3.560 \(\int \frac{x^{-1-\frac{n}{3}}}{a+b x^n+c x^{2 n}} \, dx\)

Optimal. Leaf size=699 \[ \frac{\left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{a} x^{-n/3}\right )}{\sqrt [3]{2} a^{4/3} n \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}+\frac{\left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) \log \left (\sqrt [3]{\sqrt{b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{a} x^{-n/3}\right )}{\sqrt [3]{2} a^{4/3} n \left (\sqrt{b^2-4 a c}+b\right )^{2/3}}-\frac{\left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \log \left (2^{2/3} a^{2/3} x^{-2 n/3}-\sqrt [3]{2} \sqrt [3]{a} x^{-n/3} \sqrt [3]{b-\sqrt{b^2-4 a c}}+\left (b-\sqrt{b^2-4 a c}\right )^{2/3}\right )}{2 \sqrt [3]{2} a^{4/3} n \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}-\frac{\left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) \log \left (2^{2/3} a^{2/3} x^{-2 n/3}-\sqrt [3]{2} \sqrt [3]{a} x^{-n/3} \sqrt [3]{\sqrt{b^2-4 a c}+b}+\left (\sqrt{b^2-4 a c}+b\right )^{2/3}\right )}{2 \sqrt [3]{2} a^{4/3} n \left (\sqrt{b^2-4 a c}+b\right )^{2/3}}-\frac{\sqrt{3} \left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{a} x^{-n/3}}{\sqrt [3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right )}{\sqrt [3]{2} a^{4/3} n \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}-\frac{\sqrt{3} \left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{a} x^{-n/3}}{\sqrt [3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right )}{\sqrt [3]{2} a^{4/3} n \left (\sqrt{b^2-4 a c}+b\right )^{2/3}}-\frac{3 x^{-n/3}}{a n} \]

[Out]

-3/(a*n*x^(n/3)) - (Sqrt[3]*(b - (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(1 - (2*2^(1/3)*a^(1/3))/((b - Sqrt[b
^2 - 4*a*c])^(1/3)*x^(n/3)))/Sqrt[3]])/(2^(1/3)*a^(4/3)*(b - Sqrt[b^2 - 4*a*c])^(2/3)*n) - (Sqrt[3]*(b + (b^2
- 2*a*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(1 - (2*2^(1/3)*a^(1/3))/((b + Sqrt[b^2 - 4*a*c])^(1/3)*x^(n/3)))/Sqrt[3]])
/(2^(1/3)*a^(4/3)*(b + Sqrt[b^2 - 4*a*c])^(2/3)*n) + ((b - (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*Log[(b - Sqrt[b^2
- 4*a*c])^(1/3) + (2^(1/3)*a^(1/3))/x^(n/3)])/(2^(1/3)*a^(4/3)*(b - Sqrt[b^2 - 4*a*c])^(2/3)*n) + ((b + (b^2 -
2*a*c)/Sqrt[b^2 - 4*a*c])*Log[(b + Sqrt[b^2 - 4*a*c])^(1/3) + (2^(1/3)*a^(1/3))/x^(n/3)])/(2^(1/3)*a^(4/3)*(b
+ Sqrt[b^2 - 4*a*c])^(2/3)*n) - ((b - (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*Log[(b - Sqrt[b^2 - 4*a*c])^(2/3) + (2
^(2/3)*a^(2/3))/x^((2*n)/3) - (2^(1/3)*a^(1/3)*(b - Sqrt[b^2 - 4*a*c])^(1/3))/x^(n/3)])/(2*2^(1/3)*a^(4/3)*(b
- Sqrt[b^2 - 4*a*c])^(2/3)*n) - ((b + (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*Log[(b + Sqrt[b^2 - 4*a*c])^(2/3) + (2^
(2/3)*a^(2/3))/x^((2*n)/3) - (2^(1/3)*a^(1/3)*(b + Sqrt[b^2 - 4*a*c])^(1/3))/x^(n/3)])/(2*2^(1/3)*a^(4/3)*(b +
Sqrt[b^2 - 4*a*c])^(2/3)*n)

