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

Optimal. Leaf size=610 \[ \frac{2^{2/3} c^{2/3} \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{n \sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}-\frac{2^{2/3} c^{2/3} \log \left (\sqrt [3]{\sqrt{b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{n \sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}+b\right )^{2/3}}-\frac{c^{2/3} \log \left (-\sqrt [3]{2} \sqrt [3]{c} x^{n/3} \sqrt [3]{b-\sqrt{b^2-4 a c}}+\left (b-\sqrt{b^2-4 a c}\right )^{2/3}+2^{2/3} c^{2/3} x^{2 n/3}\right )}{\sqrt [3]{2} n \sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}+\frac{c^{2/3} \log \left (-\sqrt [3]{2} \sqrt [3]{c} x^{n/3} \sqrt [3]{\sqrt{b^2-4 a c}+b}+\left (\sqrt{b^2-4 a c}+b\right )^{2/3}+2^{2/3} c^{2/3} x^{2 n/3}\right )}{\sqrt [3]{2} n \sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}+b\right )^{2/3}}-\frac{2^{2/3} \sqrt{3} c^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x^{n/3}}{\sqrt [3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right )}{n \sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}+\frac{2^{2/3} \sqrt{3} c^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x^{n/3}}{\sqrt [3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right )}{n \sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}+b\right )^{2/3}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.15229, antiderivative size = 610, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {1381, 1347, 200, 31, 634, 617, 204, 628} \[ \frac{2^{2/3} c^{2/3} \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{n \sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}-\frac{2^{2/3} c^{2/3} \log \left (\sqrt [3]{\sqrt{b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{n \sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}+b\right )^{2/3}}-\frac{c^{2/3} \log \left (-\sqrt [3]{2} \sqrt [3]{c} x^{n/3} \sqrt [3]{b-\sqrt{b^2-4 a c}}+\left (b-\sqrt{b^2-4 a c}\right )^{2/3}+2^{2/3} c^{2/3} x^{2 n/3}\right )}{\sqrt [3]{2} n \sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}+\frac{c^{2/3} \log \left (-\sqrt [3]{2} \sqrt [3]{c} x^{n/3} \sqrt [3]{\sqrt{b^2-4 a c}+b}+\left (\sqrt{b^2-4 a c}+b\right )^{2/3}+2^{2/3} c^{2/3} x^{2 n/3}\right )}{\sqrt [3]{2} n \sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}+b\right )^{2/3}}-\frac{2^{2/3} \sqrt{3} c^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x^{n/3}}{\sqrt [3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right )}{n \sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}+\frac{2^{2/3} \sqrt{3} c^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x^{n/3}}{\sqrt [3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right )}{n \sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}+b\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((2^(2/3)*Sqrt[3]*c^(2/3)*ArcTan[(1 - (2*2^(1/3)*c^(1/3)*x^(n/3))/(b - Sqrt[b^2 - 4*a*c])^(1/3))/Sqrt[3]])/(S
qrt[b^2 - 4*a*c]*(b - Sqrt[b^2 - 4*a*c])^(2/3)*n)) + (2^(2/3)*Sqrt[3]*c^(2/3)*ArcTan[(1 - (2*2^(1/3)*c^(1/3)*x
^(n/3))/(b + Sqrt[b^2 - 4*a*c])^(1/3))/Sqrt[3]])/(Sqrt[b^2 - 4*a*c]*(b + Sqrt[b^2 - 4*a*c])^(2/3)*n) + (2^(2/3
)*c^(2/3)*Log[(b - Sqrt[b^2 - 4*a*c])^(1/3) + 2^(1/3)*c^(1/3)*x^(n/3)])/(Sqrt[b^2 - 4*a*c]*(b - Sqrt[b^2 - 4*a
