3.3.22 \(\int \frac {\sqrt {1-x} x (1+x)^{2/3}}{-(1-x)^{5/6} \sqrt [3]{1+x}+(1-x)^{2/3} \sqrt {1+x}} \, dx\) [222]

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

[Out]

-1/12*(1-3*x)*(1-x)^(2/3)*(1+x)^(1/3)-1/4*(1-x)*(3+x)+1/12*(1-x)^(1/3)*(1+x)^(2/3)*(1+3*x)+1/12*(1-x)^(1/6)*(1
+x)^(5/6)*(2+3*x)-1/12*(1-x)^(5/6)*(1+x)^(1/6)*(10+3*x)+1/6*arctan((1+x)^(1/6)/(1-x)^(1/6))-5/6*arctan(((1-x)^
(1/3)-(1+x)^(1/3))/(1-x)^(1/6)/(1+x)^(1/6))-4/9*arctan(1/3*((1-x)^(1/3)-2*(1+x)^(1/3))/(1-x)^(1/3)*3^(1/2))*3^
(1/2)+1/18*arctanh((1-x)^(1/6)*(1+x)^(1/6)*3^(1/2)/((1-x)^(1/3)+(1+x)^(1/3)))*3^(1/2)+1/4*x*(1-x)^(1/2)*(1+x)^
(1/2)

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Rubi [A]
time = 1.35, 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 = {6820, 6874, 52, 62, 531, 201, 222, 689, 904, 65, 246, 215, 648, 632, 210, 642, 209, 26, 21, 338, 301} \begin {gather*} \frac {\text {ArcSin}(x)}{4}-\frac {2}{3} \text {ArcTan}\left (\frac {\sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )+\frac {2 \text {ArcTan}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{x+1}}\right )}{3 \sqrt {3}}+\frac {1}{3} \text {ArcTan}\left (\sqrt {3}-\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )-\frac {1}{3} \text {ArcTan}\left (\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+\sqrt {3}\right )-\frac {2 \text {ArcTan}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x+1}}{\sqrt {3} \sqrt [3]{1-x}}\right )}{3 \sqrt {3}}+\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 ) \end {gather*}

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 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 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 52

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 62

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

Rule 65

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 201

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 209

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

Rule 210

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

Rule 215

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/(a*n))*Int[1/
(r^2 + s^2*x^2), x] + 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 222

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 246

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 301

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)/(a*n*s^m))*Int[1/(r^2 + s^2*x^2), x] +
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]

Rule 338

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 531

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 632

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 642

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 648

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 689

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c/e)*x)^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 904

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/e)*x)^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 6820

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

Rule 6874

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

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} \text {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} \text {Subst}\left (\int \frac {1}{1+x^6} \, dx,x,\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac {7}{6} \text {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} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac {5}{18} \text {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} \text {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} \text {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} \text {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} \text {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} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac {7}{18} \text {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} \text {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 \text {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 \text {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} \text {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} \text {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} \text {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} \text {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 \text {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 \text {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} \text {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} \text {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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 50.06, size = 455, normalized size = 1.56 \begin {gather*} -\frac {1}{12} \sqrt [3]{1+x} \left ((1-3 x) (1-x)^{2/3}-\frac {3 \sqrt [3]{1-x} x (2+x)}{\sqrt [3]{1-x^2}}-3 \sqrt [3]{1-x} x \sqrt [6]{1-x^2}-(1+3 x) \sqrt [3]{1-x^2}-\frac {(2+3 x) \sqrt {1-x^2}}{\sqrt [3]{1-x}}+\frac {(10+3 x) \left (1-x^2\right )^{5/6}}{1+x}-4\ 2^{2/3} \, _2F_1\left (\frac {1}{3},\frac {1}{3};\frac {4}{3};\frac {1+x}{2}\right )\right )+\frac {1}{36} \left (9 \sin ^{-1}(x)+14 \tan ^{-1}\left (\frac {\sqrt [3]{1+x}}{\sqrt [6]{1-x^2}}\right )+7 \left (1+i \sqrt {3}\right ) \tan ^{-1}\left (\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{1+x}}{2 \sqrt [6]{1-x^2}}\right )+7 \left (1-i \sqrt {3}\right ) \tan ^{-1}\left (\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{1+x}}{2 \sqrt [6]{1-x^2}}\right )+8 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{1-x^2}}{(1+x)^{2/3}}}{\sqrt {3}}\right )-\frac {15\ 2^{5/6} \sqrt {1-x^2} \, _2F_1\left (\frac {1}{6},\frac {1}{6};\frac {7}{6};\frac {1-x}{2}\right )}{\sqrt [3]{1-x} \sqrt {1+x}}-8 \log \left ((1+x)^{2/3}+\sqrt [3]{1-x^2}\right )+4 \log \left (\sqrt [3]{1+x}+x \sqrt [3]{1+x}-(1+x)^{2/3} \sqrt [3]{1-x^2}+\left (1-x^2\right )^{2/3}\right )\right ) \end {gather*}