________________________________________________________________________________________

Rubi [A]  time = 1.49293, antiderivative size = 699, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 10, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {1381, 1340, 1367, 1422, 200, 31, 634, 617, 204, 628} \[ \frac{\left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{a} x^{-n/3}\right )}{\sqrt [3]{2} a^{4/3} n \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}+\frac{\left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) \log \left (\sqrt [3]{\sqrt{b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{a} x^{-n/3}\right )}{\sqrt [3]{2} a^{4/3} n \left (\sqrt{b^2-4 a c}+b\right )^{2/3}}-\frac{\left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \log \left (2^{2/3} a^{2/3} x^{-2 n/3}-\sqrt [3]{2} \sqrt [3]{a} x^{-n/3} \sqrt [3]{b-\sqrt{b^2-4 a c}}+\left (b-\sqrt{b^2-4 a c}\right )^{2/3}\right )}{2 \sqrt [3]{2} a^{4/3} n \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}-\frac{\left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) \log \left (2^{2/3} a^{2/3} x^{-2 n/3}-\sqrt [3]{2} \sqrt [3]{a} x^{-n/3} \sqrt [3]{\sqrt{b^2-4 a c}+b}+\left (\sqrt{b^2-4 a c}+b\right )^{2/3}\right )}{2 \sqrt [3]{2} a^{4/3} n \left (\sqrt{b^2-4 a c}+b\right )^{2/3}}-\frac{\sqrt{3} \left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{a} x^{-n/3}}{\sqrt [3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right )}{\sqrt [3]{2} a^{4/3} n \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}-\frac{\sqrt{3} \left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{a} x^{-n/3}}{\sqrt [3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right )}{\sqrt [3]{2} a^{4/3} n \left (\sqrt{b^2-4 a c}+b\right )^{2/3}}-\frac{3 x^{-n/3}}{a n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - n/3)/(a + b*x^n + c*x^(2*n)),x]

[Out]

-3/(a*n*x^(n/3)) - (Sqrt[3]*(b - (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(1 - (2*2^(1/3)*a^(1/3))/((b - Sqrt[b
^2 - 4*a*c])^(1/3)*x^(n/3)))/Sqrt[3]])/(2^(1/3)*a^(4/3)*(b - Sqrt[b^2 - 4*a*c])^(2/3)*n) - (Sqrt[3]*(b + (b^2
- 2*a*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(1 - (2*2^(1/3)*a^(1/3))/((b + Sqrt[b^2 - 4*a*c])^(1/3)*x^(n/3)))/Sqrt[3]])
/(2^(1/3)*a^(4/3)*(b + Sqrt[b^2 - 4*a*c])^(2/3)*n) + ((b - (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*Log[(b - Sqrt[b^2
- 4*a*c])^(1/3) + (2^(1/3)*a^(1/3))/x^(n/3)])/(2^(1/3)*a^(4/3)*(b - Sqrt[b^2 - 4*a*c])^(2/3)*n) + ((b + (b^2 -
2*a*c)/Sqrt[b^2 - 4*a*c])*Log[(b + Sqrt[b^2 - 4*a*c])^(1/3) + (2^(1/3)*a^(1/3))/x^(n/3)])/(2^(1/3)*a^(4/3)*(b
+ Sqrt[b^2 - 4*a*c])^(2/3)*n) - ((b - (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*Log[(b - Sqrt[b^2 - 4*a*c])^(2/3) + (2
^(2/3)*a^(2/3))/x^((2*n)/3) - (2^(1/3)*a^(1/3)*(b - Sqrt[b^2 - 4*a*c])^(1/3))/x^(n/3)])/(2*2^(1/3)*a^(4/3)*(b
- Sqrt[b^2 - 4*a*c])^(2/3)*n) - ((b + (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*Log[(b + Sqrt[b^2 - 4*a*c])^(2/3) + (2^
(2/3)*a^(2/3))/x^((2*n)/3) - (2^(1/3)*a^(1/3)*(b + Sqrt[b^2 - 4*a*c])^(1/3))/x^(n/3)])/(2*2^(1/3)*a^(4/3)*(b +
Sqrt[b^2 - 4*a*c])^(2/3)*n)