*c])^(2/3)*n) - (2^(2/3)*c^(2/3)*Log[(b + Sqrt[b^2 - 4*a*c])^(1/3) + 2^(1/3)*c^(1/3)*x^(n/3)])/(Sqrt[b^2 - 4*a
*c]*(b + Sqrt[b^2 - 4*a*c])^(2/3)*n) - (c^(2/3)*Log[(b - Sqrt[b^2 - 4*a*c])^(2/3) - 2^(1/3)*c^(1/3)*(b - Sqrt[
b^2 - 4*a*c])^(1/3)*x^(n/3) + 2^(2/3)*c^(2/3)*x^((2*n)/3)])/(2^(1/3)*Sqrt[b^2 - 4*a*c]*(b - Sqrt[b^2 - 4*a*c])
^(2/3)*n) + (c^(2/3)*Log[(b + Sqrt[b^2 - 4*a*c])^(2/3) - 2^(1/3)*c^(1/3)*(b + Sqrt[b^2 - 4*a*c])^(1/3)*x^(n/3)
+ 2^(2/3)*c^(2/3)*x^((2*n)/3)])/(2^(1/3)*Sqrt[b^2 - 4*a*c]*(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 1347

Int[((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, In
t[1/(b/2 - q/2 + c*x^n), x], x] - Dist[c/q, Int[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c}, x] && EqQ[n
2, 2*n] && NeQ[b^2 - 4*a*c, 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+b x^3+c x^6} \, dx,x,x^{n/3}\right )}{n}\\ &=\frac{(3 c) \operatorname{Subst}\left (\int \frac{1}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^3} \, dx,x,x^{n/3}\right )}{\sqrt{b^2-4 a c} n}-\frac{(3 c) \operatorname{Subst}\left (\int \frac{1}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^3} \, dx,x,x^{n/3}\right )}{\sqrt{b^2-4 a c} n}\\ &=\frac{\left (2^{2/3} c\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt [3]{b-\sqrt{b^2-4 a c}}}{\sqrt [3]{2}}+\sqrt [3]{c} x} \, dx,x,x^{n/3}\right )}{\sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right )^{2/3} n}+\frac{\left (2^{2/3} c\right ) \operatorname{Subst}\left (\int \frac{2^{2/3} \sqrt [3]{b-\sqrt{b^2-4 a c}}-\sqrt [3]{c} x}{\frac{\left (b-\sqrt{b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac{\sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx,x,x^{n/3}\right )}{\sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right )^{2/3} n}-\frac{\left (2^{2/3} c\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt [3]{b+\sqrt{b^2-4 a c}}}{\sqrt [3]{2}}+\sqrt [3]{c} x} \, dx,x,x^{n/3}\right )}{\sqrt{b^2-4 a c} \left (b+\sqrt{b^2-4 a c}\right )^{2/3} n}-\frac{\left (2^{2/3} c\right ) \operatorname{Subst}\left (\int \frac{2^{2/3} \sqrt [3]{b+\sqrt{b^2-4 a c}}-\sqrt [3]{c} x}{\frac{\left (b+\sqrt{b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac{\sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx,x,x^{n/3}\right )}{\sqrt{b^2-4 a c} \left (b+\sqrt{b^2-4 a c}\right )^{2/3} n}\\ &=\frac{2^{2/3} c^{2/3} \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{\sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right )^{2/3} n}-\frac{2^{2/3} c^{2/3} \log \left (\sqrt [3]{b+\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{\sqrt{b^2-4 a c} \left (b+\sqrt{b^2-4 a c}\right )^{2/3} n}-\frac{c^{2/3} \operatorname{Subst}\left (\int \frac{-\frac{\sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}}}{\sqrt [3]{2}}+2 c^{2/3} x}{\frac{\left (b-\sqrt{b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac{\sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx,x,x^{n/3}\right )}{\sqrt [3]{2} \sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right )^{2/3} n}+\frac{(3 c) \operatorname{Subst}\left (\int \frac{1}{\frac{\left (b-\sqrt{b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac{\sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx,x,x^{n/3}\right )}{2^{2/3} \sqrt{b^2-4 a c} \sqrt [3]{b-\sqrt{b^2-4 a c}} n}+\frac{c^{2/3} \operatorname{Subst}\left (\int \frac{-\frac{\sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}}}{\sqrt [3]{2}}+2 c^{2/3} x}{\frac{\left (b+\sqrt{b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac{\sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx,x,x^{n/3}\right )}{\sqrt [3]{2} \sqrt{b^2-4 a c} \left (b+\sqrt{b^2-4 a c}\right )^{2/3} n}-\frac{(3 c) \operatorname{Subst}\left (\int \frac{1}{\frac{\left (b+\sqrt{b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac{\sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx,x,x^{n/3}\right )}{2^{2/3} \sqrt{b^2-4 a c} \sqrt [3]{b+\sqrt{b^2-4 a c}} n}\\ &=\frac{2^{2/3} c^{2/3} \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{\sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right )^{2/3} n}-\frac{2^{2/3} c^{2/3} \log \left (\sqrt [3]{b+\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{\sqrt{b^2-4 a c} \left (b+\sqrt{b^2-4 a c}\right )^{2/3} n}-\frac{c^{2/3} \log \left (\left (b-\sqrt{b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}} x^{n/3}+2^{2/3} c^{2/3} x^{2 n/3}\right )}{\sqrt [3]{2} \sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right )^{2/3} n}+\frac{c^{2/3} \log \left (\left (b+\sqrt{b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}} x^{n/3}+2^{2/3} c^{2/3} x^{2 n/3}\right )}{\sqrt [3]{2} \sqrt{b^2-4 a c} \left (b+\sqrt{b^2-4 a c}\right )^{2/3} n}+\frac{\left (3\ 2^{2/3} c^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x^{n/3}}{\sqrt [3]{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right )^{2/3} n}-\frac{\left (3\ 2^{2/3} c^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x^{n/3}}{\sqrt [3]{b+\sqrt{b^2-4 a c}}}\right )}{\sqrt{b^2-4 a c} \left (b+\sqrt{b^2-4 a c}\right )^{2/3} n}\\ &=-\frac{2^{2/3} \sqrt{3} c^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x^{n/3}}{\sqrt [3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right )}{\sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right )^{2/3} n}+\frac{2^{2/3} \sqrt{3} c^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x^{n/3}}{\sqrt [3]{b+\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right )}{\sqrt{b^2-4 a c} \left (b+\sqrt{b^2-4 a c}\right )^{2/3} n}+\frac{2^{2/3} c^{2/3} \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{\sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right )^{2/3} n}-\frac{2^{2/3} c^{2/3} \log \left (\sqrt [3]{b+\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{\sqrt{b^2-4 a c} \left (b+\sqrt{b^2-4 a c}\right )^{2/3} n}-\frac{c^{2/3} \log \left (\left (b-\sqrt{b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}} x^{n/3}+2^{2/3} c^{2/3} x^{2 n/3}\right )}{\sqrt [3]{2} \sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right )^{2/3} n}+\frac{c^{2/3} \log \left (\left (b+\sqrt{b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}} x^{n/3}+2^{2/3} c^{2/3} x^{2 n/3}\right )}{\sqrt [3]{2} \sqrt{b^2-4 a c} \left (b+\sqrt{b^2-4 a c}\right )^{2/3} n}\\ \end{align*}

