3.222 \(\int \frac{\sqrt{1-x} x (1+x)^{2/3}}{-(1-x)^{5/6} \sqrt [3]{1+x}+(1-x)^{2/3} \sqrt{1+x}} \, dx\)

Optimal. Leaf size=292 \[ -\frac{1}{12} (1-x)^{2/3} \sqrt [3]{x+1} (1-3 x)-\frac{1}{4} (1-x) (x+3)+\frac{1}{12} \sqrt [3]{1-x} (x+1)^{2/3} (3 x+1)+\frac{1}{12} \sqrt [6]{1-x} (x+1)^{5/6} (3 x+2)-\frac{1}{12} (1-x)^{5/6} \sqrt [6]{x+1} (3 x+10)+\frac{1}{4} \sqrt{1-x} x \sqrt{x+1}+\frac{1}{6} \tan ^{-1}\left (\frac{\sqrt [6]{x+1}}{\sqrt [6]{1-x}}\right )-\frac{4 \tan ^{-1}\left (\frac{\sqrt [3]{1-x}-2 \sqrt [3]{x+1}}{\sqrt{3} \sqrt [3]{1-x}}\right )}{3 \sqrt{3}}-\frac{5}{6} \tan ^{-1}\left (\frac{\sqrt [3]{1-x}-\sqrt [3]{x+1}}{\sqrt [6]{1-x} \sqrt [6]{x+1}}\right )+\frac{\tanh ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{1-x} \sqrt [6]{x+1}}{\sqrt [3]{1-x}+\sqrt [3]{x+1}}\right )}{6 \sqrt{3}} \]

[Out]

-((1 - 3*x)*(1 - x)^(2/3)*(1 + x)^(1/3))/12 + (Sqrt[1 - x]*x*Sqrt[1 + x])/4 - ((1 - x)*(3 + x))/4 + ((1 - x)^(
1/3)*(1 + x)^(2/3)*(1 + 3*x))/12 + ((1 - x)^(1/6)*(1 + x)^(5/6)*(2 + 3*x))/12 - ((1 - x)^(5/6)*(1 + x)^(1/6)*(
10 + 3*x))/12 + ArcTan[(1 + x)^(1/6)/(1 - x)^(1/6)]/6 - (4*ArcTan[((1 - x)^(1/3) - 2*(1 + x)^(1/3))/(Sqrt[3]*(
1 - x)^(1/3))])/(3*Sqrt[3]) - (5*ArcTan[((1 - x)^(1/3) - (1 + x)^(1/3))/((1 - x)^(1/6)*(1 + x)^(1/6))])/6 + Ar
cTanh[(Sqrt[3]*(1 - x)^(1/6)*(1 + x)^(1/6))/((1 - x)^(1/3) + (1 + x)^(1/3))]/(6*Sqrt[3])

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Rubi [A]  time = 1.84631, antiderivative size = 522, normalized size of antiderivative = 1.79, number of steps used = 46, number of rules used = 21, integrand size = 56, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6688, 6742, 50, 60, 517, 195, 216, 675, 890, 63, 240, 209, 634, 618, 204, 628, 203, 26, 21, 331, 295} \[ \frac{x^2}{4}+\frac{1}{4} \sqrt{1-x^2} x+\frac{x}{2}-\frac{1}{4} (1-x)^{5/6} (x+1)^{7/6}-\frac{1}{4} (1-x)^{7/6} (x+1)^{5/6}+\frac{5}{12} \sqrt [6]{1-x} (x+1)^{5/6}-\frac{1}{4} (1-x)^{4/3} (x+1)^{2/3}+\frac{1}{3} \sqrt [3]{1-x} (x+1)^{2/3}-\frac{1}{4} (1-x)^{5/3} \sqrt [3]{x+1}+\frac{1}{6} (1-x)^{2/3} \sqrt [3]{x+1}-\frac{7}{12} (1-x)^{5/6} \sqrt [6]{x+1}-\frac{1}{9} \log (1-x)+\frac{1}{9} \log (x+1)+\frac{1}{3} \log \left (\frac{\sqrt [3]{1-x}}{\sqrt [3]{x+1}}+1\right )-\frac{\log \left (\frac{\sqrt [3]{1-x}}{\sqrt [3]{x+1}}-\frac{\sqrt{3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1\right )}{12 \sqrt{3}}+\frac{\log \left (\frac{\sqrt [3]{1-x}}{\sqrt [3]{x+1}}+\frac{\sqrt{3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1\right )}{12 \sqrt{3}}-\frac{1}{3} \log \left (\frac{\sqrt [3]{x+1}}{\sqrt [3]{1-x}}+1\right )+\frac{1}{4} \sin ^{-1}(x)-\frac{2}{3} \tan ^{-1}\left (\frac{\sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )+\frac{2 \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{1-x}}{\sqrt{3} \sqrt [3]{x+1}}\right )}{3 \sqrt{3}}+\frac{1}{3} \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )-\frac{1}{3} \tan ^{-1}\left (\frac{2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+\sqrt{3}\right )-\frac{2 \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{x+1}}{\sqrt{3} \sqrt [3]{1-x}}\right )}{3 \sqrt{3}} \]

Warning: Unable to verify antiderivative.