Rule 1381

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/(m + 1), Subst[Int[(a +
b*x^Simplify[n/(m + 1)] + c*x^Simplify[(2*n)/(m + 1)])^p, x], x, x^(m + 1)], x] /; FreeQ[{a, b, c, m, n, p}, x
] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[Simplify[n/(m + 1)]] &&  !IntegerQ[n]

Rule 1340

Int[((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[x^(2*n*p)*(c + b/x^n + a/x^(2*n))^p,
x] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && LtQ[n, 0] && IntegerQ[p]

Rule 1367

Int[((d_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(d^(2*n - 1)*(d*x)
^(m - 2*n + 1)*(a + b*x^n + c*x^(2*n))^(p + 1))/(c*(m + 2*n*p + 1)), x] - Dist[d^(2*n)/(c*(m + 2*n*p + 1)), In
t[(d*x)^(m - 2*n)*Simp[a*(m - 2*n + 1) + b*(m + n*(p - 1) + 1)*x^n, x]*(a + b*x^n + c*x^(2*n))^p, x], x] /; Fr
eeQ[{a, b, c, d, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1] && NeQ[m + 2*n
*p + 1, 0] && IntegerQ[p]

Rule 1422

Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*
c, 2]}, Dist[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^n), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), In
t[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ
[c*d^2 - b*d*e + a*e^2, 0] && (PosQ[b^2 - 4*a*c] ||  !IGtQ[n/2, 0])

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

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

Rule 628

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

Rubi steps

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

Mathematica [C]  time = 0.149777, size = 127, normalized size = 0.18 \[ \frac{6 c x^{-n/3} \left (\frac{\, _2F_1\left (-\frac{1}{3},1;\frac{2}{3};\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )}{-b \sqrt{b^2-4 a c}-4 a c+b^2}+\frac{\, _2F_1\left (-\frac{1}{3},1;\frac{2}{3};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{b \sqrt{b^2-4 a c}-4 a c+b^2}\right )}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - n/3)/(a + b*x^n + c*x^(2*n)),x]

[Out]

(6*c*(Hypergeometric2F1[-1/3, 1, 2/3, (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])]/(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])
+ Hypergeometric2F1[-1/3, 1, 2/3, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]/(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])))/(n
*x^(n/3))