Mathematica [A]  time = 0.77329, size = 526, normalized size = 0.86 \[ \frac{c^{2/3} \left (-\left (\sqrt{b^2-4 a c}+b\right )^{2/3} \log \left (-\sqrt [3]{2} \sqrt [3]{c} x^{n/3} \sqrt [3]{b-\sqrt{b^2-4 a c}}+\left (b-\sqrt{b^2-4 a c}\right )^{2/3}+2^{2/3} c^{2/3} x^{2 n/3}\right )+\left (b-\sqrt{b^2-4 a c}\right )^{2/3} \log \left (-\sqrt [3]{2} \sqrt [3]{c} x^{n/3} \sqrt [3]{\sqrt{b^2-4 a c}+b}+\left (\sqrt{b^2-4 a c}+b\right )^{2/3}+2^{2/3} c^{2/3} x^{2 n/3}\right )+2 \left (\sqrt{b^2-4 a c}+b\right )^{2/3} \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )-2 \left (b-\sqrt{b^2-4 a c}\right )^{2/3} \log \left (\sqrt [3]{\sqrt{b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )-2 \sqrt{3} \left (\sqrt{b^2-4 a c}+b\right )^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x^{n/3}}{\sqrt [3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right )+2 \sqrt{3} \left (b-\sqrt{b^2-4 a c}\right )^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x^{n/3}}{\sqrt [3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right )\right )}{\sqrt [3]{2} n \sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right )^{2/3} \left (\sqrt{b^2-4 a c}+b\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.215, size = 260, normalized size = 0.4 \begin{align*} \sum _{{\it \_R}={\it RootOf} \left ( \left ( 64\,{a}^{5}{c}^{3}{n}^{6}-48\,{a}^{4}{b}^{2}{c}^{2}{n}^{6}+12\,{a}^{3}{b}^{4}c{n}^{6}-{a}^{2}{b}^{6}{n}^{6} \right ){{\it \_Z}}^{6}+ \left ( 16\,{a}^{2}b{c}^{2}{n}^{3}-8\,a{b}^{3}c{n}^{3}+{b}^{5}{n}^{3} \right ){{\it \_Z}}^{3}+{c}^{2} \right ) }{\it \_R}\,\ln \left ({x}^{{\frac{n}{3}}}+ \left ( -16\,{\frac{{n}^{4}b{a}^{4}{c}^{2}}{2\,{c}^{2}a-{b}^{2}c}}+8\,{\frac{{n}^{4}{b}^{3}{a}^{3}c}{2\,{c}^{2}a-{b}^{2}c}}-{\frac{{n}^{4}{b}^{5}{a}^{2}}{2\,{c}^{2}a-{b}^{2}c}} \right ){{\it \_R}}^{4}+ \left ( 4\,{\frac{{a}^{2}{c}^{2}n}{2\,{c}^{2}a-{b}^{2}c}}-5\,{\frac{a{b}^{2}cn}{2\,{c}^{2}a-{b}^{2}c}}+{\frac{{b}^{4}n}{2\,{c}^{2}a-{b}^{2}c}} \right ){\it \_R} \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]

sum(_R*ln(x^(1/3*n)+(-16/(2*a*c^2-b^2*c)*n^4*b*a^4*c^2+8/(2*a*c^2-b^2*c)*n^4*b^3*a^3*c-1/(2*a*c^2-b^2*c)*n^4*b
^5*a^2)*_R^4+(4/(2*a*c^2-b^2*c)*n*a^2*c^2-5/(2*a*c^2-b^2*c)*n*b^2*a*c+1/(2*a*c^2-b^2*c)*n*b^4)*_R),_R=RootOf((
64*a^5*c^3*n^6-48*a^4*b^2*c^2*n^6+12*a^3*b^4*c*n^6-a^2*b^6*n^6)*_Z^6+(16*a^2*b*c^2*n^3-8*a*b^3*c*n^3+b^5*n^3)*
_Z^3+c^2))

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Maxima [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="maxima")

[Out]

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

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Fricas [B]  time = 4.07959, size = 9999, normalized size = 16.39 \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]

2*sqrt(3)*(1/2)^(1/3)*(((a^2*b^2 - 4*a^3*c)*n^3*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c +
48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) + b)/((a^2*b^2 - 4*a^3*c)*n^3))^(1/3)*arctan(-1/6*(2*(1/2)^(2/3)*(sqrt(3)*(
a^2*b^8*c - 14*a^3*b^6*c^2 + 72*a^4*b^4*c^3 - 160*a^5*b^2*c^4 + 128*a^6*c^5)*n^5*x*sqrt((b^4 - 4*a*b^2*c + 4*a
^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) - sqrt(3)*(b^7*c - 8*a*b^5*c^2 + 20*a^2*
b^3*c^3 - 16*a^3*b*c^4)*n^2*x)*x^(1/3*n - 1)*(((a^2*b^2 - 4*a^3*c)*n^3*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^