[In]

Int[(Sqrt[1 - x]*x*(1 + x)^(2/3))/(-((1 - x)^(5/6)*(1 + x)^(1/3)) + (1 - x)^(2/3)*Sqrt[1 + x]),x]

[Out]

x/2 + x^2/4 - (7*(1 - x)^(5/6)*(1 + x)^(1/6))/12 + ((1 - x)^(2/3)*(1 + x)^(1/3))/6 - ((1 - x)^(5/3)*(1 + x)^(1
/3))/4 + ((1 - x)^(1/3)*(1 + x)^(2/3))/3 - ((1 - x)^(4/3)*(1 + x)^(2/3))/4 + (5*(1 - x)^(1/6)*(1 + x)^(5/6))/1
2 - ((1 - x)^(7/6)*(1 + x)^(5/6))/4 - ((1 - x)^(5/6)*(1 + x)^(7/6))/4 + (x*Sqrt[1 - x^2])/4 + ArcSin[x]/4 - (2
*ArcTan[(1 - x)^(1/6)/(1 + x)^(1/6)])/3 + (2*ArcTan[1/Sqrt[3] - (2*(1 - x)^(1/3))/(Sqrt[3]*(1 + x)^(1/3))])/(3
*Sqrt[3]) + ArcTan[Sqrt[3] - (2*(1 - x)^(1/6))/(1 + x)^(1/6)]/3 - ArcTan[Sqrt[3] + (2*(1 - x)^(1/6))/(1 + x)^(
1/6)]/3 - (2*ArcTan[1/Sqrt[3] - (2*(1 + x)^(1/3))/(Sqrt[3]*(1 - x)^(1/3))])/(3*Sqrt[3]) - Log[1 - x]/9 + Log[1
+ x]/9 + Log[1 + (1 - x)^(1/3)/(1 + x)^(1/3)]/3 - Log[1 + (1 - x)^(1/3)/(1 + x)^(1/3) - (Sqrt[3]*(1 - x)^(1/6
))/(1 + x)^(1/6)]/(12*Sqrt[3]) + Log[1 + (1 - x)^(1/3)/(1 + x)^(1/3) + (Sqrt[3]*(1 - x)^(1/6))/(1 + x)^(1/6)]/
(12*Sqrt[3]) - Log[1 + (1 + x)^(1/3)/(1 - x)^(1/3)]/3

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 60

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

Rule 517

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.)*(x_)^(non2_.))^
(p_.), x_Symbol] :> Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n)^q, x] /; FreeQ[{a1, b1, a2, b2, c, d, n, p, q}, x]
&& EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && (IntegerQ[p] || (GtQ[a1, 0] && GtQ[a2, 0]))

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 675

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c*x)/e)^p,
x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && GtQ[a, 0] && GtQ[d, 0] &&  !I
GtQ[m, 0]

Rule 890

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Int[(d + e*x)^
(m + p)*(f + g*x)^n*(a/d + (c*x)/e)^p, x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[e*f - d*g, 0] && EqQ[c*
d^2 + a*e^2, 0] &&  !IntegerQ[p] && GtQ[a, 0] && GtQ[d, 0] &&  !IGtQ[m, 0] &&  !IGtQ[n, 0]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 240

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[1/(1 - b*x^n)^(p + 1/n + 1), x], x
, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p
+ 1/n]

Rule 209

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/b, n]]
, k, u, v}, Simp[u = Int[(r - s*Cos[((2*k - 1)*Pi)/n]*x)/(r^2 - 2*r*s*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x] +
Int[(r + s*Cos[((2*k - 1)*Pi)/n]*x)/(r^2 + 2*r*s*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x]; (2*r^2*Int[1/(r^2 +
s^2*x^2), x])/(a*n) + Dist[(2*r)/(a*n), Sum[u, {k, 1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)
/4, 0] && PosQ[a/b]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 26

Int[(u_.)*((a_) + (b_.)*(x_)^(n_.))^(m_.)*((c_) + (d_.)*(x_)^(j_))^(p_.), x_Symbol] :> Dist[(-(b^2/d))^m, Int[
u/(a - b*x^n)^m, x], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[j, 2*n] && EqQ[p, -m] && EqQ[b^2*c + a^2*d,
0] && GtQ[a, 0] && LtQ[d, 0]

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 331

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + (m + 1)/n), Subst[Int[x^m/(1 - b*x^n)^(
p + (m + 1)/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[
p, -2^(-1)] && IntegersQ[m, p + (m + 1)/n]