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Maple [C]  time = 0.398, size = 534, normalized size = 0.8 \begin{align*} -3\,{\frac{1}{an{x}^{n/3}}}+\sum _{{\it \_R}={\it RootOf} \left ( \left ( 64\,{a}^{7}{c}^{3}{n}^{6}-48\,{a}^{6}{b}^{2}{c}^{2}{n}^{6}+12\,{a}^{5}{b}^{4}c{n}^{6}-{a}^{4}{b}^{6}{n}^{6} \right ){{\it \_Z}}^{6}+ \left ( -32\,{a}^{3}b{c}^{3}{n}^{3}+32\,{a}^{2}{b}^{3}{c}^{2}{n}^{3}-10\,a{b}^{5}c{n}^{3}+{b}^{7}{n}^{3} \right ){{\it \_Z}}^{3}+{c}^{4} \right ) }{\it \_R}\,\ln \left ({x}^{{\frac{n}{3}}}+ \left ( -64\,{\frac{{a}^{8}{n}^{5}{c}^{4}}{2\,{a}^{2}{c}^{5}-4\,a{b}^{2}{c}^{4}+{b}^{4}{c}^{3}}}+112\,{\frac{{n}^{5}{b}^{2}{a}^{7}{c}^{3}}{2\,{a}^{2}{c}^{5}-4\,a{b}^{2}{c}^{4}+{b}^{4}{c}^{3}}}-60\,{\frac{{n}^{5}{b}^{4}{a}^{6}{c}^{2}}{2\,{a}^{2}{c}^{5}-4\,a{b}^{2}{c}^{4}+{b}^{4}{c}^{3}}}+13\,{\frac{{n}^{5}{b}^{6}{a}^{5}c}{2\,{a}^{2}{c}^{5}-4\,a{b}^{2}{c}^{4}+{b}^{4}{c}^{3}}}-{\frac{{n}^{5}{b}^{8}{a}^{4}}{2\,{a}^{2}{c}^{5}-4\,a{b}^{2}{c}^{4}+{b}^{4}{c}^{3}}} \right ){{\it \_R}}^{5}+ \left ( 28\,{\frac{b{n}^{2}{a}^{4}{c}^{4}}{2\,{a}^{2}{c}^{5}-4\,a{b}^{2}{c}^{4}+{b}^{4}{c}^{3}}}-63\,{\frac{{b}^{3}{n}^{2}{a}^{3}{c}^{3}}{2\,{a}^{2}{c}^{5}-4\,a{b}^{2}{c}^{4}+{b}^{4}{c}^{3}}}+42\,{\frac{{b}^{5}{n}^{2}{a}^{2}{c}^{2}}{2\,{a}^{2}{c}^{5}-4\,a{b}^{2}{c}^{4}+{b}^{4}{c}^{3}}}-11\,{\frac{{n}^{2}{b}^{7}ac}{2\,{a}^{2}{c}^{5}-4\,a{b}^{2}{c}^{4}+{b}^{4}{c}^{3}}}+{\frac{{n}^{2}{b}^{9}}{2\,{a}^{2}{c}^{5}-4\,a{b}^{2}{c}^{4}+{b}^{4}{c}^{3}}} \right ){{\it \_R}}^{2} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1-1/3*n)/(a+b*x^n+c*x^(2*n)),x)

[Out]

-3/a/n/(x^(1/3*n))+sum(_R*ln(x^(1/3*n)+(-64/(2*a^2*c^5-4*a*b^2*c^4+b^4*c^3)*n^5*a^8*c^4+112/(2*a^2*c^5-4*a*b^2
*c^4+b^4*c^3)*n^5*b^2*a^7*c^3-60/(2*a^2*c^5-4*a*b^2*c^4+b^4*c^3)*n^5*b^4*a^6*c^2+13/(2*a^2*c^5-4*a*b^2*c^4+b^4
*c^3)*n^5*b^6*a^5*c-1/(2*a^2*c^5-4*a*b^2*c^4+b^4*c^3)*n^5*b^8*a^4)*_R^5+(28/(2*a^2*c^5-4*a*b^2*c^4+b^4*c^3)*n^
2*b*a^4*c^4-63/(2*a^2*c^5-4*a*b^2*c^4+b^4*c^3)*n^2*b^3*a^3*c^3+42/(2*a^2*c^5-4*a*b^2*c^4+b^4*c^3)*n^2*b^5*a^2*
c^2-11/(2*a^2*c^5-4*a*b^2*c^4+b^4*c^3)*n^2*b^7*a*c+1/(2*a^2*c^5-4*a*b^2*c^4+b^4*c^3)*n^2*b^9)*_R^2),_R=RootOf(
(64*a^7*c^3*n^6-48*a^6*b^2*c^2*n^6+12*a^5*b^4*c*n^6-a^4*b^6*n^6)*_Z^6+(-32*a^3*b*c^3*n^3+32*a^2*b^3*c^2*n^3-10
*a*b^5*c*n^3+b^7*n^3)*_Z^3+c^4))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{3}{a n x^{\frac{1}{3} \, n}} - \int \frac{c x^{\frac{5}{3} \, n} + b x^{\frac{2}{3} \, n}}{a c x x^{2 \, n} + a b x x^{n} + a^{2} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-1/3*n)/(a+b*x^n+c*x^(2*n)),x, algorithm="maxima")

