3.9.82 \(\int \frac {1}{\sqrt [3]{1-x} \sqrt [3]{2-x} x^3} \, dx\) [882]

Optimal. Leaf size=821 \[ -\frac {(1-x)^{2/3} (2-x)^{2/3}}{4 x^2}-\frac {(1-x)^{2/3} (2-x)^{2/3}}{2 x}-\frac {\sqrt {(3-2 x)^2} \sqrt {(-3+2 x)^2} \sqrt [3]{2-3 x+x^2}}{2 \sqrt [3]{2} (3-2 x) \sqrt [3]{1-x} \sqrt [3]{2-x} \left (1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}\right )}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {\sqrt [3]{2} (2-x)^{2/3}}{\sqrt {3} \sqrt [3]{1-x}}\right )}{2 \sqrt [3]{2} \sqrt {3}}+\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt {(-3+2 x)^2} \sqrt [3]{2-3 x+x^2} \left (1+2^{2/3} \sqrt [3]{2-3 x+x^2}\right ) \sqrt {\frac {1-2^{2/3} \sqrt [3]{2-3 x+x^2}+2 \sqrt [3]{2} \left (2-3 x+x^2\right )^{2/3}}{\left (1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}}{1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}}\right )|-7-4 \sqrt {3}\right )}{4 \sqrt [3]{2} (3-2 x) \sqrt {(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt {\frac {1+2^{2/3} \sqrt [3]{2-3 x+x^2}}{\left (1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}\right )^2}}}-\frac {\sqrt {(-3+2 x)^2} \sqrt [3]{2-3 x+x^2} \left (1+2^{2/3} \sqrt [3]{2-3 x+x^2}\right ) \sqrt {\frac {1-2^{2/3} \sqrt [3]{2-3 x+x^2}+2 \sqrt [3]{2} \left (2-3 x+x^2\right )^{2/3}}{\left (1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}}{1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}}\right )|-7-4 \sqrt {3}\right )}{2^{5/6} \sqrt [4]{3} (3-2 x) \sqrt {(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt {\frac {1+2^{2/3} \sqrt [3]{2-3 x+x^2}}{\left (1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}\right )^2}}}+\frac {\log \left (-\sqrt [3]{1-x}+\frac {(2-x)^{2/3}}{2^{2/3}}\right )}{4 \sqrt [3]{2}}-\frac {\log (x)}{6 \sqrt [3]{2}} \]

[Out]

-1/4*(1-x)^(2/3)*(2-x)^(2/3)/x^2-1/2*(1-x)^(2/3)*(2-x)^(2/3)/x+1/8*ln(-(1-x)^(1/3)+1/2*(2-x)^(2/3)*2^(1/3))*2^
(2/3)-1/12*ln(x)*2^(2/3)-1/12*arctan(1/3*3^(1/2)+1/3*2^(1/3)*(2-x)^(2/3)/(1-x)^(1/3)*3^(1/2))*3^(1/2)*2^(2/3)-
1/4*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))-1/6*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)+1/8*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)

________________________________________________________________________________________

Rubi [A]
time = 0.40, antiderivative size = 821, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {125, 156, 163, 63, 637, 309, 224, 1891, 124} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt [3]{2} (2-x)^{2/3}}{\sqrt {3} \sqrt [3]{1-x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}+\frac {\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 {\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} \sqrt [4]{3} (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 {\log \left (\frac {(2-x)^{2/3}}{2^{2/3}}-\sqrt [3]{1-x}\right )}{4 \sqrt [3]{2}}-\frac {\log (x)}{6 \sqrt [3]{2}}-\frac {(1-x)^{2/3} (2-x)^{2/3}}{2 x}-\frac {\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 {(1-x)^{2/3} (2-x)^{2/3}}{4 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*((1 - x)^(2/3)*(2 - x)^(2/3))/x^2 - ((1 - x)^(2/3)*(2 - x)^(2/3))/(2*x) - (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))) - ArcTan[1/Sqrt[3] + (2^(1/3)*(2 - x)^(2/3))/(Sqrt[3]*(1 - x)^(1/3))]/(2*2^(1/3)*Sqrt[3]) + (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]])/(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]) - (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^(1/4)*(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]) + Log[-(1 - x)^(1/3) + (2 - x)^(2/3)/2^(2/3)]/(4*2^(1/3)) - Log[x]/(6*2^(1/3))

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 124

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)*((e_.) + (f_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[
b*((b*e - a*f)/(b*c - a*d)^2), 3]}, Simp[-Log[a + b*x]/(2*q*(b*c - a*d)), x] + (-Simp[Sqrt[3]*(ArcTan[1/Sqrt[3
] + 2*q*((c + d*x)^(2/3)/(Sqrt[3]*(e + f*x)^(1/3)))]/(2*q*(b*c - a*d))), x] + Simp[3*(Log[q*(c + d*x)^(2/3) -
(e + f*x)^(1/3)]/(4*q*(b*c - a*d))), x])] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[2*b*d*e - b*c*f - a*d*f, 0]

