3.32.38 \(\int \frac {\sqrt [6]{\frac {1-b x}{c+x}} (1+d x^2)}{(1+b x) (1+c x)} \, dx\) [3138]

Optimal. Leaf size=887 \[ \frac {d (c+x) \sqrt [6]{\frac {1-b x}{c+x}}}{b c}-\frac {\left (6 b+7 c+b c^2\right ) d \text {ArcTan}\left (\frac {\sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}\right )}{3 b^{11/6} c^2}-\frac {2 \sqrt [6]{b+c} \left (c^2+d\right ) \text {ArcTan}\left (\frac {\sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b+c}}\right )}{c^2 (-b+c) \sqrt [6]{1-c^2}}-\frac {\sqrt [6]{2} \sqrt {3} \left (b^2+d\right ) \text {ArcTan}\left (\frac {-\sqrt [6]{b}+2^{5/6} \sqrt [6]{-1+b c} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt {3} \sqrt [6]{b}}\right )}{b^{11/6} (b-c) \sqrt [6]{-1+b c}}-\frac {\sqrt [6]{2} \sqrt {3} \left (b^2+d\right ) \text {ArcTan}\left (\frac {\sqrt [6]{b}+2^{5/6} \sqrt [6]{-1+b c} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt {3} \sqrt [6]{b}}\right )}{b^{11/6} (b-c) \sqrt [6]{-1+b c}}+\frac {\left (6 b+7 c+b c^2\right ) d \text {ArcTan}\left (\frac {\sqrt [6]{b} \sqrt [6]{\frac {1-b x}{c+x}}}{-\sqrt [3]{b}+\sqrt [3]{\frac {1-b x}{c+x}}}\right )}{6 b^{11/6} c^2}-\frac {\sqrt [6]{b+c} \left (c^2+d\right ) \text {ArcTan}\left (\frac {\sqrt [6]{b+c} \sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [3]{b+c}-\sqrt [3]{1-c^2} \sqrt [3]{\frac {1-b x}{c+x}}}\right )}{c^2 (-b+c) \sqrt [6]{1-c^2}}-\frac {2 \sqrt [6]{2} \left (b^2+d\right ) \tanh ^{-1}\left (\frac {\sqrt [6]{-1+b c} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{2} \sqrt [6]{b}}\right )}{b^{11/6} (b-c) \sqrt [6]{-1+b c}}-\frac {\left (6 b+7 c+b c^2\right ) d \tanh ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [3]{b}+\sqrt [3]{\frac {1-b x}{c+x}}}\right )}{2 \sqrt {3} b^{11/6} c^2}-\frac {\sqrt [6]{2} \left (b^2+d\right ) \tanh ^{-1}\left (\frac {2^{5/6} \sqrt [6]{b} \sqrt [6]{-1+b c} \sqrt [6]{\frac {1-b x}{c+x}}}{2 \sqrt [3]{b}+2^{2/3} \sqrt [3]{-1+b c} \sqrt [3]{\frac {1-b x}{c+x}}}\right )}{b^{11/6} (b-c) \sqrt [6]{-1+b c}}-\frac {\sqrt {3} \sqrt [6]{b+c} \left (c^2+d\right ) \tanh ^{-1}\left (\frac {\sqrt [3]{b+c}+\sqrt [3]{1-c^2} \sqrt [3]{\frac {1-b x}{c+x}}}{\sqrt {3} \sqrt [6]{b+c} \sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}}\right )}{c^2 (-b+c) \sqrt [6]{1-c^2}} \]

[Out]

d*(c+x)*((-b*x+1)/(c+x))^(1/6)/b/c-1/3*(b*c^2+6*b+7*c)*d*arctan(((-b*x+1)/(c+x))^(1/6)/b^(1/6))/b^(11/6)/c^2-2
*(b+c)^(1/6)*(c^2+d)*arctan((-c^2+1)^(1/6)*((-b*x+1)/(c+x))^(1/6)/(b+c)^(1/6))/c^2/(-b+c)/(-c^2+1)^(1/6)-2^(1/
6)*3^(1/2)*(b^2+d)*arctan(1/3*(-b^(1/6)+2^(5/6)*(b*c-1)^(1/6)*((-b*x+1)/(c+x))^(1/6))*3^(1/2)/b^(1/6))/b^(11/6
)/(b-c)/(b*c-1)^(1/6)-2^(1/6)*3^(1/2)*(b^2+d)*arctan(1/3*(b^(1/6)+2^(5/6)*(b*c-1)^(1/6)*((-b*x+1)/(c+x))^(1/6)
)*3^(1/2)/b^(1/6))/b^(11/6)/(b-c)/(b*c-1)^(1/6)+1/6*(b*c^2+6*b+7*c)*d*arctan(b^(1/6)*((-b*x+1)/(c+x))^(1/6)/(-
b^(1/3)+((-b*x+1)/(c+x))^(1/3)))/b^(11/6)/c^2-(b+c)^(1/6)*(c^2+d)*arctan((b+c)^(1/6)*(-c^2+1)^(1/6)*((-b*x+1)/
(c+x))^(1/6)/((b+c)^(1/3)-(-c^2+1)^(1/3)*((-b*x+1)/(c+x))^(1/3)))/c^2/(-b+c)/(-c^2+1)^(1/6)-2*2^(1/6)*(b^2+d)*
arctanh(1/2*(b*c-1)^(1/6)*((-b*x+1)/(c+x))^(1/6)*2^(5/6)/b^(1/6))/b^(11/6)/(b-c)/(b*c-1)^(1/6)-1/6*(b*c^2+6*b+
7*c)*d*arctanh(3^(1/2)*b^(1/6)*((-b*x+1)/(c+x))^(1/6)/(b^(1/3)+((-b*x+1)/(c+x))^(1/3)))*3^(1/2)/b^(11/6)/c^2-2
^(1/6)*(b^2+d)*arctanh(2^(5/6)*b^(1/6)*(b*c-1)^(1/6)*((-b*x+1)/(c+x))^(1/6)/(2*b^(1/3)+2^(2/3)*(b*c-1)^(1/3)*(
(-b*x+1)/(c+x))^(1/3)))/b^(11/6)/(b-c)/(b*c-1)^(1/6)-3^(1/2)*(b+c)^(1/6)*(c^2+d)*arctanh(1/3*((b+c)^(1/3)+(-c^
2+1)^(1/3)*((-b*x+1)/(c+x))^(1/3))*3^(1/2)/(b+c)^(1/6)/(-c^2+1)^(1/6)/((-b*x+1)/(c+x))^(1/6))/c^2/(-b+c)/(-c^2
+1)^(1/6)