[Out]

-3/(a*n*x^(1/3*n)) - integrate((c*x^(5/3*n) + b*x^(2/3*n))/(a*c*x*x^(2*n) + a*b*x*x^n + a^2*x), x)

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Fricas [B]  time = 14.3671, size = 13524, normalized size = 19.35 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-1/3*n)/(a+b*x^n+c*x^(2*n)),x, algorithm="fricas")

[Out]

1/2*(4*sqrt(3)*(1/2)^(1/3)*a*n*(((a^4*b^2 - 4*a^5*c)*n^3*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c
^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)*n^6)) + b^3 - 2*a*b*c)/((a^4*b^2 - 4
*a^5*c)*n^3))^(1/3)*arctan(-1/6*(2*(1/2)^(2/3)*(sqrt(3)*(a^4*b^12*c - 17*a^5*b^10*c^2 + 114*a^6*b^8*c^3 - 378*
a^7*b^6*c^4 + 632*a^8*b^4*c^5 - 480*a^9*b^2*c^6 + 128*a^10*c^7)*n^5*x*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 -
16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)*n^6)) - sqrt(3)*(b^13*c
- 15*a*b^11*c^2 + 88*a^2*b^9*c^3 - 252*a^3*b^7*c^4 + 356*a^4*b^5*c^5 - 220*a^5*b^3*c^6 + 48*a^6*b*c^7)*n^2*x)
*x^(-1/3*n - 1)*(((a^4*b^2 - 4*a^5*c)*n^3*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)
/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)*n^6)) + b^3 - 2*a*b*c)/((a^4*b^2 - 4*a^5*c)*n^3))^(
2/3) - sqrt(2)*(1/2)^(2/3)*(sqrt(3)*(a^4*b^8 - 13*a^5*b^6*c + 60*a^6*b^4*c^2 - 112*a^7*b^2*c^3 + 64*a^8*c^4)*n
^5*x*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b
^2*c^2 - 64*a^11*c^3)*n^6)) - sqrt(3)*(b^9 - 11*a*b^7*c + 42*a^2*b^5*c^2 - 62*a^3*b^3*c^3 + 24*a^4*b*c^4)*n^2*
x)*sqrt((2*(b^8*c^2 - 8*a*b^6*c^3 + 20*a^2*b^4*c^4 - 16*a^3*b^2*c^5 + 4*a^4*c^6)*x^2*x^(-2/3*n - 2) - (1/2)^(1
/3)*((a^4*b^9*c - 12*a^5*b^7*c^2 + 50*a^6*b^5*c^3 - 80*a^7*b^3*c^4 + 32*a^8*b*c^5)*n^4*x*sqrt((b^8 - 8*a*b^6*c
+ 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)*n^6)
) - (b^10*c - 12*a*b^8*c^2 + 52*a^2*b^6*c^3 - 96*a^3*b^4*c^4 + 68*a^4*b^2*c^5 - 16*a^5*c^6)*n*x)*x^(-1/3*n - 1
)*(((a^4*b^2 - 4*a^5*c)*n^3*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 1
2*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)*n^6)) + b^3 - 2*a*b*c)/((a^4*b^2 - 4*a^5*c)*n^3))^(1/3) - (1/2)^(
2/3)*((a^4*b^11 - 16*a^5*b^9*c + 98*a^6*b^7*c^2 - 280*a^7*b^5*c^3 + 352*a^8*b^3*c^4 - 128*a^9*b*c^5)*n^5*sqrt(
(b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 6
4*a^11*c^3)*n^6)) - (b^12 - 14*a*b^10*c + 76*a^2*b^8*c^2 - 200*a^3*b^6*c^3 + 260*a^4*b^4*c^4 - 152*a^5*b^2*c^5
+ 32*a^6*c^6)*n^2)*(((a^4*b^2 - 4*a^5*c)*n^3*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*