Rule 125

Int[((a_.) + (b_.)*(x_))^(m_)/(((c_.) + (d_.)*(x_))^(1/3)*((e_.) + (f_.)*(x_))^(1/3)), x_Symbol] :> Simp[b*(a
+ b*x)^(m + 1)*(c + d*x)^(2/3)*((e + f*x)^(2/3)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Dist[f/(6*(m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*((a*d*(3*m + 1) - 3*b*c*(3*m + 5) - 2*b*d*(3*m + 7)*x)/((c + d*x)
^(1/3)*(e + f*x)^(1/3))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[2*b*d*e - b*c*f - a*d*f, 0] && ILtQ[m,
-1]

Rule 156

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]

Rule 163

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

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 {1}{\sqrt [3]{1-x} \sqrt [3]{2-x} x^3} \, dx &=-\frac {(1-x)^{2/3} (2-x)^{2/3}}{4 x^2}+\frac {1}{24} \int \frac {24-4 x}{\sqrt [3]{1-x} \sqrt [3]{2-x} x^2} \, dx\\ &=-\frac {(1-x)^{2/3} (2-x)^{2/3}}{4 x^2}-\frac {(1-x)^{2/3} (2-x)^{2/3}}{2 x}-\frac {1}{48} \int \frac {-16-8 x}{\sqrt [3]{1-x} \sqrt [3]{2-x} x} \, dx\\ &=-\frac {(1-x)^{2/3} (2-x)^{2/3}}{4 x^2}-\frac {(1-x)^{2/3} (2-x)^{2/3}}{2 x}+\frac {1}{6} \int \frac {1}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx+\frac {1}{3} \int \frac {1}{\sqrt [3]{1-x} \sqrt [3]{2-x} x} \, dx\\ &=-\frac {(1-x)^{2/3} (2-x)^{2/3}}{4 x^2}-\frac {(1-x)^{2/3} (2-x)^{2/3}}{2 x}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {\sqrt [3]{2} (2-x)^{2/3}}{\sqrt {3} \sqrt [3]{1-x}}\right )}{2 \sqrt [3]{2} \sqrt {3}}+\frac {\log \left (-\sqrt [3]{1-x}+\frac {(2-x)^{2/3}}{2^{2/3}}\right )}{4 \sqrt [3]{2}}-\frac {\log (x)}{6 \sqrt [3]{2}}+\frac {\sqrt [3]{2-3 x+x^2} \int \frac {1}{\sqrt [3]{2-3 x+x^2}} \, dx}{6 \sqrt [3]{1-x} \sqrt [3]{2-x}}\\ &=-\frac {(1-x)^{2/3} (2-x)^{2/3}}{4 x^2}-\frac {(1-x)^{2/3} (2-x)^{2/3}}{2 x}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {\sqrt [3]{2} (2-x)^{2/3}}{\sqrt {3} \sqrt [3]{1-x}}\right )}{2 \sqrt [3]{2} \sqrt {3}}+\frac {\log \left (-\sqrt [3]{1-x}+\frac {(2-x)^{2/3}}{2^{2/3}}\right )}{4 \sqrt [3]{2}}-\frac {\log (x)}{6 \sqrt [3]{2}}+\frac {\left (\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 {(1-x)^{2/3} (2-x)^{2/3}}{4 x^2}-\frac {(1-x)^{2/3} (2-x)^{2/3}}{2 x}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {\sqrt [3]{2} (2-x)^{2/3}}{\sqrt {3} \sqrt [3]{1-x}}\right )}{2 \sqrt [3]{2} \sqrt {3}}+\frac {\log \left (-\sqrt [3]{1-x}+\frac {(2-x)^{2/3}}{2^{2/3}}\right )}{4 \sqrt [3]{2}}-\frac {\log (x)}{6 \sqrt [3]{2}}+\frac {\left (\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 (\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 {(1-x)^{2/3} (2-x)^{2/3}}{4 x^2}-\frac {(1-x)^{2/3} (2-x)^{2/3}}{2 x}-\frac {\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 {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {\sqrt [3]{2} (2-x)^{2/3}}{\sqrt {3} \sqrt [3]{1-x}}\right )}{2 \sqrt [3]{2} \sqrt {3}}+\frac {\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 {\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} \sqrt [4]{3} (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 {\log \left (-\sqrt [3]{1-x}+\frac {(2-x)^{2/3}}{2^{2/3}}\right )}{4 \sqrt [3]{2}}-\frac {\log (x)}{6 \sqrt [3]{2}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
time = 30.06, size = 84, normalized size = 0.10 \begin {gather*} -\frac {(1-x)^{2/3} \left (5 (2-x)^{2/3} (1+2 x)+15 x^2 F_1\left (\frac {2}{3};\frac {1}{3},1;\frac {5}{3};-1+x,1-x\right )+2 (-1+x) x^2 F_1\left (\frac {5}{3};\frac {1}{3},1;\frac {8}{3};-1+x,1-x\right )\right )}{20 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/20*((1 - x)^(2/3)*(5*(2 - x)^(2/3)*(1 + 2*x) + 15*x^2*AppellF1[2/3, 1/3, 1, 5/3, -1 + x, 1 - x] + 2*(-1 + x
)*x^2*AppellF1[5/3, 1/3, 1, 8/3, -1 + x, 1 - x]))/x^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

integral((-x + 2)^(2/3)*(-x + 1)^(2/3)/(x^5 - 3*x^4 + 2*x^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

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