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]

-1/12*((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])) + (9*ArcSin[x] + 14*ArcTan[(1 + x)^
(1/3)/(1 - x^2)^(1/6)] + 7*(1 + I*Sqrt[3])*ArcTan[((1 - I*Sqrt[3])*(1 + x)^(1/3))/(2*(1 - x^2)^(1/6))] + 7*(1
- I*Sqrt[3])*ArcTan[((1 + I*Sqrt[3])*(1 + x)^(1/3))/(2*(1 - x^2)^(1/6))] + 8*Sqrt[3]*ArcTan[(1 - (2*(1 - x^2)^
(1/3))/(1 + x)^(2/3))/Sqrt[3]] - (15*2^(5/6)*Sqrt[1 - x^2]*Hypergeometric2F1[1/6, 1/6, 7/6, (1 - x)/2])/((1 -
x)^(1/3)*Sqrt[1 + x]) - 8*Log[(1 + x)^(2/3) + (1 - x^2)^(1/3)] + 4*Log[(1 + x)^(1/3) + x*(1 + x)^(1/3) - (1 +
x)^(2/3)*(1 - x^2)^(1/3) + (1 - x^2)^(2/3)])/36

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x \left (1+x \right )^{\frac {2}{3}} \sqrt {1-x}}{-\left (1-x \right )^{\frac {5}{6}} \left (1+x \right )^{\frac {1}{3}}+\left (1-x \right )^{\frac {2}{3}} \sqrt {1+x}}\, dx\]