c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)*n^6)) + b^3 - 2*a*b*c)/((a^4*b^2 - 4*a^5*c)*n^3
))^(2/3))/x^2)*(((a^4*b^2 - 4*a^5*c)*n^3*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/
((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)*n^6)) + b^3 - 2*a*b*c)/((a^4*b^2 - 4*a^5*c)*n^3))^(2
/3) + 2*sqrt(3)*(b^8*c^4 - 8*a*b^6*c^5 + 20*a^2*b^4*c^6 - 16*a^3*b^2*c^7 + 4*a^4*c^8))/(b^8*c^4 - 8*a*b^6*c^5
+ 20*a^2*b^4*c^6 - 16*a^3*b^2*c^7 + 4*a^4*c^8)) - 4*sqrt(3)*(1/2)^(1/3)*a*n*(-((a^4*b^2 - 4*a^5*c)*n^3*sqrt((b
^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*
a^11*c^3)*n^6)) - b^3 + 2*a*b*c)/((a^4*b^2 - 4*a^5*c)*n^3))^(1/3)*arctan(-1/6*(2*(1/2)^(2/3)*(sqrt(3)*(a^4*b^1
2*c - 17*a^5*b^10*c^2 + 114*a^6*b^8*c^3 - 378*a^7*b^6*c^4 + 632*a^8*b^4*c^5 - 480*a^9*b^2*c^6 + 128*a^10*c^7)*
n^5*x*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*
b^2*c^2 - 64*a^11*c^3)*n^6)) + sqrt(3)*(b^13*c - 15*a*b^11*c^2 + 88*a^2*b^9*c^3 - 252*a^3*b^7*c^4 + 356*a^4*b^
5*c^5 - 220*a^5*b^3*c^6 + 48*a^6*b*c^7)*n^2*x)*x^(-1/3*n - 1)*(-((a^4*b^2 - 4*a^5*c)*n^3*sqrt((b^8 - 8*a*b^6*c
+ 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)*n^6)
) - b^3 + 2*a*b*c)/((a^4*b^2 - 4*a^5*c)*n^3))^(2/3) - sqrt(2)*(1/2)^(2/3)*(sqrt(3)*(a^4*b^8 - 13*a^5*b^6*c + 6
0*a^6*b^4*c^2 - 112*a^7*b^2*c^3 + 64*a^8*c^4)*n^5*x*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 +
4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)*n^6)) + sqrt(3)*(b^9 - 11*a*b^7*c + 42*a^
2*b^5*c^2 - 62*a^3*b^3*c^3 + 24*a^4*b*c^4)*n^2*x)*sqrt((2*(b^8*c^2 - 8*a*b^6*c^3 + 20*a^2*b^4*c^4 - 16*a^3*b^2
*c^5 + 4*a^4*c^6)*x^2*x^(-2/3*n - 2) + (1/2)^(1/3)*((a^4*b^9*c - 12*a^5*b^7*c^2 + 50*a^6*b^5*c^3 - 80*a^7*b^3*
c^4 + 32*a^8*b*c^5)*n^4*x*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*
a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)*n^6)) + (b^10*c - 12*a*b^8*c^2 + 52*a^2*b^6*c^3 - 96*a^3*b^4*c^4 +
68*a^4*b^2*c^5 - 16*a^5*c^6)*n*x)*x^(-1/3*n - 1)*(-((a^4*b^2 - 4*a^5*c)*n^3*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4
*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)*n^6)) - b^3 + 2*a
*b*c)/((a^4*b^2 - 4*a^5*c)*n^3))^(1/3) + (1/2)^(2/3)*((a^4*b^11 - 16*a^5*b^9*c + 98*a^6*b^7*c^2 - 280*a^7*b^5*
c^3 + 352*a^8*b^3*c^4 - 128*a^9*b*c^5)*n^5*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4
)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)*n^6)) + (b^12 - 14*a*b^10*c + 76*a^2*b^8*c^2 - 200