________________________________________________________________________________________

Rubi [A]
time = 4.12, antiderivative size = 1549, normalized size of antiderivative = 1.75, number of steps used = 45, number of rules used = 10, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6, 6857, 205, 215, 648, 632, 210, 642, 209, 211} \begin {gather*} \frac {d \sqrt [6]{\frac {1-b x}{c+x}} (c+x)}{b c}+\frac {5 (b c+1) d \text {ArcTan}\left (\frac {\sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}\right )}{3 b^{11/6} c}-\frac {2 \left (b c^2+2 c+b\right ) d \text {ArcTan}\left (\frac {\sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}\right )}{b^{11/6} c^2}-\frac {2 \sqrt [6]{2} \left (b^2+d\right ) \text {ArcTan}\left (\frac {\sqrt [6]{1-b c} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{2} \sqrt [6]{b}}\right )}{b^{11/6} (b-c) \sqrt [6]{1-b c}}+\frac {2 \sqrt [6]{b+c} \left (c^2+d\right ) \text {ArcTan}\left (\frac {\sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b+c}}\right )}{(b-c) c^2 \sqrt [6]{1-c^2}}-\frac {5 (b c+1) d \text {ArcTan}\left (\sqrt {3}-\frac {2 \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}\right )}{6 b^{11/6} c}+\frac {\left (b c^2+2 c+b\right ) d \text {ArcTan}\left (\sqrt {3}-\frac {2 \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}\right )}{b^{11/6} c^2}+\frac {5 (b c+1) d \text {ArcTan}\left (\frac {2 \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}+\sqrt {3}\right )}{6 b^{11/6} c}-\frac {\left (b c^2+2 c+b\right ) d \text {ArcTan}\left (\frac {2 \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}+\sqrt {3}\right )}{b^{11/6} c^2}+\frac {\sqrt [6]{2} \left (b^2+d\right ) \text {ArcTan}\left (\sqrt {3}-\frac {2^{5/6} \sqrt [6]{1-b c} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}\right )}{b^{11/6} (b-c) \sqrt [6]{1-b c}}-\frac {\sqrt [6]{2} \left (b^2+d\right ) \text {ArcTan}\left (\frac {2^{5/6} \sqrt [6]{1-b c} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}+\sqrt {3}\right )}{b^{11/6} (b-c) \sqrt [6]{1-b c}}-\frac {\sqrt [6]{b+c} \left (c^2+d\right ) \text {ArcTan}\left (\sqrt {3}-\frac {2 \sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b+c}}\right )}{(b-c) c^2 \sqrt [6]{1-c^2}}+\frac {\sqrt [6]{b+c} \left (c^2+d\right ) \text {ArcTan}\left (\frac {2 \sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b+c}}+\sqrt {3}\right )}{(b-c) c^2 \sqrt [6]{1-c^2}}-\frac {5 (b c+1) d \log \left (\sqrt [3]{b}-\sqrt {3} \sqrt [6]{\frac {1-b x}{c+x}} \sqrt [6]{b}+\sqrt [3]{\frac {1-b x}{c+x}}\right )}{4 \sqrt {3} b^{11/6} c}+\frac {\sqrt {3} \left (b c^2+2 c+b\right ) d \log \left (\sqrt [3]{b}-\sqrt {3} \sqrt [6]{\frac {1-b x}{c+x}} \sqrt [6]{b}+\sqrt [3]{\frac {1-b x}{c+x}}\right )}{2 b^{11/6} c^2}+\frac {5 (b c+1) d \log \left (\sqrt [3]{b}+\sqrt {3} \sqrt [6]{\frac {1-b x}{c+x}} \sqrt [6]{b}+\sqrt [3]{\frac {1-b x}{c+x}}\right )}{4 \sqrt {3} b^{11/6} c}-\frac {\sqrt {3} \left (b c^2+2 c+b\right ) d \log \left (\sqrt [3]{b}+\sqrt {3} \sqrt [6]{\frac {1-b x}{c+x}} \sqrt [6]{b}+\sqrt [3]{\frac {1-b x}{c+x}}\right )}{2 b^{11/6} c^2}+\frac {\sqrt {3} \left (b^2+d\right ) \log \left (\sqrt [3]{2} \sqrt [3]{b}-\sqrt [6]{2} \sqrt {3} \sqrt [6]{1-b c} \sqrt [6]{\frac {1-b x}{c+x}} \sqrt [6]{b}+\sqrt [3]{1-b c} \sqrt [3]{\frac {1-b x}{c+x}}\right )}{2^{5/6} b^{11/6} (b-c) \sqrt [6]{1-b c}}-\frac {\sqrt {3} \left (b^2+d\right ) \log \left (\sqrt [3]{2} \sqrt [3]{b}+\sqrt [6]{2} \sqrt {3} \sqrt [6]{1-b c} \sqrt [6]{\frac {1-b x}{c+x}} \sqrt [6]{b}+\sqrt [3]{1-b c} \sqrt [3]{\frac {1-b x}{c+x}}\right )}{2^{5/6} b^{11/6} (b-c) \sqrt [6]{1-b c}}-\frac {\sqrt {3} \sqrt [6]{b+c} \left (c^2+d\right ) \log \left (\sqrt [3]{b+c}-\sqrt {3} \sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}} \sqrt [6]{b+c}+\sqrt [3]{1-c^2} \sqrt [3]{\frac {1-b x}{c+x}}\right )}{2 (b-c) c^2 \sqrt [6]{1-c^2}}+\frac {\sqrt {3} \sqrt [6]{b+c} \left (c^2+d\right ) \log \left (\sqrt [3]{b+c}+\sqrt {3} \sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}} \sqrt [6]{b+c}+\sqrt [3]{1-c^2} \sqrt [3]{\frac {1-b x}{c+x}}\right )}{2 (b-c) c^2 \sqrt [6]{1-c^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(((1 - b*x)/(c + x))^(1/6)*(1 + d*x^2))/((1 + b*x)*(1 + c*x)),x]

[Out]

(d*(c + x)*((1 - b*x)/(c + x))^(1/6))/(b*c) + (5*(1 + b*c)*d*ArcTan[((1 - b*x)/(c + x))^(1/6)/b^(1/6)])/(3*b^(
11/6)*c) - (2*(b + 2*c + b*c^2)*d*ArcTan[((1 - b*x)/(c + x))^(1/6)/b^(1/6)])/(b^(11/6)*c^2) - (2*2^(1/6)*(b^2
+ d)*ArcTan[((1 - b*c)^(1/6)*((1 - b*x)/(c + x))^(1/6))/(2^(1/6)*b^(1/6))])/(b^(11/6)*(b - c)*(1 - b*c)^(1/6))
+ (2*(b + c)^(1/6)*(c^2 + d)*ArcTan[((1 - c^2)^(1/6)*((1 - b*x)/(c + x))^(1/6))/(b + c)^(1/6)])/((b - c)*c^2*
(1 - c^2)^(1/6)) - (5*(1 + b*c)*d*ArcTan[Sqrt[3] - (2*((1 - b*x)/(c + x))^(1/6))/b^(1/6)])/(6*b^(11/6)*c) + ((
b + 2*c + b*c^2)*d*ArcTan[Sqrt[3] - (2*((1 - b*x)/(c + x))^(1/6))/b^(1/6)])/(b^(11/6)*c^2) + (5*(1 + b*c)*d*Ar
cTan[Sqrt[3] + (2*((1 - b*x)/(c + x))^(1/6))/b^(1/6)])/(6*b^(11/6)*c) - ((b + 2*c + b*c^2)*d*ArcTan[Sqrt[3] +
(2*((1 - b*x)/(c + x))^(1/6))/b^(1/6)])/(b^(11/6)*c^2) + (2^(1/6)*(b^2 + d)*ArcTan[Sqrt[3] - (2^(5/6)*(1 - b*c
)^(1/6)*((1 - b*x)/(c + x))^(1/6))/b^(1/6)])/(b^(11/6)*(b - c)*(1 - b*c)^(1/6)) - (2^(1/6)*(b^2 + d)*ArcTan[Sq
rt[3] + (2^(5/6)*(1 - b*c)^(1/6)*((1 - b*x)/(c + x))^(1/6))/b^(1/6)])/(b^(11/6)*(b - c)*(1 - b*c)^(1/6)) - ((b
+ c)^(1/6)*(c^2 + d)*ArcTan[Sqrt[3] - (2*(1 - c^2)^(1/6)*((1 - b*x)/(c + x))^(1/6))/(b + c)^(1/6)])/((b - c)*
c^2*(1 - c^2)^(1/6)) + ((b + c)^(1/6)*(c^2 + d)*ArcTan[Sqrt[3] + (2*(1 - c^2)^(1/6)*((1 - b*x)/(c + x))^(1/6))
/(b + c)^(1/6)])/((b - c)*c^2*(1 - c^2)^(1/6)) - (5*(1 + b*c)*d*Log[b^(1/3) - Sqrt[3]*b^(1/6)*((1 - b*x)/(c +
x))^(1/6) + ((1 - b*x)/(c + x))^(1/3)])/(4*Sqrt[3]*b^(11/6)*c) + (Sqrt[3]*(b + 2*c + b*c^2)*d*Log[b^(1/3) - Sq
rt[3]*b^(1/6)*((1 - b*x)/(c + x))^(1/6) + ((1 - b*x)/(c + x))^(1/3)])/(2*b^(11/6)*c^2) + (5*(1 + b*c)*d*Log[b^
(1/3) + Sqrt[3]*b^(1/6)*((1 - b*x)/(c + x))^(1/6) + ((1 - b*x)/(c + x))^(1/3)])/(4*Sqrt[3]*b^(11/6)*c) - (Sqrt
[3]*(b + 2*c + b*c^2)*d*Log[b^(1/3) + Sqrt[3]*b^(1/6)*((1 - b*x)/(c + x))^(1/6) + ((1 - b*x)/(c + x))^(1/3)])/
(2*b^(11/6)*c^2) + (Sqrt[3]*(b^2 + d)*Log[2^(1/3)*b^(1/3) - 2^(1/6)*Sqrt[3]*b^(1/6)*(1 - b*c)^(1/6)*((1 - b*x)
/(c + x))^(1/6) + (1 - b*c)^(1/3)*((1 - b*x)/(c + x))^(1/3)])/(2^(5/6)*b^(11/6)*(b - c)*(1 - b*c)^(1/6)) - (Sq
rt[3]*(b^2 + d)*Log[2^(1/3)*b^(1/3) + 2^(1/6)*Sqrt[3]*b^(1/6)*(1 - b*c)^(1/6)*((1 - b*x)/(c + x))^(1/6) + (1 -
b*c)^(1/3)*((1 - b*x)/(c + x))^(1/3)])/(2^(5/6)*b^(11/6)*(b - c)*(1 - b*c)^(1/6)) - (Sqrt[3]*(b + c)^(1/6)*(c
^2 + d)*Log[(b + c)^(1/3) - Sqrt[3]*(b + c)^(1/6)*(1 - c^2)^(1/6)*((1 - b*x)/(c + x))^(1/6) + (1 - c^2)^(1/3)*
((1 - b*x)/(c + x))^(1/3)])/(2*(b - c)*c^2*(1 - c^2)^(1/6)) + (Sqrt[3]*(b + c)^(1/6)*(c^2 + d)*Log[(b + c)^(1/
3) + Sqrt[3]*(b + c)^(1/6)*(1 - c^2)^(1/6)*((1 - b*x)/(c + x))^(1/6) + (1 - c^2)^(1/3)*((1 - b*x)/(c + x))^(1/
3)])/(2*(b - c)*c^2*(1 - c^2)^(1/6))