4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) + b)/((a^2*b^2 - 4*a^3*c)*n^3))^(2/3) + sqrt(2)*(1/2
)^(2/3)*(sqrt(3)*(a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3)*n^5*x*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^
2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) - sqrt(3)*(b^5 - 6*a*b^3*c + 8*a^2*b*c^2)*n^2
*x)*sqrt((2*(b^4*c^2 - 4*a*b^2*c^3 + 4*a^2*c^4)*x^2*x^(2/3*n - 2) - (1/2)^(1/3)*((a^2*b^7*c - 10*a^3*b^5*c^2 +
32*a^4*b^3*c^3 - 32*a^5*b*c^4)*n^4*x*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2
*c^2 - 64*a^7*c^3)*n^6)) - (b^6*c - 8*a*b^4*c^2 + 20*a^2*b^2*c^3 - 16*a^3*c^4)*n*x)*x^(1/3*n - 1)*(((a^2*b^2 -
4*a^3*c)*n^3*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6))
+ b)/((a^2*b^2 - 4*a^3*c)*n^3))^(1/3) - (1/2)^(2/3)*((a^2*b^9 - 14*a^3*b^7*c + 72*a^4*b^5*c^2 - 160*a^5*b^3*c
^3 + 128*a^6*b*c^4)*n^5*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*
c^3)*n^6)) - (b^8 - 10*a*b^6*c + 36*a^2*b^4*c^2 - 56*a^3*b^2*c^3 + 32*a^4*c^4)*n^2)*(((a^2*b^2 - 4*a^3*c)*n^3*
sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) + b)/((a^2*b^
2 - 4*a^3*c)*n^3))^(2/3))/x^2)*(((a^2*b^2 - 4*a^3*c)*n^3*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5
*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) + b)/((a^2*b^2 - 4*a^3*c)*n^3))^(2/3) + 2*sqrt(3)*(b^4*c^2 - 4*a*b
^2*c^3 + 4*a^2*c^4))/(b^4*c^2 - 4*a*b^2*c^3 + 4*a^2*c^4)) - 2*sqrt(3)*(1/2)^(1/3)*(-((a^2*b^2 - 4*a^3*c)*n^3*s
qrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) - b)/((a^2*b^2
- 4*a^3*c)*n^3))^(1/3)*arctan(-1/6*(2*(1/2)^(2/3)*(sqrt(3)*(a^2*b^8*c - 14*a^3*b^6*c^2 + 72*a^4*b^4*c^3 - 160
*a^5*b^2*c^4 + 128*a^6*c^5)*n^5*x*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2
- 64*a^7*c^3)*n^6)) + sqrt(3)*(b^7*c - 8*a*b^5*c^2 + 20*a^2*b^3*c^3 - 16*a^3*b*c^4)*n^2*x)*x^(1/3*n - 1)*(-((
a^2*b^2 - 4*a^3*c)*n^3*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c
^3)*n^6)) - b)/((a^2*b^2 - 4*a^3*c)*n^3))^(2/3) + sqrt(2)*(1/2)^(2/3)*(sqrt(3)*(a^2*b^6 - 12*a^3*b^4*c + 48*a^
4*b^2*c^2 - 64*a^5*c^3)*n^5*x*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 6
4*a^7*c^3)*n^6)) + sqrt(3)*(b^5 - 6*a*b^3*c + 8*a^2*b*c^2)*n^2*x)*sqrt((2*(b^4*c^2 - 4*a*b^2*c^3 + 4*a^2*c^4)*
x^2*x^(2/3*n - 2) + (1/2)^(1/3)*((a^2*b^7*c - 10*a^3*b^5*c^2 + 32*a^4*b^3*c^3 - 32*a^5*b*c^4)*n^4*x*sqrt((b^4
- 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) + (b^6*c - 8*a*b^4*c^2
+ 20*a^2*b^2*c^3 - 16*a^3*c^4)*n*x)*x^(1/3*n - 1)*(-((a^2*b^2 - 4*a^3*c)*n^3*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2
)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) - b)/((a^2*b^2 - 4*a^3*c)*n^3))^(1/3) + (1/2)^
(2/3)*((a^2*b^9 - 14*a^3*b^7*c + 72*a^4*b^5*c^2 - 160*a^5*b^3*c^3 + 128*a^6*b*c^4)*n^5*sqrt((b^4 - 4*a*b^2*c +
4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) + (b^8 - 10*a*b^6*c + 36*a^2*b^4*c^2
- 56*a^3*b^2*c^3 + 32*a^4*c^4)*n^2)*(-((a^2*b^2 - 4*a^3*c)*n^3*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 -
12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) - b)/((a^2*b^2 - 4*a^3*c)*n^3))^(2/3))/x^2)*(-((a^2*b^2 - 4
*a^3*c)*n^3*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) -
b)/((a^2*b^2 - 4*a^3*c)*n^3))^(2/3) - 2*sqrt(3)*(b^4*c^2 - 4*a*b^2*c^3 + 4*a^2*c^4))/(b^4*c^2 - 4*a*b^2*c^3 +