Rule 295

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/
b, n]], k, u}, Simp[u = Int[(r*Cos[((2*k - 1)*m*Pi)/n] - s*Cos[((2*k - 1)*(m + 1)*Pi)/n]*x)/(r^2 - 2*r*s*Cos[(
(2*k - 1)*Pi)/n]*x + s^2*x^2), x] + Int[(r*Cos[((2*k - 1)*m*Pi)/n] + s*Cos[((2*k - 1)*(m + 1)*Pi)/n]*x)/(r^2 +
2*r*s*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x]; (2*(-1)^(m/2)*r^(m + 2)*Int[1/(r^2 + s^2*x^2), x])/(a*n*s^m) +
Dist[(2*r^(m + 1))/(a*n*s^m), Sum[u, {k, 1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] &&
IGtQ[m, 0] && LtQ[m, n - 1] && PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{\sqrt{1-x} x (1+x)^{2/3}}{-(1-x)^{5/6} \sqrt [3]{1+x}+(1-x)^{2/3} \sqrt{1+x}} \, dx &=\int \frac{x \sqrt [3]{1+x}}{-\sqrt [3]{1-x}+\sqrt [6]{1-x^2}} \, dx\\ &=\int \left (\frac{1}{2} (1-x)^{2/3} \sqrt [3]{1+x}+\frac{1}{2} \sqrt [3]{1-x} \sqrt [3]{1+x} \sqrt [6]{1-x^2}+\frac{1}{2} \sqrt [3]{1+x} \sqrt [3]{1-x^2}+\frac{\sqrt [3]{1+x} \sqrt{1-x^2}}{2 \sqrt [3]{1-x}}+\frac{\sqrt [3]{1+x} \left (1-x^2\right )^{2/3}}{2 (1-x)^{2/3}}-\frac{\sqrt [3]{1+x} \left (1-x^2\right )^{5/6}}{2 (-1+x)}\right ) \, dx\\ &=\frac{1}{2} \int (1-x)^{2/3} \sqrt [3]{1+x} \, dx+\frac{1}{2} \int \sqrt [3]{1-x} \sqrt [3]{1+x} \sqrt [6]{1-x^2} \, dx+\frac{1}{2} \int \sqrt [3]{1+x} \sqrt [3]{1-x^2} \, dx+\frac{1}{2} \int \frac{\sqrt [3]{1+x} \sqrt{1-x^2}}{\sqrt [3]{1-x}} \, dx+\frac{1}{2} \int \frac{\sqrt [3]{1+x} \left (1-x^2\right )^{2/3}}{(1-x)^{2/3}} \, dx-\frac{1}{2} \int \frac{\sqrt [3]{1+x} \left (1-x^2\right )^{5/6}}{-1+x} \, dx\\ &=-\frac{1}{4} (1-x)^{5/3} \sqrt [3]{1+x}+\frac{1}{6} \int \frac{(1-x)^{2/3}}{(1+x)^{2/3}} \, dx+\frac{1}{2} \int \sqrt [3]{1-x} (1+x)^{2/3} \, dx+\frac{1}{2} \int \sqrt [6]{1-x} (1+x)^{5/6} \, dx+\frac{1}{2} \int (1+x) \, dx-\frac{1}{2} \int \frac{(1-x)^{5/6} (1+x)^{7/6}}{-1+x} \, dx+\frac{1}{2} \int \sqrt{1-x^2} \, dx\\ &=\frac{x}{2}+\frac{x^2}{4}+\frac{1}{6} (1-x)^{2/3} \sqrt [3]{1+x}-\frac{1}{4} (1-x)^{5/3} \sqrt [3]{1+x}-\frac{1}{4} (1-x)^{4/3} (1+x)^{2/3}-\frac{1}{4} (1-x)^{7/6} (1+x)^{5/6}+\frac{1}{4} x \sqrt{1-x^2}+\frac{2}{9} \int \frac{1}{\sqrt [3]{1-x} (1+x)^{2/3}} \, dx+\frac{1}{4} \int \frac{1}{\sqrt{1-x^2}} \, dx+\frac{1}{3} \int \frac{\sqrt [3]{1-x}}{\sqrt [3]{1+x}} \, dx+\frac{5}{12} \int \frac{\sqrt [6]{1-x}}{\sqrt [6]{1+x}} \, dx+\frac{1}{2} \int \frac{(1+x)^{7/6}}{\sqrt [6]{1-x}} \, dx\\ &=\frac{x}{2}+\frac{x^2}{4}+\frac{1}{6} (1-x)^{2/3} \sqrt [3]{1+x}-\frac{1}{4} (1-x)^{5/3} \sqrt [3]{1+x}+\frac{1}{3} \sqrt [3]{1-x} (1+x)^{2/3}-\frac{1}{4} (1-x)^{4/3} (1+x)^{2/3}+\frac{5}{12} \sqrt [6]{1-x} (1+x)^{5/6}-\frac{1}{4} (1-x)^{7/6} (1+x)^{5/6}-\frac{1}{4} (1-x)^{5/6} (1+x)^{7/6}+\frac{1}{4} x \sqrt{1-x^2}+\frac{1}{4} \sin ^{-1}(x)+\frac{2 \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{1-x}}{\sqrt{3} \sqrt [3]{1+x}}\right )}{3 \sqrt{3}}+\frac{1}{9} \log (1+x)+\frac{1}{3} \log \left (1+\frac{\sqrt [3]{1-x}}{\sqrt [3]{1+x}}\right )+\frac{5}{36} \int \frac{1}{(1-x)^{5/6} \sqrt [6]{1+x}} \, dx+\frac{2}{9} \int \frac{1}{(1-x)^{2/3} \sqrt [3]{1+x}} \, dx+\frac{7}{12} \int \frac{\sqrt [6]{1+x}}{\sqrt [6]{1-x}} \, dx\\ &=\frac{x}{2}+\frac{x^2}{4}-\frac{7}{12} (1-x)^{5/6} \sqrt [6]{1+x}+\frac{1}{6} (1-x)^{2/3} \sqrt [3]{1+x}-\frac{1}{4} (1-x)^{5/3} \sqrt [3]{1+x}+\frac{1}{3} \sqrt [3]{1-x} (1+x)^{2/3}-\frac{1}{4} (1-x)^{4/3} (1+x)^{2/3}+\frac{5}{12} \sqrt [6]{1-x} (1+x)^{5/6}-\frac{1}{4} (1-x)^{7/6} (1+x)^{5/6}-\frac{1}{4} (1-x)^{5/6} (1+x)^{7/6}+\frac{1}{4} x \sqrt{1-x^2}+\frac{1}{4} \sin ^{-1}(x)+\frac{2 \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{1-x}}{\sqrt{3} \sqrt [3]{1+x}}\right )}{3 \sqrt{3}}-\frac{2 \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{1+x}}{\sqrt{3} \sqrt [3]{1-x}}\right )}{3 \sqrt{3}}-\frac{1}{9} \log (1-x)+\frac{1}{9} \log (1+x)+\frac{1}{3} \log \left (1+\frac{\sqrt [3]{1-x}}{\sqrt [3]{1+x}}\right )-\frac{1}{3} \log \left (1+\frac{\sqrt [3]{1+x}}{\sqrt [3]{1-x}}\right )+\frac{7}{36} \int \frac{1}{\sqrt [6]{1-x} (1+x)^{5/6}} \, dx-\frac{5}{6} \operatorname{Subst}\left (\int \frac{1}{\sqrt [6]{2-x^6}} \, dx,x,\sqrt [6]{1-x}\right )\\ &=\frac{x}{2}+\frac{x^2}{4}-\frac{7}{12} (1-x)^{5/6} \sqrt [6]{1+x}+\frac{1}{6} (1-x)^{2/3} \sqrt [3]{1+x}-\frac{1}{4} (1-x)^{5/3} \sqrt [3]{1+x}+\frac{1}{3} \sqrt [3]{1-x} (1+x)^{2/3}-\frac{1}{4} (1-x)^{4/3} (1+x)^{2/3}+\frac{5}{12} \sqrt [6]{1-x} (1+x)^{5/6}-\frac{1}{4} (1-x)^{7/6} (1+x)^{5/6}-\frac{1}{4} (1-x)^{5/6} (1+x)^{7/6}+\frac{1}{4} x \sqrt{1-x^2}+\frac{1}{4} \sin ^{-1}(x)+\frac{2 \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{1-x}}{\sqrt{3} \sqrt [3]{1+x}}\right )}{3 \sqrt{3}}-\frac{2 \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{1+x}}{\sqrt{3} \sqrt [3]{1-x}}\right )}{3 \sqrt{3}}-\frac{1}{9} \log (1-x)+\frac{1}{9} \log (1+x)+\frac{1}{3} \log \left (1+\frac{\sqrt [3]{1-x}}{\sqrt [3]{1+x}}\right )-\frac{1}{3} \log \left (1+\frac{\sqrt [3]{1+x}}{\sqrt [3]{1-x}}\right )-\frac{5}{6} \operatorname{Subst}\left (\int \frac{1}{1+x^6} \, dx,x,\frac{\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac{7}{6} \operatorname{Subst}\left (\int \frac{x^4}{\left (2-x^6\right )^{5/6}} \, dx,x,\sqrt [6]{1-x}\right )\\ &=\frac{x}{2}+\frac{x^2}{4}-\frac{7}{12} (1-x)^{5/6} \sqrt [6]{1+x}+\frac{1}{6} (1-x)^{2/3} \sqrt [3]{1+x}-\frac{1}{4} (1-x)^{5/3} \sqrt [3]{1+x}+\frac{1}{3} \sqrt [3]{1-x} (1+x)^{2/3}-\frac{1}{4} (1-x)^{4/3} (1+x)^{2/3}+\frac{5}{12} \sqrt [6]{1-x} (1+x)^{5/6}-\frac{1}{4} (1-x)^{7/6} (1+x)^{5/6}-\frac{1}{4} (1-x)^{5/6} (1+x)^{7/6}+\frac{1}{4} x \sqrt{1-x^2}+\frac{1}{4} \sin ^{-1}(x)+\frac{2 \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{1-x}}{\sqrt{3} \sqrt [3]{1+x}}\right )}{3 \sqrt{3}}-\frac{2 \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{1+x}}{\sqrt{3} \sqrt [3]{1-x}}\right )}{3 \sqrt{3}}-\frac{1}{9} \log (1-x)+\frac{1}{9} \log (1+x)+\frac{1}{3} \log \left (1+\frac{\sqrt [3]{1-x}}{\sqrt [3]{1+x}}\right )-\frac{1}{3} \log \left (1+\frac{\sqrt [3]{1+x}}{\sqrt [3]{1-x}}\right )-\frac{5}{18} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac{5}{18} \operatorname{Subst}\left (\int \frac{1-\frac{\sqrt{3} x}{2}}{1-\sqrt{3} x+x^2} \, dx,x,\frac{\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac{5}{18} \operatorname{Subst}\left (\int \frac{1+\frac{\sqrt{3} x}{2}}{1+\sqrt{3} x+x^2} \, dx,x,\frac{\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac{7}{6} \operatorname{Subst}\left (\int \frac{x^4}{1+x^6} \, dx,x,\frac{\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )\\ &=\frac{x}{2}+\frac{x^2}{4}-\frac{7}{12} (1-x)^{5/6} \sqrt [6]{1+x}+\frac{1}{6} (1-x)^{2/3} \sqrt [3]{1+x}-\frac{1}{4} (1-x)^{5/3} \sqrt [3]{1+x}+\frac{1}{3} \sqrt [3]{1-x} (1+x)^{2/3}-\frac{1}{4} (1-x)^{4/3} (1+x)^{2/3}+\frac{5}{12} \sqrt [6]{1-x} (1+x)^{5/6}-\frac{1}{4} (1-x)^{7/6} (1+x)^{5/6}-\frac{1}{4} (1-x)^{5/6} (1+x)^{7/6}+\frac{1}{4} x \sqrt{1-x^2}+\frac{1}{4} \sin ^{-1}(x)-\frac{5}{18} \tan ^{-1}\left (\frac{\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\frac{2 \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{1-x}}{\sqrt{3} \sqrt [3]{1+x}}\right )}{3 \sqrt{3}}-\frac{2 \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{1+x}}{\sqrt{3} \sqrt [3]{1-x}}\right )}{3 \sqrt{3}}-\frac{1}{9} \log (1-x)+\frac{1}{9} \log (1+x)+\frac{1}{3} \log \left (1+\frac{\sqrt [3]{1-x}}{\sqrt [3]{1+x}}\right )-\frac{1}{3} \log \left (1+\frac{\sqrt [3]{1+x}}{\sqrt [3]{1-x}}\right )-\frac{5}{72} \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{3} x+x^2} \, dx,x,\frac{\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac{5}{72} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{3} x+x^2} \, dx,x,\frac{\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac{7}{18} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac{7}{18} \operatorname{Subst}\left (\int \frac{-\frac{1}{2}+\frac{\sqrt{3} x}{2}}{1-\sqrt{3} x+x^2} \, dx,x,\frac{\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac{7}{18} \operatorname{Subst}\left (\int \frac{-\frac{1}{2}-\frac{\sqrt{3} x}{2}}{1+\sqrt{3} x+x^2} \, dx,x,\frac{\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\frac{5 \operatorname{Subst}\left (\int \frac{-\sqrt{3}+2 x}{1-\sqrt{3} x+x^2} \, dx,x,\frac{\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )}{24 \sqrt{3}}-\frac{5 \operatorname{Subst}\left (\int \frac{\sqrt{3}+2 x}{1+\sqrt{3} x+x^2} \, dx,x,\frac{\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )}{24 \sqrt{3}}\\ &=\frac{x}{2}+\frac{x^2}{4}-\frac{7}{12} (1-x)^{5/6} \sqrt [6]{1+x}+\frac{1}{6} (1-x)^{2/3} \sqrt [3]{1+x}-\frac{1}{4} (1-x)^{5/3} \sqrt [3]{1+x}+\frac{1}{3} \sqrt [3]{1-x} (1+x)^{2/3}-\frac{1}{4} (1-x)^{4/3} (1+x)^{2/3}+\frac{5}{12} \sqrt [6]{1-x} (1+x)^{5/6}-\frac{1}{4} (1-x)^{7/6} (1+x)^{5/6}-\frac{1}{4} (1-x)^{5/6} (1+x)^{7/6}+\frac{1}{4} x \sqrt{1-x^2}+\frac{1}{4} \sin ^{-1}(x)-\frac{2}{3} \tan ^{-1}\left (\frac{\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\frac{2 \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{1-x}}{\sqrt{3} \sqrt [3]{1+x}}\right )}{3 \sqrt{3}}-\frac{2 \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{1+x}}{\sqrt{3} \sqrt [3]{1-x}}\right )}{3 \sqrt{3}}-\frac{1}{9} \log (1-x)+\frac{1}{9} \log (1+x)+\frac{1}{3} \log \left (1+\frac{\sqrt [3]{1-x}}{\sqrt [3]{1+x}}\right )+\frac{5 \log \left (1+\frac{\sqrt [3]{1-x}}{\sqrt [3]{1+x}}-\frac{\sqrt{3} \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )}{24 \sqrt{3}}-\frac{5 \log \left (1+\frac{\sqrt [3]{1-x}}{\sqrt [3]{1+x}}+\frac{\sqrt{3} \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )}{24 \sqrt{3}}-\frac{1}{3} \log \left (1+\frac{\sqrt [3]{1+x}}{\sqrt [3]{1-x}}\right )-\frac{7}{72} \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{3} x+x^2} \, dx,x,\frac{\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac{7}{72} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{3} x+x^2} \, dx,x,\frac{\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\frac{5}{36} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,-\sqrt{3}+\frac{2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\frac{5}{36} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,\sqrt{3}+\frac{2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac{7 \operatorname{Subst}\left (\int \frac{-\sqrt{3}+2 x}{1-\sqrt{3} x+x^2} \, dx,x,\frac{\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )}{24 \sqrt{3}}+\frac{7 \operatorname{Subst}\left (\int \frac{\sqrt{3}+2 x}{1+\sqrt{3} x+x^2} \, dx,x,\frac{\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )}{24 \sqrt{3}}\\ &=\frac{x}{2}+\frac{x^2}{4}-\frac{7}{12} (1-x)^{5/6} \sqrt [6]{1+x}+\frac{1}{6} (1-x)^{2/3} \sqrt [3]{1+x}-\frac{1}{4} (1-x)^{5/3} \sqrt [3]{1+x}+\frac{1}{3} \sqrt [3]{1-x} (1+x)^{2/3}-\frac{1}{4} (1-x)^{4/3} (1+x)^{2/3}+\frac{5}{12} \sqrt [6]{1-x} (1+x)^{5/6}-\frac{1}{4} (1-x)^{7/6} (1+x)^{5/6}-\frac{1}{4} (1-x)^{5/6} (1+x)^{7/6}+\frac{1}{4} x \sqrt{1-x^2}+\frac{1}{4} \sin ^{-1}(x)-\frac{2}{3} \tan ^{-1}\left (\frac{\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\frac{2 \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{1-x}}{\sqrt{3} \sqrt [3]{1+x}}\right )}{3 \sqrt{3}}+\frac{5}{36} \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac{5}{36} \tan ^{-1}\left (\sqrt{3}+\frac{2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac{2 \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{1+x}}{\sqrt{3} \sqrt [3]{1-x}}\right )}{3 \sqrt{3}}-\frac{1}{9} \log (1-x)+\frac{1}{9} \log (1+x)+\frac{1}{3} \log \left (1+\frac{\sqrt [3]{1-x}}{\sqrt [3]{1+x}}\right )-\frac{\log \left (1+\frac{\sqrt [3]{1-x}}{\sqrt [3]{1+x}}-\frac{\sqrt{3} \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )}{12 \sqrt{3}}+\frac{\log \left (1+\frac{\sqrt [3]{1-x}}{\sqrt [3]{1+x}}+\frac{\sqrt{3} \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )}{12 \sqrt{3}}-\frac{1}{3} \log \left (1+\frac{\sqrt [3]{1+x}}{\sqrt [3]{1-x}}\right )+\frac{7}{36} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,-\sqrt{3}+\frac{2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\frac{7}{36} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,\sqrt{3}+\frac{2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )\\ &=\frac{x}{2}+\frac{x^2}{4}-\frac{7}{12} (1-x)^{5/6} \sqrt [6]{1+x}+\frac{1}{6} (1-x)^{2/3} \sqrt [3]{1+x}-\frac{1}{4} (1-x)^{5/3} \sqrt [3]{1+x}+\frac{1}{3} \sqrt [3]{1-x} (1+x)^{2/3}-\frac{1}{4} (1-x)^{4/3} (1+x)^{2/3}+\frac{5}{12} \sqrt [6]{1-x} (1+x)^{5/6}-\frac{1}{4} (1-x)^{7/6} (1+x)^{5/6}-\frac{1}{4} (1-x)^{5/6} (1+x)^{7/6}+\frac{1}{4} x \sqrt{1-x^2}+\frac{1}{4} \sin ^{-1}(x)-\frac{2}{3} \tan ^{-1}\left (\frac{\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\frac{2 \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{1-x}}{\sqrt{3} \sqrt [3]{1+x}}\right )}{3 \sqrt{3}}+\frac{1}{3} \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac{1}{3} \tan ^{-1}\left (\sqrt{3}+\frac{2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac{2 \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{1+x}}{\sqrt{3} \sqrt [3]{1-x}}\right )}{3 \sqrt{3}}-\frac{1}{9} \log (1-x)+\frac{1}{9} \log (1+x)+\frac{1}{3} \log \left (1+\frac{\sqrt [3]{1-x}}{\sqrt [3]{1+x}}\right )-\frac{\log \left (1+\frac{\sqrt [3]{1-x}}{\sqrt [3]{1+x}}-\frac{\sqrt{3} \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )}{12 \sqrt{3}}+\frac{\log \left (1+\frac{\sqrt [3]{1-x}}{\sqrt [3]{1+x}}+\frac{\sqrt{3} \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )}{12 \sqrt{3}}-\frac{1}{3} \log \left (1+\frac{\sqrt [3]{1+x}}{\sqrt [3]{1-x}}\right )\\ \end{align*}