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || 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 211

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

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 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 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {\sqrt [6]{\frac {1-b x}{c+x}} \left (1+d x^2\right )}{(1+b x) (1+c x)} \, dx &=(6 (1+b c)) \text {Subst}\left (\int \frac {x^6 \left (b^2+2 b x^6+x^{12}+d \left (-1+c x^6\right )^2\right )}{\left (b+x^6\right )^2 \left (b+c+x^6-c^2 x^6\right ) \left (-x^6+b \left (-2+c x^6\right )\right )} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )\\ &=(6 (1+b c)) \text {Subst}\left (\int \frac {x^6 \left (b^2+2 b x^6+x^{12}+d \left (-1+c x^6\right )^2\right )}{\left (b+x^6\right )^2 \left (b+c+\left (1-c^2\right ) x^6\right ) \left (-x^6+b \left (-2+c x^6\right )\right )} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )\\ &=(6 (1+b c)) \text {Subst}\left (\int \left (\frac {d}{c \left (b+x^6\right )^2}-\frac {\left (b+2 c+b c^2\right ) d}{b c^2 (1+b c) \left (b+x^6\right )}+\frac {2 \left (-b^2-d\right )}{b (b-c) (1+b c) \left (2 b+(1-b c) x^6\right )}+\frac {(b+c) \left (c^2+d\right )}{(b-c) c^2 (1+b c) \left (b+c+\left (1-c^2\right ) x^6\right )}\right ) \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )\\ &=\frac {(6 (1+b c) d) \text {Subst}\left (\int \frac {1}{\left (b+x^6\right )^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{c}-\frac {\left (6 \left (b+2 c+b c^2\right ) d\right ) \text {Subst}\left (\int \frac {1}{b+x^6} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{b c^2}-\frac {\left (12 \left (b^2+d\right )\right ) \text {Subst}\left (\int \frac {1}{2 b+(1-b c) x^6} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{b (b-c)}+\frac {\left (6 (b+c) \left (c^2+d\right )\right ) \text {Subst}\left (\int \frac {1}{b+c+\left (1-c^2\right ) x^6} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{(b-c) c^2}\\ &=\frac {d (c+x) \sqrt [6]{\frac {1-b x}{c+x}}}{b c}+\frac {(5 (1+b c) d) \text {Subst}\left (\int \frac {1}{b+x^6} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{b c}-\frac {\left (2 \left (b+2 c+b c^2\right ) d\right ) \text {Subst}\left (\int \frac {\sqrt [6]{b}-\frac {\sqrt {3} x}{2}}{\sqrt [3]{b}-\sqrt {3} \sqrt [6]{b} x+x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{b^{11/6} c^2}-\frac {\left (2 \left (b+2 c+b c^2\right ) d\right ) \text {Subst}\left (\int \frac {\sqrt [6]{b}+\frac {\sqrt {3} x}{2}}{\sqrt [3]{b}+\sqrt {3} \sqrt [6]{b} x+x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{b^{11/6} c^2}-\frac {\left (2 \left (b+2 c+b c^2\right ) d\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{b}+x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{b^{5/3} c^2}-\frac {\left (2 \sqrt [6]{2} \left (b^2+d\right )\right ) \text {Subst}\left (\int \frac {\sqrt [6]{2} \sqrt [6]{b}-\frac {1}{2} \sqrt {3} \sqrt [6]{1-b c} x}{\sqrt [3]{2} \sqrt [3]{b}-\sqrt [6]{2} \sqrt {3} \sqrt [6]{b} \sqrt [6]{1-b c} x+\sqrt [3]{1-b c} x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{b^{11/6} (b-c)}-\frac {\left (2 \sqrt [6]{2} \left (b^2+d\right )\right ) \text {Subst}\left (\int \frac {\sqrt [6]{2} \sqrt [6]{b}+\frac {1}{2} \sqrt {3} \sqrt [6]{1-b c} x}{\sqrt [3]{2} \sqrt [3]{b}+\sqrt [6]{2} \sqrt {3} \sqrt [6]{b} \sqrt [6]{1-b c} x+\sqrt [3]{1-b c} x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{b^{11/6} (b-c)}-\frac {\left (2 \sqrt [3]{2} \left (b^2+d\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{2} \sqrt [3]{b}+\sqrt [3]{1-b c} x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{b^{5/3} (b-c)}+\frac {\left (2 \sqrt [6]{b+c} \left (c^2+d\right )\right ) \text {Subst}\left (\int \frac {\sqrt [6]{b+c}-\frac {1}{2} \sqrt {3} \sqrt [6]{1-c^2} x}{\sqrt [3]{b+c}-\sqrt {3} \sqrt [6]{b+c} \sqrt [6]{1-c^2} x+\sqrt [3]{1-c^2} x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{(b-c) c^2}+\frac {\left (2 \sqrt [6]{b+c} \left (c^2+d\right )\right ) \text {Subst}\left (\int \frac {\sqrt [6]{b+c}+\frac {1}{2} \sqrt {3} \sqrt [6]{1-c^2} x}{\sqrt [3]{b+c}+\sqrt {3} \sqrt [6]{b+c} \sqrt [6]{1-c^2} x+\sqrt [3]{1-c^2} x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{(b-c) c^2}+\frac {\left (2 \sqrt [3]{b+c} \left (c^2+d\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{b+c}+\sqrt [3]{1-c^2} x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{(b-c) c^2}\\ &=\frac {d (c+x) \sqrt [6]{\frac {1-b x}{c+x}}}{b c}-\frac {2 \left (b+2 c+b c^2\right ) d \tan ^{-1}\left (\frac {\sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}\right )}{b^{11/6} c^2}-\frac {2 \sqrt [6]{2} \left (b^2+d\right ) \tan ^{-1}\left (\frac {\sqrt [6]{1-b c} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{2} \sqrt [6]{b}}\right )}{b^{11/6} (b-c) \sqrt [6]{1-b c}}+\frac {2 \sqrt [6]{b+c} \left (c^2+d\right ) \tan ^{-1}\left (\frac {\sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b+c}}\right )}{(b-c) c^2 \sqrt [6]{1-c^2}}+\frac {(5 (1+b c) d) \text {Subst}\left (\int \frac {\sqrt [6]{b}-\frac {\sqrt {3} x}{2}}{\sqrt [3]{b}-\sqrt {3} \sqrt [6]{b} x+x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{3 b^{11/6} c}+\frac {(5 (1+b c) d) \text {Subst}\left (\int \frac {\sqrt [6]{b}+\frac {\sqrt {3} x}{2}}{\sqrt [3]{b}+\sqrt {3} \sqrt [6]{b} x+x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{3 b^{11/6} c}+\frac {(5 (1+b c) d) \text {Subst}\left (\int \frac {1}{\sqrt [3]{b}+x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{3 b^{5/3} c}+\frac {\left (\sqrt {3} \left (b+2 c+b c^2\right ) d\right ) \text {Subst}\left (\int \frac {-\sqrt {3} \sqrt [6]{b}+2 x}{\sqrt [3]{b}-\sqrt {3} \sqrt [6]{b} x+x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{2 b^{11/6} c^2}-\frac {\left (\sqrt {3} \left (b+2 c+b c^2\right ) d\right ) \text {Subst}\left (\int \frac {\sqrt {3} \sqrt [6]{b}+2 x}{\sqrt [3]{b}+\sqrt {3} \sqrt [6]{b} x+x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{2 b^{11/6} c^2}-\frac {\left (\left (b+2 c+b c^2\right ) d\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{b}-\sqrt {3} \sqrt [6]{b} x+x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{2 b^{5/3} c^2}-\frac {\left (\left (b+2 c+b c^2\right ) d\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{b}+\sqrt {3} \sqrt [6]{b} x+x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{2 b^{5/3} c^2}-\frac {\left (b^2+d\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{2} \sqrt [3]{b}-\sqrt [6]{2} \sqrt {3} \sqrt [6]{b} \sqrt [6]{1-b c} x+\sqrt [3]{1-b c} x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{2^{2/3} b^{5/3} (b-c)}-\frac {\left (b^2+d\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{2} \sqrt [3]{b}+\sqrt [6]{2} \sqrt {3} \sqrt [6]{b} \sqrt [6]{1-b c} x+\sqrt [3]{1-b c} x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{2^{2/3} b^{5/3} (b-c)}+\frac {\left (\sqrt {3} \left (b^2+d\right )\right ) \text {Subst}\left (\int \frac {-\sqrt [6]{2} \sqrt {3} \sqrt [6]{b} \sqrt [6]{1-b c}+2 \sqrt [3]{1-b c} x}{\sqrt [3]{2} \sqrt [3]{b}-\sqrt [6]{2} \sqrt {3} \sqrt [6]{b} \sqrt [6]{1-b c} x+\sqrt [3]{1-b c} x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{2^{5/6} b^{11/6} (b-c) \sqrt [6]{1-b c}}-\frac {\left (\sqrt {3} \left (b^2+d\right )\right ) \text {Subst}\left (\int \frac {\sqrt [6]{2} \sqrt {3} \sqrt [6]{b} \sqrt [6]{1-b c}+2 \sqrt [3]{1-b c} x}{\sqrt [3]{2} \sqrt [3]{b}+\sqrt [6]{2} \sqrt {3} \sqrt [6]{b} \sqrt [6]{1-b c} x+\sqrt [3]{1-b c} x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{2^{5/6} b^{11/6} (b-c) \sqrt [6]{1-b c}}+\frac {\left (\sqrt [3]{b+c} \left (c^2+d\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{b+c}-\sqrt {3} \sqrt [6]{b+c} \sqrt [6]{1-c^2} x+\sqrt [3]{1-c^2} x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{2 (b-c) c^2}+\frac {\left (\sqrt [3]{b+c} \left (c^2+d\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{b+c}+\sqrt {3} \sqrt [6]{b+c} \sqrt [6]{1-c^2} x+\sqrt [3]{1-c^2} x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{2 (b-c) c^2}-\frac {\left (\sqrt {3} \sqrt [6]{b+c} \left (c^2+d\right )\right ) \text {Subst}\left (\int \frac {-\sqrt {3} \sqrt [6]{b+c} \sqrt [6]{1-c^2}+2 \sqrt [3]{1-c^2} x}{\sqrt [3]{b+c}-\sqrt {3} \sqrt [6]{b+c} \sqrt [6]{1-c^2} x+\sqrt [3]{1-c^2} x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{2 (b-c) c^2 \sqrt [6]{1-c^2}}+\frac {\left (\sqrt {3} \sqrt [6]{b+c} \left (c^2+d\right )\right ) \text {Subst}\left (\int \frac {\sqrt {3} \sqrt [6]{b+c} \sqrt [6]{1-c^2}+2 \sqrt [3]{1-c^2} x}{\sqrt [3]{b+c}+\sqrt {3} \sqrt [6]{b+c} \sqrt [6]{1-c^2} x+\sqrt [3]{1-c^2} x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{2 (b-c) c^2 \sqrt [6]{1-c^2}}\\ &=\frac {d (c+x) \sqrt [6]{\frac {1-b x}{c+x}}}{b c}+\frac {5 (1+b c) d \tan ^{-1}\left (\frac {\sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}\right )}{3 b^{11/6} c}-\frac {2 \left (b+2 c+b c^2\right ) d \tan ^{-1}\left (\frac {\sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}\right )}{b^{11/6} c^2}-\frac {2 \sqrt [6]{2} \left (b^2+d\right ) \tan ^{-1}\left (\frac {\sqrt [6]{1-b c} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{2} \sqrt [6]{b}}\right )}{b^{11/6} (b-c) \sqrt [6]{1-b c}}+\frac {2 \sqrt [6]{b+c} \left (c^2+d\right ) \tan ^{-1}\left (\frac {\sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b+c}}\right )}{(b-c) c^2 \sqrt [6]{1-c^2}}+\frac {\sqrt {3} \left (b+2 c+b c^2\right ) d \log \left (\sqrt [3]{b}-\sqrt {3} \sqrt [6]{b} \sqrt [6]{\frac {1-b x}{c+x}}+\sqrt [3]{\frac {1-b x}{c+x}}\right )}{2 b^{11/6} c^2}-\frac {\sqrt {3} \left (b+2 c+b c^2\right ) d \log \left (\sqrt [3]{b}+\sqrt {3} \sqrt [6]{b} \sqrt [6]{\frac {1-b x}{c+x}}+\sqrt [3]{\frac {1-b x}{c+x}}\right )}{2 b^{11/6} c^2}+\frac {\sqrt {3} \left (b^2+d\right ) \log \left (\sqrt [3]{2} \sqrt [3]{b}-\sqrt [6]{2} \sqrt {3} \sqrt [6]{b} \sqrt [6]{1-b c} \sqrt [6]{\frac {1-b x}{c+x}}+\sqrt [3]{1-b c} \sqrt [3]{\frac {1-b x}{c+x}}\right )}{2^{5/6} b^{11/6} (b-c) \sqrt [6]{1-b c}}-\frac {\sqrt {3} \left (b^2+d\right ) \log \left (\sqrt [3]{2} \sqrt [3]{b}+\sqrt [6]{2} \sqrt {3} \sqrt [6]{b} \sqrt [6]{1-b c} \sqrt [6]{\frac {1-b x}{c+x}}+\sqrt [3]{1-b c} \sqrt [3]{\frac {1-b x}{c+x}}\right )}{2^{5/6} b^{11/6} (b-c) \sqrt [6]{1-b c}}-\frac {\sqrt {3} \sqrt [6]{b+c} \left (c^2+d\right ) \log \left (\sqrt [3]{b+c}-\sqrt {3} \sqrt [6]{b+c} \sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}+\sqrt [3]{1-c^2} \sqrt [3]{\frac {1-b x}{c+x}}\right )}{2 (b-c) c^2 \sqrt [6]{1-c^2}}+\frac {\sqrt {3} \sqrt [6]{b+c} \left (c^2+d\right ) \log \left (\sqrt [3]{b+c}+\sqrt {3} \sqrt [6]{b+c} \sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}+\sqrt [3]{1-c^2} \sqrt [3]{\frac {1-b x}{c+x}}\right )}{2 (b-c) c^2 \sqrt [6]{1-c^2}}-\frac {(5 (1+b c) d) \text {Subst}\left (\int \frac {-\sqrt {3} \sqrt [6]{b}+2 x}{\sqrt [3]{b}-\sqrt {3} \sqrt [6]{b} x+x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{4 \sqrt {3} b^{11/6} c}+\frac {(5 (1+b c) d) \text {Subst}\left (\int \frac {\sqrt {3} \sqrt [6]{b}+2 x}{\sqrt [3]{b}+\sqrt {3} \sqrt [6]{b} x+x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{4 \sqrt {3} b^{11/6} c}+\frac {(5 (1+b c) d) \text {Subst}\left (\int \frac {1}{\sqrt [3]{b}-\sqrt {3} \sqrt [6]{b} x+x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{12 b^{5/3} c}+\frac {(5 (1+b c) d) \text {Subst}\left (\int \frac {1}{\sqrt [3]{b}+\sqrt {3} \sqrt [6]{b} x+x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{12 b^{5/3} c}-\frac {\left (\left (b+2 c+b c^2\right ) d\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1-\frac {2 \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt {3} \sqrt [6]{b}}\right )}{\sqrt {3} b^{11/6} c^2}+\frac {\left (\left (b+2 c+b c^2\right ) d\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1+\frac {2 \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt {3} \sqrt [6]{b}}\right )}{\sqrt {3} b^{11/6} c^2}-\frac {\left (\sqrt [6]{2} \left (b^2+d\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1-\frac {2^{5/6} \sqrt [6]{1-b c} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt {3} \sqrt [6]{b}}\right )}{\sqrt {3} b^{11/6} (b-c) \sqrt [6]{1-b c}}+\frac {\left (\sqrt [6]{2} \left (b^2+d\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1+\frac {2^{5/6} \sqrt [6]{1-b c} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt {3} \sqrt [6]{b}}\right )}{\sqrt {3} b^{11/6} (b-c) \sqrt [6]{1-b c}}+\frac {\left (\sqrt [6]{b+c} \left (c^2+d\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1-\frac {2 \sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt {3} \sqrt [6]{b+c}}\right )}{\sqrt {3} (b-c) c^2 \sqrt [6]{1-c^2}}-\frac {\left (\sqrt [6]{b+c} \left (c^2+d\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1+\frac {2 \sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt {3} \sqrt [6]{b+c}}\right )}{\sqrt {3} (b-c) c^2 \sqrt [6]{1-c^2}}\\ &=\frac {d (c+x) \sqrt [6]{\frac {1-b x}{c+x}}}{b c}+\frac {5 (1+b c) d \tan ^{-1}\left (\frac {\sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}\right )}{3 b^{11/6} c}-\frac {2 \left (b+2 c+b c^2\right ) d \tan ^{-1}\left (\frac {\sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}\right )}{b^{11/6} c^2}-\frac {2 \sqrt [6]{2} \left (b^2+d\right ) \tan ^{-1}\left (\frac {\sqrt [6]{1-b c} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{2} \sqrt [6]{b}}\right )}{b^{11/6} (b-c) \sqrt [6]{1-b c}}+\frac {2 \sqrt [6]{b+c} \left (c^2+d\right ) \tan ^{-1}\left (\frac {\sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b+c}}\right )}{(b-c) c^2 \sqrt [6]{1-c^2}}+\frac {\left (b+2 c+b c^2\right ) d \tan ^{-1}\left (\frac {1}{3} \left (3 \sqrt {3}-\frac {6 \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}\right )\right )}{b^{11/6} c^2}-\frac {\left (b+2 c+b c^2\right ) d \tan ^{-1}\left (\frac {1}{3} \left (3 \sqrt {3}+\frac {6 \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}\right )\right )}{b^{11/6} c^2}+\frac {\sqrt [6]{2} \left (b^2+d\right ) \tan ^{-1}\left (\frac {1}{3} \left (3 \sqrt {3}-\frac {3\ 2^{5/6} \sqrt [6]{1-b c} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}\right )\right )}{b^{11/6} (b-c) \sqrt [6]{1-b c}}-\frac {\sqrt [6]{2} \left (b^2+d\right ) \tan ^{-1}\left (\frac {1}{3} \left (3 \sqrt {3}+\frac {3\ 2^{5/6} \sqrt [6]{1-b c} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}\right )\right )}{b^{11/6} (b-c) \sqrt [6]{1-b c}}-\frac {\sqrt [6]{b+c} \left (c^2+d\right ) \tan ^{-1}\left (\frac {1}{3} \left (3 \sqrt {3}-\frac {6 \sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b+c}}\right )\right )}{(b-c) c^2 \sqrt [6]{1-c^2}}+\frac {\sqrt [6]{b+c} \left (c^2+d\right ) \tan ^{-1}\left (\frac {1}{3} \left (3 \sqrt {3}+\frac {6 \sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b+c}}\right )\right )}{(b-c) c^2 \sqrt [6]{1-c^2}}-\frac {5 (1+b c) d \log \left (\sqrt [3]{b}-\sqrt {3} \sqrt [6]{b} \sqrt [6]{\frac {1-b x}{c+x}}+\sqrt [3]{\frac {1-b x}{c+x}}\right )}{4 \sqrt {3} b^{11/6} c}+\frac {\sqrt {3} \left (b+2 c+b c^2\right ) d \log \left (\sqrt [3]{b}-\sqrt {3} \sqrt [6]{b} \sqrt [6]{\frac {1-b x}{c+x}}+\sqrt [3]{\frac {1-b x}{c+x}}\right )}{2 b^{11/6} c^2}+\frac {5 (1+b c) d \log \left (\sqrt [3]{b}+\sqrt {3} \sqrt [6]{b} \sqrt [6]{\frac {1-b x}{c+x}}+\sqrt [3]{\frac {1-b x}{c+x}}\right )}{4 \sqrt {3} b^{11/6} c}-\frac {\sqrt {3} \left (b+2 c+b c^2\right ) d \log \left (\sqrt [3]{b}+\sqrt {3} \sqrt [6]{b} \sqrt [6]{\frac {1-b x}{c+x}}+\sqrt [3]{\frac {1-b x}{c+x}}\right )}{2 b^{11/6} c^2}+\frac {\sqrt {3} \left (b^2+d\right ) \log \left (\sqrt [3]{2} \sqrt [3]{b}-\sqrt [6]{2} \sqrt {3} \sqrt [6]{b} \sqrt [6]{1-b c} \sqrt [6]{\frac {1-b x}{c+x}}+\sqrt [3]{1-b c} \sqrt [3]{\frac {1-b x}{c+x}}\right )}{2^{5/6} b^{11/6} (b-c) \sqrt [6]{1-b c}}-\frac {\sqrt {3} \left (b^2+d\right ) \log \left (\sqrt [3]{2} \sqrt [3]{b}+\sqrt [6]{2} \sqrt {3} \sqrt [6]{b} \sqrt [6]{1-b c} \sqrt [6]{\frac {1-b x}{c+x}}+\sqrt [3]{1-b c} \sqrt [3]{\frac {1-b x}{c+x}}\right )}{2^{5/6} b^{11/6} (b-c) \sqrt [6]{1-b c}}-\frac {\sqrt {3} \sqrt [6]{b+c} \left (c^2+d\right ) \log \left (\sqrt [3]{b+c}-\sqrt {3} \sqrt [6]{b+c} \sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}+\sqrt [3]{1-c^2} \sqrt [3]{\frac {1-b x}{c+x}}\right )}{2 (b-c) c^2 \sqrt [6]{1-c^2}}+\frac {\sqrt {3} \sqrt [6]{b+c} \left (c^2+d\right ) \log \left (\sqrt [3]{b+c}+\sqrt {3} \sqrt [6]{b+c} \sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}+\sqrt [3]{1-c^2} \sqrt [3]{\frac {1-b x}{c+x}}\right )}{2 (b-c) c^2 \sqrt [6]{1-c^2}}+\frac {(5 (1+b c) d) \text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1-\frac {2 \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt {3} \sqrt [6]{b}}\right )}{6 \sqrt {3} b^{11/6} c}-\frac {(5 (1+b c) d) \text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1+\frac {2 \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt {3} \sqrt [6]{b}}\right )}{6 \sqrt {3} b^{11/6} c}\\ &=\frac {d (c+x) \sqrt [6]{\frac {1-b x}{c+x}}}{b c}+\frac {5 (1+b c) d \tan ^{-1}\left (\frac {\sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}\right )}{3 b^{11/6} c}-\frac {2 \left (b+2 c+b c^2\right ) d \tan ^{-1}\left (\frac {\sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}\right )}{b^{11/6} c^2}-\frac {2 \sqrt [6]{2} \left (b^2+d\right ) \tan ^{-1}\left (\frac {\sqrt [6]{1-b c} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{2} \sqrt [6]{b}}\right )}{b^{11/6} (b-c) \sqrt [6]{1-b c}}+\frac {2 \sqrt [6]{b+c} \left (c^2+d\right ) \tan ^{-1}\left (\frac {\sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b+c}}\right )}{(b-c) c^2 \sqrt [6]{1-c^2}}-\frac {5 (1+b c) d \tan ^{-1}\left (\frac {1}{3} \left (3 \sqrt {3}-\frac {6 \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}\right )\right )}{6 b^{11/6} c}+\frac {\left (b+2 c+b c^2\right ) d \tan ^{-1}\left (\frac {1}{3} \left (3 \sqrt {3}-\frac {6 \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}\right )\right )}{b^{11/6} c^2}+\frac {5 (1+b c) d \tan ^{-1}\left (\frac {1}{3} \left (3 \sqrt {3}+\frac {6 \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}\right )\right )}{6 b^{11/6} c}-\frac {\left (b+2 c+b c^2\right ) d \tan ^{-1}\left (\frac {1}{3} \left (3 \sqrt {3}+\frac {6 \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}\right )\right )}{b^{11/6} c^2}+\frac {\sqrt [6]{2} \left (b^2+d\right ) \tan ^{-1}\left (\frac {1}{3} \left (3 \sqrt {3}-\frac {3\ 2^{5/6} \sqrt [6]{1-b c} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}\right )\right )}{b^{11/6} (b-c) \sqrt [6]{1-b c}}-\frac {\sqrt [6]{2} \left (b^2+d\right ) \tan ^{-1}\left (\frac {1}{3} \left (3 \sqrt {3}+\frac {3\ 2^{5/6} \sqrt [6]{1-b c} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}\right )\right )}{b^{11/6} (b-c) \sqrt [6]{1-b c}}-\frac {\sqrt [6]{b+c} \left (c^2+d\right ) \tan ^{-1}\left (\frac {1}{3} \left (3 \sqrt {3}-\frac {6 \sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b+c}}\right )\right )}{(b-c) c^2 \sqrt [6]{1-c^2}}+\frac {\sqrt [6]{b+c} \left (c^2+d\right ) \tan ^{-1}\left (\frac {1}{3} \left (3 \sqrt {3}+\frac {6 \sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b+c}}\right )\right )}{(b-c) c^2 \sqrt [6]{1-c^2}}-\frac {5 (1+b c) d \log \left (\sqrt [3]{b}-\sqrt {3} \sqrt [6]{b} \sqrt [6]{\frac {1-b x}{c+x}}+\sqrt [3]{\frac {1-b x}{c+x}}\right )}{4 \sqrt {3} b^{11/6} c}+\frac {\sqrt {3} \left (b+2 c+b c^2\right ) d \log \left (\sqrt [3]{b}-\sqrt {3} \sqrt [6]{b} \sqrt [6]{\frac {1-b x}{c+x}}+\sqrt [3]{\frac {1-b x}{c+x}}\right )}{2 b^{11/6} c^2}+\frac {5 (1+b c) d \log \left (\sqrt [3]{b}+\sqrt {3} \sqrt [6]{b} \sqrt [6]{\frac {1-b x}{c+x}}+\sqrt [3]{\frac {1-b x}{c+x}}\right )}{4 \sqrt {3} b^{11/6} c}-\frac {\sqrt {3} \left (b+2 c+b c^2\right ) d \log \left (\sqrt [3]{b}+\sqrt {3} \sqrt [6]{b} \sqrt [6]{\frac {1-b x}{c+x}}+\sqrt [3]{\frac {1-b x}{c+x}}\right )}{2 b^{11/6} c^2}+\frac {\sqrt {3} \left (b^2+d\right ) \log \left (\sqrt [3]{2} \sqrt [3]{b}-\sqrt [6]{2} \sqrt {3} \sqrt [6]{b} \sqrt [6]{1-b c} \sqrt [6]{\frac {1-b x}{c+x}}+\sqrt [3]{1-b c} \sqrt [3]{\frac {1-b x}{c+x}}\right )}{2^{5/6} b^{11/6} (b-c) \sqrt [6]{1-b c}}-\frac {\sqrt {3} \left (b^2+d\right ) \log \left (\sqrt [3]{2} \sqrt [3]{b}+\sqrt [6]{2} \sqrt {3} \sqrt [6]{b} \sqrt [6]{1-b c} \sqrt [6]{\frac {1-b x}{c+x}}+\sqrt [3]{1-b c} \sqrt [3]{\frac {1-b x}{c+x}}\right )}{2^{5/6} b^{11/6} (b-c) \sqrt [6]{1-b c}}-\frac {\sqrt {3} \sqrt [6]{b+c} \left (c^2+d\right ) \log \left (\sqrt [3]{b+c}-\sqrt {3} \sqrt [6]{b+c} \sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}+\sqrt [3]{1-c^2} \sqrt [3]{\frac {1-b x}{c+x}}\right )}{2 (b-c) c^2 \sqrt [6]{1-c^2}}+\frac {\sqrt {3} \sqrt [6]{b+c} \left (c^2+d\right ) \log \left (\sqrt [3]{b+c}+\sqrt {3} \sqrt [6]{b+c} \sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}+\sqrt [3]{1-c^2} \sqrt [3]{\frac {1-b x}{c+x}}\right )}{2 (b-c) c^2 \sqrt [6]{1-c^2}}\\ \end {align*}