*a^3*b^6*c^3 + 260*a^4*b^4*c^4 - 152*a^5*b^2*c^5 + 32*a^6*c^6)*n^2)*(-((a^4*b^2 - 4*a^5*c)*n^3*sqrt((b^8 - 8*a
*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3
)*n^6)) - b^3 + 2*a*b*c)/((a^4*b^2 - 4*a^5*c)*n^3))^(2/3))/x^2)*(-((a^4*b^2 - 4*a^5*c)*n^3*sqrt((b^8 - 8*a*b^6
*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)*n^
6)) - b^3 + 2*a*b*c)/((a^4*b^2 - 4*a^5*c)*n^3))^(2/3) - 2*sqrt(3)*(b^8*c^4 - 8*a*b^6*c^5 + 20*a^2*b^4*c^6 - 16
*a^3*b^2*c^7 + 4*a^4*c^8))/(b^8*c^4 - 8*a*b^6*c^5 + 20*a^2*b^4*c^6 - 16*a^3*b^2*c^7 + 4*a^4*c^8)) + 2*(1/2)^(1
/3)*a*n*(((a^4*b^2 - 4*a^5*c)*n^3*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b
^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)*n^6)) + b^3 - 2*a*b*c)/((a^4*b^2 - 4*a^5*c)*n^3))^(1/3)*log
((2*(b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*x*x^(-1/3*n - 1) + (1/2)^(1/3)*((a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*n
^4*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2
*c^2 - 64*a^11*c^3)*n^6)) - (b^6 - 8*a*b^4*c + 18*a^2*b^2*c^2 - 8*a^3*c^3)*n)*(((a^4*b^2 - 4*a^5*c)*n^3*sqrt((
b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64
*a^11*c^3)*n^6)) + b^3 - 2*a*b*c)/((a^4*b^2 - 4*a^5*c)*n^3))^(1/3))/x) + 2*(1/2)^(1/3)*a*n*(-((a^4*b^2 - 4*a^5
*c)*n^3*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^1
0*b^2*c^2 - 64*a^11*c^3)*n^6)) - b^3 + 2*a*b*c)/((a^4*b^2 - 4*a^5*c)*n^3))^(1/3)*log((2*(b^4*c - 4*a*b^2*c^2 +
2*a^2*c^3)*x*x^(-1/3*n - 1) - (1/2)^(1/3)*((a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*n^4*sqrt((b^8 - 8*a*b^6*c +
20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)*n^6))
+ (b^6 - 8*a*b^4*c + 18*a^2*b^2*c^2 - 8*a^3*c^3)*n)*(-((a^4*b^2 - 4*a^5*c)*n^3*sqrt((b^8 - 8*a*b^6*c + 20*a^2*
b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)*n^6)) - b^3 +
2*a*b*c)/((a^4*b^2 - 4*a^5*c)*n^3))^(1/3))/x) - (1/2)^(1/3)*a*n*(((a^4*b^2 - 4*a^5*c)*n^3*sqrt((b^8 - 8*a*b^6*
c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)*n^6
)) + b^3 - 2*a*b*c)/((a^4*b^2 - 4*a^5*c)*n^3))^(1/3)*log(8*(2*(b^8*c^2 - 8*a*b^6*c^3 + 20*a^2*b^4*c^4 - 16*a^3
*b^2*c^5 + 4*a^4*c^6)*x^2*x^(-2/3*n - 2) - (1/2)^(1/3)*((a^4*b^9*c - 12*a^5*b^7*c^2 + 50*a^6*b^5*c^3 - 80*a^7*
b^3*c^4 + 32*a^8*b*c^5)*n^4*x*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 -
12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)*n^6)) - (b^10*c - 12*a*b^8*c^2 + 52*a^2*b^6*c^3 - 96*a^3*b^4*c^