Mathematica [C]  time = 0.675763, size = 348, normalized size = 1.19 \[ -\frac{5 \sqrt{1-x^2} \, _2F_1\left (\frac{1}{6},\frac{1}{6};\frac{7}{6};\frac{1-x}{2}\right )}{6 \sqrt [6]{2} \sqrt [3]{1-x} \sqrt{x+1}}-\frac{2^{2/3} \sqrt [3]{1-x^2} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{1-x}{2}\right )}{3 \sqrt [3]{x+1}}-\frac{1}{12} \sqrt [3]{x+1} \left (-4\ 2^{2/3} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{x+1}{2}\right )+\frac{(3 x+10) \left (1-x^2\right )^{5/6}}{x+1}-\frac{(3 x+2) \sqrt{1-x^2}}{\sqrt [3]{1-x}}-(3 x+1) \sqrt [3]{1-x^2}-3 \sqrt [3]{1-x} x \sqrt [6]{1-x^2}-\frac{3 \sqrt [3]{1-x} x (x+2)}{\sqrt [3]{1-x^2}}+(1-x)^{2/3} (1-3 x)\right )-\frac{7 \left (1-x^2\right )^{5/6} \, _2F_1\left (\frac{5}{6},\frac{5}{6};\frac{11}{6};\frac{1-x}{2}\right )}{30\ 2^{5/6} (x+1)^{5/6}}+\frac{(1-x)^{5/6} (x+1)^{5/6} \sin ^{-1}(x)}{4 \left (1-x^2\right )^{5/6}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(Sqrt[1 - x]*x*(1 + x)^(2/3))/(-((1 - x)^(5/6)*(1 + x)^(1/3)) + (1 - x)^(2/3)*Sqrt[1 + x]),x]