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Mathematica [A]
time = 16.32, size = 853, normalized size = 0.96 \begin {gather*} \frac {6 b^{5/6} c d (c+x) \sqrt [6]{\frac {1-b x}{c+x}}-2 \left (7 c+b \left (6+c^2\right )\right ) d \text {ArcTan}\left (\frac {\sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}\right )+\frac {12 b^{11/6} \sqrt [6]{b+c} \left (c^2+d\right ) \text {ArcTan}\left (\frac {\sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b+c}}\right )}{(b-c) \sqrt [6]{1-c^2}}+\frac {6 \sqrt [6]{2} \sqrt {3} c^2 \left (b^2+d\right ) \text {ArcTan}\left (\frac {1-\frac {2^{5/6} \sqrt [6]{-1+b c} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}}{\sqrt {3}}\right )}{(b-c) \sqrt [6]{-1+b c}}-\frac {6 \sqrt [6]{2} \sqrt {3} c^2 \left (b^2+d\right ) \text {ArcTan}\left (\frac {1+\frac {2^{5/6} \sqrt [6]{-1+b c} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}}{\sqrt {3}}\right )}{(b-c) \sqrt [6]{-1+b c}}+\left (7 c+b \left (6+c^2\right )\right ) d \text {ArcTan}\left (\frac {\sqrt [3]{b}-\sqrt [3]{\frac {1-b x}{c+x}}}{\sqrt [6]{b} \sqrt [6]{\frac {1-b x}{c+x}}}\right )+\frac {6 b^{11/6} \sqrt [6]{b+c} \left (c^2+d\right ) \text {ArcTan}\left (\frac {\sqrt [3]{b+c}-\sqrt [3]{1-c^2} \sqrt [3]{\frac {1-b x}{c+x}}}{\sqrt [6]{b+c} \sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}}\right )}{(-b+c) \sqrt [6]{1-c^2}}-\frac {12 \sqrt [6]{2} c^2 \left (b^2+d\right ) \tanh ^{-1}\left (\frac {\sqrt [6]{-1+b c} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{2} \sqrt [6]{b}}\right )}{(b-c) \sqrt [6]{-1+b c}}-\sqrt {3} \left (7 c+b \left (6+c^2\right )\right ) d \tanh ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [3]{b}+\sqrt [3]{\frac {1-b x}{c+x}}}\right )-\frac {6 \sqrt [6]{2} c^2 \left (b^2+d\right ) \tanh ^{-1}\left (\frac {2^{5/6} \sqrt [6]{b} \sqrt [6]{-1+b c} \sqrt [6]{\frac {1-b x}{c+x}}}{2 \sqrt [3]{b}+2^{2/3} \sqrt [3]{-1+b c} \sqrt [3]{\frac {1-b x}{c+x}}}\right )}{(b-c) \sqrt [6]{-1+b c}}+\frac {6 \sqrt {3} b^{11/6} \sqrt [6]{b+c} \left (c^2+d\right ) \tanh ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{b+c} \sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [3]{b+c}+\sqrt [3]{1-c^2} \sqrt [3]{\frac {1-b x}{c+x}}}\right )}{(b-c) \sqrt [6]{1-c^2}}}{6 b^{11/6} c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(((1 - b*x)/(c + x))^(1/6)*(1 + d*x^2))/((1 + b*x)*(1 + c*x)),x]

