2} (3-2 x) \sqrt {(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt {\frac {1+2^{2/3} \sqrt [3]{2-3 x+x^2}}{\left (1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}\right )^2}}}-\frac {3\ 3^{3/4} \sqrt {(-3+2 x)^2} \sqrt [3]{2-3 x+x^2} \left (1+2^{2/3} \sqrt [3]{2-3 x+x^2}\right ) \sqrt {\frac {1-2^{2/3} \sqrt [3]{2-3 x+x^2}+2 \sqrt [3]{2} \left (2-3 x+x^2\right )^{2/3}}{\left (1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}}{1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}}\right )|-7-4 \sqrt {3}\right )}{2^{5/6} (3-2 x) \sqrt {(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt {\frac {1+2^{2/3} \sqrt [3]{2-3 x+x^2}}{\left (1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}\right )^2}}} \]

[Out]

3/4*(1-x)^(2/3)*(2-x)^(2/3)-9/4*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))-3/2*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)+9/8*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)

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Rubi [A]
time = 0.27, antiderivative size = 695, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {81, 63, 637, 309, 224, 1891} \begin {gather*} -\frac {3\ 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 )}{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 {9 \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 )}{4 \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 \sqrt {(3-2 x)^2} \sqrt {(2 x-3)^2} \sqrt [3]{x^2-3 x+2}}{2 \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 {3}{4} (1-x)^{2/3} (2-x)^{2/3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(1 - x)^(2/3)*(2 - x)^(2/3))/4 - (9*Sqrt[(3 - 2*x)^2]*Sqrt[(-3 + 2*x)^2]*(2 - 3*x + x^2)^(1/3))/(2*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))) + (9*3^(1/4)*Sqrt[2 - Sqr
t[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[Arc
Sin[(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]])/(4*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]) - (3*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]])/(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 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 81

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 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}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx &=\frac {3}{4} (1-x)^{2/3} (2-x)^{2/3}+\frac {3}{2} \int \frac {1}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx\\ &=\frac {3}{4} (1-x)^{2/3} (2-x)^{2/3}+\frac {\left (3 \sqrt [3]{2-3 x+x^2}\right ) \int \frac {1}{\sqrt [3]{2-3 x+x^2}} \, dx}{2 \sqrt [3]{1-x} \sqrt [3]{2-x}}\\ &=\frac {3}{4} (1-x)^{2/3} (2-x)^{2/3}+\frac {\left (9 \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 )}{2 \sqrt [3]{1-x} \sqrt [3]{2-x} (-3+2 x)}\\ &=\frac {3}{4} (1-x)^{2/3} (2-x)^{2/3}+\frac {\left (9 \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 )}{2\ 2^{2/3} \sqrt [3]{1-x} \sqrt [3]{2-x} (-3+2 x)}+\frac {\left (9 \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 )}{2 \sqrt [6]{2} \sqrt {2+\sqrt {3}} \sqrt [3]{1-x} \sqrt [3]{2-x} (-3+2 x)}\\ &=\frac {3}{4} (1-x)^{2/3} (2-x)^{2/3}-\frac {9 \sqrt {(3-2 x)^2} \sqrt {(-3+2 x)^2} \sqrt [3]{2-3 x+x^2}}{2 \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 {9 \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 )}{4 \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 {3\ 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 )}{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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.01, size = 38, normalized size = 0.05 \begin {gather*} \frac {3}{4} (1-x)^{2/3} \left ((2-x)^{2/3}-3 \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};-1+x\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(x/((-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}{{\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/((1 - x)^(1/3)*(2 - x)^(1/3)),x)

[Out]

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

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