4*a^2*c^4)) + (1/2)^(1/3)*(((a^2*b^2 - 4*a^3*c)*n^3*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4
*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) + b)/((a^2*b^2 - 4*a^3*c)*n^3))^(1/3)*log(-(2*(b^2*c - 2*a*c^2)*x*x^(1
/3*n - 1) + (1/2)^(1/3)*((a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*n^4*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b
^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) - (b^4 - 6*a*b^2*c + 8*a^2*c^2)*n)*(((a^2*b^2 - 4*a^3*c
)*n^3*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) + b)/((
a^2*b^2 - 4*a^3*c)*n^3))^(1/3))/x) + (1/2)^(1/3)*(-((a^2*b^2 - 4*a^3*c)*n^3*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)
/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) - b)/((a^2*b^2 - 4*a^3*c)*n^3))^(1/3)*log(-(2*(
b^2*c - 2*a*c^2)*x*x^(1/3*n - 1) - (1/2)^(1/3)*((a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*n^4*sqrt((b^4 - 4*a*b^2
*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) + (b^4 - 6*a*b^2*c + 8*a^2*c^2)*
n)*(-((a^2*b^2 - 4*a^3*c)*n^3*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 6
4*a^7*c^3)*n^6)) - b)/((a^2*b^2 - 4*a^3*c)*n^3))^(1/3))/x) - 1/2*(1/2)^(1/3)*(((a^2*b^2 - 4*a^3*c)*n^3*sqrt((b
^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) + b)/((a^2*b^2 - 4*a
^3*c)*n^3))^(1/3)*log(8*(2*(b^4*c^2 - 4*a*b^2*c^3 + 4*a^2*c^4)*x^2*x^(2/3*n - 2) - (1/2)^(1/3)*((a^2*b^7*c - 1
0*a^3*b^5*c^2 + 32*a^4*b^3*c^3 - 32*a^5*b*c^4)*n^4*x*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4
*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) - (b^6*c - 8*a*b^4*c^2 + 20*a^2*b^2*c^3 - 16*a^3*c^4)*n*x)*x^(1/3*n -
1)*(((a^2*b^2 - 4*a^3*c)*n^3*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64
*a^7*c^3)*n^6)) + b)/((a^2*b^2 - 4*a^3*c)*n^3))^(1/3) - (1/2)^(2/3)*((a^2*b^9 - 14*a^3*b^7*c + 72*a^4*b^5*c^2
- 160*a^5*b^3*c^3 + 128*a^6*b*c^4)*n^5*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^
2*c^2 - 64*a^7*c^3)*n^6)) - (b^8 - 10*a*b^6*c + 36*a^2*b^4*c^2 - 56*a^3*b^2*c^3 + 32*a^4*c^4)*n^2)*(((a^2*b^2
- 4*a^3*c)*n^3*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)
) + b)/((a^2*b^2 - 4*a^3*c)*n^3))^(2/3))/x^2) - 1/2*(1/2)^(1/3)*(-((a^2*b^2 - 4*a^3*c)*n^3*sqrt((b^4 - 4*a*b^2
*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) - b)/((a^2*b^2 - 4*a^3*c)*n^3))^
(1/3)*log(8*(2*(b^4*c^2 - 4*a*b^2*c^3 + 4*a^2*c^4)*x^2*x^(2/3*n - 2) + (1/2)^(1/3)*((a^2*b^7*c - 10*a^3*b^5*c^
2 + 32*a^4*b^3*c^3 - 32*a^5*b*c^4)*n^4*x*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*
b^2*c^2 - 64*a^7*c^3)*n^6)) + (b^6*c - 8*a*b^4*c^2 + 20*a^2*b^2*c^3 - 16*a^3*c^4)*n*x)*x^(1/3*n - 1)*(-((a^2*b
^2 - 4*a^3*c)*n^3*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)*n
^6)) - b)/((a^2*b^2 - 4*a^3*c)*n^3))^(1/3) + (1/2)^(2/3)*((a^2*b^9 - 14*a^3*b^7*c + 72*a^4*b^5*c^2 - 160*a^5*b
^3*c^3 + 128*a^6*b*c^4)*n^5*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*
a^7*c^3)*n^6)) + (b^8 - 10*a*b^6*c + 36*a^2*b^4*c^2 - 56*a^3*b^2*c^3 + 32*a^4*c^4)*n^2)*(-((a^2*b^2 - 4*a^3*c)
*n^3*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) - b)/((a
^2*b^2 - 4*a^3*c)*n^3))^(2/3))/x^2)

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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)