[Out]

(6*b^(5/6)*c*d*(c + x)*((1 - b*x)/(c + x))^(1/6) - 2*(7*c + b*(6 + c^2))*d*ArcTan[((1 - b*x)/(c + x))^(1/6)/b^
(1/6)] + (12*b^(11/6)*(b + c)^(1/6)*(c^2 + d)*ArcTan[((1 - c^2)^(1/6)*((1 - b*x)/(c + x))^(1/6))/(b + c)^(1/6)
])/((b - c)*(1 - c^2)^(1/6)) + (6*2^(1/6)*Sqrt[3]*c^2*(b^2 + d)*ArcTan[(1 - (2^(5/6)*(-1 + b*c)^(1/6)*((1 - b*
x)/(c + x))^(1/6))/b^(1/6))/Sqrt[3]])/((b - c)*(-1 + b*c)^(1/6)) - (6*2^(1/6)*Sqrt[3]*c^2*(b^2 + d)*ArcTan[(1
+ (2^(5/6)*(-1 + b*c)^(1/6)*((1 - b*x)/(c + x))^(1/6))/b^(1/6))/Sqrt[3]])/((b - c)*(-1 + b*c)^(1/6)) + (7*c +
b*(6 + c^2))*d*ArcTan[(b^(1/3) - ((1 - b*x)/(c + x))^(1/3))/(b^(1/6)*((1 - b*x)/(c + x))^(1/6))] + (6*b^(11/6)
*(b + c)^(1/6)*(c^2 + d)*ArcTan[((b + c)^(1/3) - (1 - c^2)^(1/3)*((1 - b*x)/(c + x))^(1/3))/((b + c)^(1/6)*(1
- c^2)^(1/6)*((1 - b*x)/(c + x))^(1/6))])/((-b + c)*(1 - c^2)^(1/6)) - (12*2^(1/6)*c^2*(b^2 + d)*ArcTanh[((-1
+ b*c)^(1/6)*((1 - b*x)/(c + x))^(1/6))/(2^(1/6)*b^(1/6))])/((b - c)*(-1 + b*c)^(1/6)) - Sqrt[3]*(7*c + b*(6 +
c^2))*d*ArcTanh[(Sqrt[3]*b^(1/6)*((1 - b*x)/(c + x))^(1/6))/(b^(1/3) + ((1 - b*x)/(c + x))^(1/3))] - (6*2^(1/
6)*c^2*(b^2 + d)*ArcTanh[(2^(5/6)*b^(1/6)*(-1 + b*c)^(1/6)*((1 - b*x)/(c + x))^(1/6))/(2*b^(1/3) + 2^(2/3)*(-1
+ b*c)^(1/3)*((1 - b*x)/(c + x))^(1/3))])/((b - c)*(-1 + b*c)^(1/6)) + (6*Sqrt[3]*b^(11/6)*(b + c)^(1/6)*(c^2
+ d)*ArcTanh[(Sqrt[3]*(b + c)^(1/6)*(1 - c^2)^(1/6)*((1 - b*x)/(c + x))^(1/6))/((b + c)^(1/3) + (1 - c^2)^(1/
3)*((1 - b*x)/(c + x))^(1/3))])/((b - c)*(1 - c^2)^(1/6)))/(6*b^(11/6)*c^2)

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (\frac {-b x +1}{c +x}\right )^{\frac {1}{6}} \left (d \,x^{2}+1\right )}{\left (b x +1\right ) \left (c x +1\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-b*x+1)/(c+x))^(1/6)*(d*x^2+1)/(b*x+1)/(c*x+1),x)

[Out]

int(((-b*x+1)/(c+x))^(1/6)*(d*x^2+1)/(b*x+1)/(c*x+1),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-b*x+1)/(c+x))^(1/6)*(d*x^2+1)/(b*x+1)/(c*x+1),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b*c-1>0)', see `assume?` for m
ore details)

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Fricas [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(((-b*x+1)/(c+x))^(1/6)*(d*x^2+1)/(b*x+1)/(c*x+1),x, algorithm="fricas")

[Out]

Timed out

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Sympy [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(((-b*x+1)/(c+x))**(1/6)*(d*x**2+1)/(b*x+1)/(c*x+1),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2549 vs. \(2 (732) = 1464\).
time = 85.70, size = 2549, normalized size = 2.87 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-b*x+1)/(c+x))^(1/6)*(d*x^2+1)/(b*x+1)/(c*x+1),x, algorithm="giac")

[Out]