4 + 68*a^4*b^2*c^5 - 16*a^5*c^6)*n*x)*x^(-1/3*n - 1)*(((a^4*b^2 - 4*a^5*c)*n^3*sqrt((b^8 - 8*a*b^6*c + 20*a^2*
b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)*n^6)) + b^3 -
2*a*b*c)/((a^4*b^2 - 4*a^5*c)*n^3))^(1/3) - (1/2)^(2/3)*((a^4*b^11 - 16*a^5*b^9*c + 98*a^6*b^7*c^2 - 280*a^7*b
^5*c^3 + 352*a^8*b^3*c^4 - 128*a^9*b*c^5)*n^5*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*
c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)*n^6)) - (b^12 - 14*a*b^10*c + 76*a^2*b^8*c^2 -
200*a^3*b^6*c^3 + 260*a^4*b^4*c^4 - 152*a^5*b^2*c^5 + 32*a^6*c^6)*n^2)*(((a^4*b^2 - 4*a^5*c)*n^3*sqrt((b^8 - 8
*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c
^3)*n^6)) + b^3 - 2*a*b*c)/((a^4*b^2 - 4*a^5*c)*n^3))^(2/3))/x^2) - (1/2)^(1/3)*a*n*(-((a^4*b^2 - 4*a^5*c)*n^3
*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c
^2 - 64*a^11*c^3)*n^6)) - b^3 + 2*a*b*c)/((a^4*b^2 - 4*a^5*c)*n^3))^(1/3)*log(8*(2*(b^8*c^2 - 8*a*b^6*c^3 + 20
*a^2*b^4*c^4 - 16*a^3*b^2*c^5 + 4*a^4*c^6)*x^2*x^(-2/3*n - 2) + (1/2)^(1/3)*((a^4*b^9*c - 12*a^5*b^7*c^2 + 50*
a^6*b^5*c^3 - 80*a^7*b^3*c^4 + 32*a^8*b*c^5)*n^4*x*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4
*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)*n^6)) + (b^10*c - 12*a*b^8*c^2 + 52*a^2*b^
6*c^3 - 96*a^3*b^4*c^4 + 68*a^4*b^2*c^5 - 16*a^5*c^6)*n*x)*x^(-1/3*n - 1)*(-((a^4*b^2 - 4*a^5*c)*n^3*sqrt((b^8
- 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^
11*c^3)*n^6)) - b^3 + 2*a*b*c)/((a^4*b^2 - 4*a^5*c)*n^3))^(1/3) + (1/2)^(2/3)*((a^4*b^11 - 16*a^5*b^9*c + 98*a
^6*b^7*c^2 - 280*a^7*b^5*c^3 + 352*a^8*b^3*c^4 - 128*a^9*b*c^5)*n^5*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 1
6*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)*n^6)) + (b^12 - 14*a*b^10
*c + 76*a^2*b^8*c^2 - 200*a^3*b^6*c^3 + 260*a^4*b^4*c^4 - 152*a^5*b^2*c^5 + 32*a^6*c^6)*n^2)*(-((a^4*b^2 - 4*a
^5*c)*n^3*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a
^10*b^2*c^2 - 64*a^11*c^3)*n^6)) - b^3 + 2*a*b*c)/((a^4*b^2 - 4*a^5*c)*n^3))^(2/3))/x^2) - 6*x*x^(-1/3*n - 1))
/(a*n)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-1-1/3*n)/(a+b*x**n+c*x**(2*n)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{-\frac{1}{3} \, n - 1}}{c x^{2 \, n} + b x^{n} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-1/3*n)/(a+b*x^n+c*x^(2*n)),x, algorithm="giac")

[Out]

integrate(x^(-1/3*n - 1)/(c*x^(2*n) + b*x^n + a), x)