[Out]

((1 - x)^(5/6)*(1 + x)^(5/6)*ArcSin[x])/(4*(1 - x^2)^(5/6)) - (5*Sqrt[1 - x^2]*Hypergeometric2F1[1/6, 1/6, 7/6
, (1 - x)/2])/(6*2^(1/6)*(1 - x)^(1/3)*Sqrt[1 + x]) - (2^(2/3)*(1 - x^2)^(1/3)*Hypergeometric2F1[1/3, 1/3, 4/3
, (1 - x)/2])/(3*(1 + x)^(1/3)) - ((1 + x)^(1/3)*((1 - 3*x)*(1 - x)^(2/3) - (3*(1 - x)^(1/3)*x*(2 + x))/(1 - x
^2)^(1/3) - 3*(1 - x)^(1/3)*x*(1 - x^2)^(1/6) - (1 + 3*x)*(1 - x^2)^(1/3) - ((2 + 3*x)*Sqrt[1 - x^2])/(1 - x)^
(1/3) + ((10 + 3*x)*(1 - x^2)^(5/6))/(1 + x) - 4*2^(2/3)*Hypergeometric2F1[1/3, 1/3, 4/3, (1 + x)/2]))/12 - (7
*(1 - x^2)^(5/6)*Hypergeometric2F1[5/6, 5/6, 11/6, (1 - x)/2])/(30*2^(5/6)*(1 + x)^(5/6))

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Maple [F]  time = 0.039, size = 0, normalized size = 0. \begin{align*} \int{x \left ( 1+x \right ) ^{{\frac{2}{3}}}\sqrt{1-x} \left ( - \left ( 1-x \right ) ^{{\frac{5}{6}}}\sqrt [3]{1+x}+ \left ( 1-x \right ) ^{{\frac{2}{3}}}\sqrt{1+x} \right ) ^{-1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(1+x)^(2/3)*(1-x)^(1/2)/(-(1-x)^(5/6)*(1+x)^(1/3)+(1-x)^(2/3)*(1+x)^(1/2)),x)

[Out]

int(x*(1+x)^(2/3)*(1-x)^(1/2)/(-(1-x)^(5/6)*(1+x)^(1/3)+(1-x)^(2/3)*(1+x)^(1/2)),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (x + 1\right )}^{\frac{2}{3}} x \sqrt{-x + 1}}{\sqrt{x + 1}{\left (-x + 1\right )}^{\frac{2}{3}} -{\left (x + 1\right )}^{\frac{1}{3}}{\left (-x + 1\right )}^{\frac{5}{6}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(1+x)^(2/3)*(1-x)^(1/2)/(-(1-x)^(5/6)*(1+x)^(1/3)+(1-x)^(2/3)*(1+x)^(1/2)),x, algorithm="maxima")

[Out]

integrate((x + 1)^(2/3)*x*sqrt(-x + 1)/(sqrt(x + 1)*(-x + 1)^(2/3) - (x + 1)^(1/3)*(-x + 1)^(5/6)), x)

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Fricas [B]  time = 2.66366, size = 2724, normalized size = 9.33 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(1+x)^(2/3)*(1-x)^(1/2)/(-(1-x)^(5/6)*(1+x)^(1/3)+(1-x)^(2/3)*(1+x)^(1/2)),x, algorithm="fricas")

[Out]