-1/12*(12*((-2*b^6*c^5 + 10*b^5*c^4 - 20*b^4*c^3 + 20*b^3*c^2 - 10*b^2*c + 2*b)^(1/6)*(b*c + 1)*d + (-2*b^6*c^
5 + 10*b^5*c^4 - 20*b^4*c^3 + 20*b^3*c^2 - 10*b^2*c + 2*b)^(1/6)*(b^3*c + b^2))*arctan(1/2*2^(5/6)*(sqrt(3)*2^
(1/6)*(-b/(b*c - 1))^(1/6) + 2*(-(b*x - 1)/(c + x))^(1/6))/(-b/(b*c - 1))^(1/6))/(b^4*c - b^3*c^2 - b^3 + b^2*
c) + 12*((-2*b^6*c^5 + 10*b^5*c^4 - 20*b^4*c^3 + 20*b^3*c^2 - 10*b^2*c + 2*b)^(1/6)*(b*c + 1)*d + (-2*b^6*c^5
+ 10*b^5*c^4 - 20*b^4*c^3 + 20*b^3*c^2 - 10*b^2*c + 2*b)^(1/6)*(b^3*c + b^2))*arctan(-1/2*2^(5/6)*(sqrt(3)*2^(
1/6)*(-b/(b*c - 1))^(1/6) - 2*(-(b*x - 1)/(c + x))^(1/6))/(-b/(b*c - 1))^(1/6))/(b^4*c - b^3*c^2 - b^3 + b^2*c
) - 12*((-b*c^10 - c^11 + 5*b*c^8 + 5*c^9 - 10*b*c^6 - 10*c^7 + 10*b*c^4 + 10*c^5 - 5*b*c^2 - 5*c^3 + b + c)^(
1/6)*(b*c + 1)*d + (-b*c^10 - c^11 + 5*b*c^8 + 5*c^9 - 10*b*c^6 - 10*c^7 + 10*b*c^4 + 10*c^5 - 5*b*c^2 - 5*c^3
+ b + c)^(1/6)*(b*c^3 + c^2))*arctan((sqrt(3)*(-(b + c)/(c^2 - 1))^(1/6) + 2*(-(b*x - 1)/(c + x))^(1/6))/(-(b
+ c)/(c^2 - 1))^(1/6))/(b*c^4 - c^5 - b*c^2 + c^3) - 12*((-b*c^10 - c^11 + 5*b*c^8 + 5*c^9 - 10*b*c^6 - 10*c^
7 + 10*b*c^4 + 10*c^5 - 5*b*c^2 - 5*c^3 + b + c)^(1/6)*(b*c + 1)*d + (-b*c^10 - c^11 + 5*b*c^8 + 5*c^9 - 10*b*
c^6 - 10*c^7 + 10*b*c^4 + 10*c^5 - 5*b*c^2 - 5*c^3 + b + c)^(1/6)*(b*c^3 + c^2))*arctan(-(sqrt(3)*(-(b + c)/(c
^2 - 1))^(1/6) - 2*(-(b*x - 1)/(c + x))^(1/6))/(-(b + c)/(c^2 - 1))^(1/6))/(b*c^4 - c^5 - b*c^2 + c^3) + 24*((
-2*b^6*c^5 + 10*b^5*c^4 - 20*b^4*c^3 + 20*b^3*c^2 - 10*b^2*c + 2*b)^(1/6)*(b*c + 1)*d + (-2*b^6*c^5 + 10*b^5*c
^4 - 20*b^4*c^3 + 20*b^3*c^2 - 10*b^2*c + 2*b)^(1/6)*(b^3*c + b^2))*arctan(1/2*2^(5/6)*(-(b*x - 1)/(c + x))^(1
/6)/(-b/(b*c - 1))^(1/6))/(b^4*c - b^3*c^2 - b^3 + b^2*c) - 24*((-b*c^10 - c^11 + 5*b*c^8 + 5*c^9 - 10*b*c^6 -
10*c^7 + 10*b*c^4 + 10*c^5 - 5*b*c^2 - 5*c^3 + b + c)^(1/6)*(b*c + 1)*d + (-b*c^10 - c^11 + 5*b*c^8 + 5*c^9 -
10*b*c^6 - 10*c^7 + 10*b*c^4 + 10*c^5 - 5*b*c^2 - 5*c^3 + b + c)^(1/6)*(b*c^3 + c^2))*arctan((-(b*x - 1)/(c +
x))^(1/6)/(-(b + c)/(c^2 - 1))^(1/6))/(b*c^4 - c^5 - b*c^2 + c^3) + 6*((-2*b^6*c^5 + 10*b^5*c^4 - 20*b^4*c^3
+ 20*b^3*c^2 - 10*b^2*c + 2*b)^(1/6)*(sqrt(3)*b*c + sqrt(3))*d + (-2*b^6*c^5 + 10*b^5*c^4 - 20*b^4*c^3 + 20*b^
3*c^2 - 10*b^2*c + 2*b)^(1/6)*(sqrt(3)*b^3*c + sqrt(3)*b^2))*log(sqrt(3)*2^(1/6)*(-b/(b*c - 1))^(1/6)*(-(b*x -
1)/(c + x))^(1/6) + 2^(1/3)*(-b/(b*c - 1))^(1/3) + (-(b*x - 1)/(c + x))^(1/3))/(b^4*c - b^3*c^2 - b^3 + b^2*c
) - 6*((-2*b^6*c^5 + 10*b^5*c^4 - 20*b^4*c^3 + 20*b^3*c^2 - 10*b^2*c + 2*b)^(1/6)*(sqrt(3)*b*c + sqrt(3))*d +
(-2*b^6*c^5 + 10*b^5*c^4 - 20*b^4*c^3 + 20*b^3*c^2 - 10*b^2*c + 2*b)^(1/6)*(sqrt(3)*b^3*c + sqrt(3)*b^2))*log(
-sqrt(3)*2^(1/6)*(-b/(b*c - 1))^(1/6)*(-(b*x - 1)/(c + x))^(1/6) + 2^(1/3)*(-b/(b*c - 1))^(1/3) + (-(b*x - 1)/
(c + x))^(1/3))/(b^4*c - b^3*c^2 - b^3 + b^2*c) - 6*((-b*c^10 - c^11 + 5*b*c^8 + 5*c^9 - 10*b*c^6 - 10*c^7 + 1
0*b*c^4 + 10*c^5 - 5*b*c^2 - 5*c^3 + b + c)^(1/6)*(sqrt(3)*b*c + sqrt(3))*d + (-b*c^10 - c^11 + 5*b*c^8 + 5*c^
9 - 10*b*c^6 - 10*c^7 + 10*b*c^4 + 10*c^5 - 5*b*c^2 - 5*c^3 + b + c)^(1/6)*(sqrt(3)*b*c^3 + sqrt(3)*c^2))*log(
sqrt(3)*(-(b + c)/(c^2 - 1))^(1/6)*(-(b*x - 1)/(c + x))^(1/6) + (-(b + c)/(c^2 - 1))^(1/3) + (-(b*x - 1)/(c +
x))^(1/3))/(b*c^4 - c^5 - b*c^2 + c^3) + 6*((-b*c^10 - c^11 + 5*b*c^8 + 5*c^9 - 10*b*c^6 - 10*c^7 + 10*b*c^4 +
10*c^5 - 5*b*c^2 - 5*c^3 + b + c)^(1/6)*(sqrt(3)*b*c + sqrt(3))*d + (-b*c^10 - c^11 + 5*b*c^8 + 5*c^9 - 10*b*
c^6 - 10*c^7 + 10*b*c^4 + 10*c^5 - 5*b*c^2 - 5*c^3 + b + c)^(1/6)*(sqrt(3)*b*c^3 + sqrt(3)*c^2))*log(-sqrt(3)*
(-(b + c)/(c^2 - 1))^(1/6)*(-(b*x - 1)/(c + x))^(1/6) + (-(b + c)/(c^2 - 1))^(1/3) + (-(b*x - 1)/(c + x))^(1/3
))/(b*c^4 - c^5 - b*c^2 + c^3) + sqrt(3)*(b^(13/6)*c^3*d + 8*b^(7/6)*c^2*d + (6*b^2 + 7)*b^(1/6)*c*d + 6*b^(7/
6)*d)*log(sqrt(3)*b^(1/6)*(-(b*x - 1)/(c + x))^(1/6) + b^(1/3) + (-(b*x - 1)/(c + x))^(1/3))/(b^2*c^2) - sqrt(
3)*(b^(13/6)*c^3*d + 8*b^(7/6)*c^2*d + (6*b^2 + 7)*b^(1/6)*c*d + 6*b^(7/6)*d)*log(-sqrt(3)*b^(1/6)*(-(b*x - 1)
/(c + x))^(1/6) + b^(1/3) + (-(b*x - 1)/(c + x))^(1/3))/(b^2*c^2) - 12*(b^2*c^2*d*(-(b*x - 1)/(c + x))^(1/6) +
2*b*c*d*(-(b*x - 1)/(c + x))^(1/6) + d*(-(b*x - 1)/(c + x))^(1/6))/((b - (b*x - 1)/(c + x))*b*c) + 2*(b^(13/6
)*c^3*d + 8*b^(7/6)*c^2*d + (6*b^2 + 7)*b^(1/6)*c*d + 6*b^(7/6)*d)*arctan((sqrt(3)*b^(1/6) + 2*(-(b*x - 1)/(c
+ x))^(1/6))/b^(1/6))/(b^2*c^2) + 2*(b^(13/6)*c^3*d + 8*b^(7/6)*c^2*d + (6*b^2 + 7)*b^(1/6)*c*d + 6*b^(7/6)*d)
*arctan(-(sqrt(3)*b^(1/6) - 2*(-(b*x - 1)/(c + x))^(1/6))/b^(1/6))/(b^2*c^2) + 4*(b^(13/6)*c^3*d + 8*b^(7/6)*c
^2*d + (6*b^2 + 7)*b^(1/6)*c*d + 6*b^(7/6)*d)*arctan((-(b*x - 1)/(c + x))^(1/6)/b^(1/6))/(b^2*c^2))*(b*c/(b*c
+ 1)^2 + 1/(b*c + 1)^2)

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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-(b*x - 1)/(c + x))^(1/6)*(d*x^2 + 1))/((b*x + 1)*(c*x + 1)),x)

[Out]

\text{Hanged}

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