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

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

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Rubi [A]  time = 5.28, 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, 6725, 199, 209, 634, 618, 204, 628, 203, 205}

result too large to display

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 199

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] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

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 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 205

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

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 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 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 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 6725

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)) \operatorname {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)) \operatorname {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)) \operatorname {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) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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) \operatorname {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) \operatorname {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) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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) \operatorname {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) \operatorname {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) \operatorname {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) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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) \operatorname {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) \operatorname {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 [C]  time = 0.48, size = 322, normalized size = 0.36 \begin {gather*} \frac {6}{5} (c+x) \sqrt [6]{\frac {1-b x}{c+x}} \left (\frac {\left (b^2+d\right ) \left (\frac {1-b x}{b c+1}\right )^{5/6} \, _2F_1\left (\frac {5}{6},\frac {5}{6};\frac {11}{6};\frac {b (c+x)}{b c+1}\right )}{b (b-c) (b x-1)}+\frac {2 \left (b^2+d\right ) \, _2F_1\left (\frac {5}{6},1;\frac {11}{6};-\frac {2 b (c+x)}{(b c-1) (b x-1)}\right )}{b (b-c) (b c-1) (b x-1)}+\frac {b \left (c^2+d\right ) \, _2F_1\left (\frac {5}{6},\frac {5}{6};\frac {11}{6};\frac {b (c+x)}{b c+1}\right )}{c^2 (b-c) (b c+1) \sqrt [6]{\frac {1-b x}{b c+1}}}-\frac {(b+c) \left (c^2+d\right ) \, _2F_1\left (\frac {5}{6},1;\frac {11}{6};-\frac {(b+c) (c+x)}{\left (c^2-1\right ) (b x-1)}\right )}{c^2 \left (c^2-1\right ) (b-c) (b x-1)}+\frac {d \, _2F_1\left (-\frac {1}{6},\frac {5}{6};\frac {11}{6};\frac {b (c+x)}{b c+1}\right )}{b c \sqrt [6]{\frac {1-b x}{b c+1}}}\right ) \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*(c + x)*((1 - b*x)/(c + x))^(1/6)*((d*Hypergeometric2F1[-1/6, 5/6, 11/6, (b*(c + x))/(1 + b*c)])/(b*c*((1 -
b*x)/(1 + b*c))^(1/6)) + (b*(c^2 + d)*Hypergeometric2F1[5/6, 5/6, 11/6, (b*(c + x))/(1 + b*c)])/((b - c)*c^2*
(1 + b*c)*((1 - b*x)/(1 + b*c))^(1/6)) + ((b^2 + d)*((1 - b*x)/(1 + b*c))^(5/6)*Hypergeometric2F1[5/6, 5/6, 11
/6, (b*(c + x))/(1 + b*c)])/(b*(b - c)*(-1 + b*x)) + (2*(b^2 + d)*Hypergeometric2F1[5/6, 1, 11/6, (-2*b*(c + x
))/((-1 + b*c)*(-1 + b*x))])/(b*(b - c)*(-1 + b*c)*(-1 + b*x)) - ((b + c)*(c^2 + d)*Hypergeometric2F1[5/6, 1,
11/6, -(((b + c)*(c + x))/((-1 + c^2)*(-1 + b*x)))])/((b - c)*c^2*(-1 + c^2)*(-1 + b*x))))/5

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IntegrateAlgebraic [A]  time = 3.48, size = 898, normalized size = 1.01 \begin {gather*} \frac {(1+b c) d \sqrt [6]{\frac {1-b x}{c+x}}}{b c \left (b+\frac {1-b x}{c+x}\right )}-\frac {\left (6 b+7 c+b c^2\right ) d \tan ^{-1}\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 ) \tan ^{-1}\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 ) \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {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 ) \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {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 \tan ^{-1}\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 )}{6 b^{11/6} c^2}+\frac {\sqrt [6]{b+c} \left (c^2+d\right ) \tan ^{-1}\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 )}{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} \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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F(-1)]  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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giac [F(-1)]  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="giac")

[Out]

Timed out

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maple [F]  time = 0.02, 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)Is b*c-1 positive or negative?

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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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sympy [F(-1)]  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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