1/4*x^2 + 1/12*(3*x + 2)*(x + 1)^(5/6)*(-x + 1)^(1/6) + 1/12*(3*x + 1)*(x + 1)^(2/3)*(-x + 1)^(1/3) + 1/4*sqrt
(x + 1)*x*sqrt(-x + 1) + 1/12*(3*x - 1)*(x + 1)^(1/3)*(-x + 1)^(2/3) - 1/12*(3*x + 10)*(x + 1)^(1/6)*(-x + 1)^
(5/6) - 2/9*sqrt(3)*arctan(-1/3*(sqrt(3)*(x + 1) - 2*sqrt(3)*(x + 1)^(2/3)*(-x + 1)^(1/3))/(x + 1)) - 2/9*sqrt
(3)*arctan(1/3*(sqrt(3)*(x - 1) + 2*sqrt(3)*(x + 1)^(1/3)*(-x + 1)^(2/3))/(x - 1)) - 5/72*sqrt(3)*log(100*(sqr
t(3)*(x + 1)^(5/6)*(-x + 1)^(1/6) + x + (x + 1)^(2/3)*(-x + 1)^(1/3) + 1)/(x + 1)) + 5/72*sqrt(3)*log(-100*(sq
rt(3)*(x + 1)^(5/6)*(-x + 1)^(1/6) - x - (x + 1)^(2/3)*(-x + 1)^(1/3) - 1)/(x + 1)) - 7/72*sqrt(3)*log(196*(sq
rt(3)*(x + 1)^(1/6)*(-x + 1)^(5/6) + x - (x + 1)^(1/3)*(-x + 1)^(2/3) - 1)/(x - 1)) + 7/72*sqrt(3)*log(-196*(s
qrt(3)*(x + 1)^(1/6)*(-x + 1)^(5/6) - x + (x + 1)^(1/3)*(-x + 1)^(2/3) + 1)/(x - 1)) + 1/2*x + 5/18*arctan(-(s
qrt(3)*(x + 1) - 2*(x + 1)*sqrt((sqrt(3)*(x + 1)^(5/6)*(-x + 1)^(1/6) + x + (x + 1)^(2/3)*(-x + 1)^(1/3) + 1)/
(x + 1)) + 2*(x + 1)^(5/6)*(-x + 1)^(1/6))/(x + 1)) + 5/18*arctan((sqrt(3)*(x + 1) + 2*(x + 1)*sqrt(-(sqrt(3)*
(x + 1)^(5/6)*(-x + 1)^(1/6) - x - (x + 1)^(2/3)*(-x + 1)^(1/3) - 1)/(x + 1)) - 2*(x + 1)^(5/6)*(-x + 1)^(1/6)
)/(x + 1)) + 7/18*arctan(-(sqrt(3)*(x - 1) - 2*(x - 1)*sqrt((sqrt(3)*(x + 1)^(1/6)*(-x + 1)^(5/6) + x - (x + 1
)^(1/3)*(-x + 1)^(2/3) - 1)/(x - 1)) + 2*(x + 1)^(1/6)*(-x + 1)^(5/6))/(x - 1)) + 7/18*arctan((sqrt(3)*(x - 1)
+ 2*(x - 1)*sqrt(-(sqrt(3)*(x + 1)^(1/6)*(-x + 1)^(5/6) - x + (x + 1)^(1/3)*(-x + 1)^(2/3) + 1)/(x - 1)) - 2*
(x + 1)^(1/6)*(-x + 1)^(5/6))/(x - 1)) - 5/18*arctan((-x + 1)^(1/6)/(x + 1)^(1/6)) - 7/18*arctan((x + 1)^(1/6)
*(-x + 1)^(5/6)/(x - 1)) - 1/2*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x) - 2/9*log((x + (x + 1)^(2/3)*(-x + 1)^
(1/3) + 1)/(x + 1)) + 1/9*log((x - (x + 1)^(2/3)*(-x + 1)^(1/3) + (x + 1)^(1/3)*(-x + 1)^(2/3) + 1)/(x + 1)) -
1/9*log((x - (x + 1)^(2/3)*(-x + 1)^(1/3) + (x + 1)^(1/3)*(-x + 1)^(2/3) - 1)/(x - 1)) + 2/9*log(-(x - (x + 1
)^(1/3)*(-x + 1)^(2/3) - 1)/(x - 1))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \sqrt{1 - x} \left (x + 1\right )^{\frac{2}{3}}}{- \left (1 - x\right )^{\frac{5}{6}} \sqrt [3]{x + 1} + \left (1 - x\right )^{\frac{2}{3}} \sqrt{x + 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(1+x)**(2/3)*(1-x)**(1/2)/(-(1-x)**(5/6)*(1+x)**(1/3)+(1-x)**(2/3)*(1+x)**(1/2)),x)

[Out]

Integral(x*sqrt(1 - x)*(x + 1)**(2/3)/(-(1 - x)**(5/6)*(x + 1)**(1/3) + (1 - x)**(2/3)*sqrt(x + 1)), x)

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(1+x)^(2/3)*(1-x)^(1/2)/(-(1-x)^(5/6)*(1+x)^(1/3)+(1-x)^(2/3)*(1+x)^(1/2)),x, algorithm="giac")

[Out]

Timed out