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 = 0.00, 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)^(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] Leaf count of result is larger than twice the leaf count of optimal. 865 vs. \(2 (220) = 440\).
time = 0.53, size = 865, normalized size = 2.96 \begin {gather*} \frac {1}{4} \, x^{2} + \frac {1}{12} \, {\left (3 \, x + 2\right )} {\left (x + 1\right )}^{\frac {5}{6}} {\left (-x + 1\right )}^{\frac {1}{6}} + \frac {1}{12} \, {\left (3 \, x + 1\right )} {\left (x + 1\right )}^{\frac {2}{3}} {\left (-x + 1\right )}^{\frac {1}{3}} + \frac {1}{4} \, \sqrt {x + 1} x \sqrt {-x + 1} + \frac {1}{12} \, {\left (3 \, x - 1\right )} {\left (x + 1\right )}^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {2}{3}} - \frac {1}{12} \, {\left (3 \, x + 10\right )} {\left (x + 1\right )}^{\frac {1}{6}} {\left (-x + 1\right )}^{\frac {5}{6}} - \frac {2}{9} \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (x + 1\right )} - 2 \, \sqrt {3} {\left (x + 1\right )}^{\frac {2}{3}} {\left (-x + 1\right )}^{\frac {1}{3}}}{3 \, {\left (x + 1\right )}}\right ) - \frac {2}{9} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (x - 1\right )} + 2 \, \sqrt {3} {\left (x + 1\right )}^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {2}{3}}}{3 \, {\left (x - 1\right )}}\right ) - \frac {5}{72} \, \sqrt {3} \log \left (\frac {100 \, {\left (\sqrt {3} {\left (x + 1\right )}^{\frac {5}{6}} {\left (-x + 1\right )}^{\frac {1}{6}} + x + {\left (x + 1\right )}^{\frac {2}{3}} {\left (-x + 1\right )}^{\frac {1}{3}} + 1\right )}}{x + 1}\right ) + \frac {5}{72} \, \sqrt {3} \log \left (-\frac {100 \, {\left (\sqrt {3} {\left (x + 1\right )}^{\frac {5}{6}} {\left (-x + 1\right )}^{\frac {1}{6}} - x - {\left (x + 1\right )}^{\frac {2}{3}} {\left (-x + 1\right )}^{\frac {1}{3}} - 1\right )}}{x + 1}\right ) - \frac {7}{72} \, \sqrt {3} \log \left (\frac {196 \, {\left (\sqrt {3} {\left (x + 1\right )}^{\frac {1}{6}} {\left (-x + 1\right )}^{\frac {5}{6}} + x - {\left (x + 1\right )}^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {2}{3}} - 1\right )}}{x - 1}\right ) + \frac {7}{72} \, \sqrt {3} \log \left (-\frac {196 \, {\left (\sqrt {3} {\left (x + 1\right )}^{\frac {1}{6}} {\left (-x + 1\right )}^{\frac {5}{6}} - x + {\left (x + 1\right )}^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {2}{3}} + 1\right )}}{x - 1}\right ) + \frac {1}{2} \, x + \frac {5}{18} \, \arctan \left (-\frac {\sqrt {3} {\left (x + 1\right )} - 2 \, {\left (x + 1\right )} \sqrt {\frac {\sqrt {3} {\left (x + 1\right )}^{\frac {5}{6}} {\left (-x + 1\right )}^{\frac {1}{6}} + x + {\left (x + 1\right )}^{\frac {2}{3}} {\left (-x + 1\right )}^{\frac {1}{3}} + 1}{x + 1}} + 2 \, {\left (x + 1\right )}^{\frac {5}{6}} {\left (-x + 1\right )}^{\frac {1}{6}}}{x + 1}\right ) + \frac {5}{18} \, \arctan \left (\frac {\sqrt {3} {\left (x + 1\right )} + 2 \, {\left (x + 1\right )} \sqrt {-\frac {\sqrt {3} {\left (x + 1\right )}^{\frac {5}{6}} {\left (-x + 1\right )}^{\frac {1}{6}} - x - {\left (x + 1\right )}^{\frac {2}{3}} {\left (-x + 1\right )}^{\frac {1}{3}} - 1}{x + 1}} - 2 \, {\left (x + 1\right )}^{\frac {5}{6}} {\left (-x + 1\right )}^{\frac {1}{6}}}{x + 1}\right ) + \frac {7}{18} \, \arctan \left (-\frac {\sqrt {3} {\left (x - 1\right )} - 2 \, {\left (x - 1\right )} \sqrt {\frac {\sqrt {3} {\left (x + 1\right )}^{\frac {1}{6}} {\left (-x + 1\right )}^{\frac {5}{6}} + x - {\left (x + 1\right )}^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {2}{3}} - 1}{x - 1}} + 2 \, {\left (x + 1\right )}^{\frac {1}{6}} {\left (-x + 1\right )}^{\frac {5}{6}}}{x - 1}\right ) + \frac {7}{18} \, \arctan \left (\frac {\sqrt {3} {\left (x - 1\right )} + 2 \, {\left (x - 1\right )} \sqrt {-\frac {\sqrt {3} {\left (x + 1\right )}^{\frac {1}{6}} {\left (-x + 1\right )}^{\frac {5}{6}} - x + {\left (x + 1\right )}^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {2}{3}} + 1}{x - 1}} - 2 \, {\left (x + 1\right )}^{\frac {1}{6}} {\left (-x + 1\right )}^{\frac {5}{6}}}{x - 1}\right ) - \frac {5}{18} \, \arctan \left (\frac {{\left (-x + 1\right )}^{\frac {1}{6}}}{{\left (x + 1\right )}^{\frac {1}{6}}}\right ) - \frac {7}{18} \, \arctan \left (\frac {{\left (x + 1\right )}^{\frac {1}{6}} {\left (-x + 1\right )}^{\frac {5}{6}}}{x - 1}\right ) - \frac {1}{2} \, \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) - \frac {2}{9} \, \log \left (\frac {x + {\left (x + 1\right )}^{\frac {2}{3}} {\left (-x + 1\right )}^{\frac {1}{3}} + 1}{x + 1}\right ) + \frac {1}{9} \, \log \left (\frac {x - {\left (x + 1\right )}^{\frac {2}{3}} {\left (-x + 1\right )}^{\frac {1}{3}} + {\left (x + 1\right )}^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {2}{3}} + 1}{x + 1}\right ) - \frac {1}{9} \, \log \left (\frac {x - {\left (x + 1\right )}^{\frac {2}{3}} {\left (-x + 1\right )}^{\frac {1}{3}} + {\left (x + 1\right )}^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {2}{3}} - 1}{x - 1}\right ) + \frac {2}{9} \, \log \left (-\frac {x - {\left (x + 1\right )}^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {2}{3}} - 1}{x - 1}\right ) \end {gather*}

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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

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]

Exception raised: SystemError >> excessive stack use: stack is 5006 deep

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x\,\sqrt {1-x}\,{\left (x+1\right )}^{2/3}}{{\left (1-x\right )}^{2/3}\,\sqrt {x+1}-{\left (1-x\right )}^{5/6}\,{\left (x+1\right )}^{1/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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