\(\int \frac {x^{-1+\frac {n}{3}}}{a+b x^n+c x^{2 n}} \, dx\) [557]

Optimal result

Integrand size = 26, antiderivative size = 610 \[ \int \frac {x^{-1+\frac {n}{3}}}{a+b x^n+c x^{2 n}} \, dx=-\frac {2^{2/3} \sqrt {3} c^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x^{n/3}}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{\sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3} n}+\frac {2^{2/3} \sqrt {3} c^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x^{n/3}}{\sqrt [3]{b+\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right )^{2/3} n}+\frac {2^{2/3} c^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{\sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3} n}-\frac {2^{2/3} c^{2/3} \log \left (\sqrt [3]{b+\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right )^{2/3} n}-\frac {c^{2/3} \log \left (\left (b-\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x^{n/3}+2^{2/3} c^{2/3} x^{2 n/3}\right )}{\sqrt [3]{2} \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3} n}+\frac {c^{2/3} \log \left (\left (b+\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x^{n/3}+2^{2/3} c^{2/3} x^{2 n/3}\right )}{\sqrt [3]{2} \sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right )^{2/3} n} \]

[Out]

2^(2/3)*c^(2/3)*ln(2^(1/3)*c^(1/3)*x^(1/3*n)+(b-(-4*a*c+b^2)^(1/2))^(1/3))/n/(b-(-4*a*c+b^2)^(1/2))^(2/3)/(-4*
a*c+b^2)^(1/2)-1/2*c^(2/3)*ln(2^(2/3)*c^(2/3)*x^(2/3*n)-2^(1/3)*c^(1/3)*x^(1/3*n)*(b-(-4*a*c+b^2)^(1/2))^(1/3)
+(b-(-4*a*c+b^2)^(1/2))^(2/3))*2^(2/3)/n/(b-(-4*a*c+b^2)^(1/2))^(2/3)/(-4*a*c+b^2)^(1/2)-2^(2/3)*c^(2/3)*arcta
n(1/3*(1-2*2^(1/3)*c^(1/3)*x^(1/3*n)/(b-(-4*a*c+b^2)^(1/2))^(1/3))*3^(1/2))*3^(1/2)/n/(b-(-4*a*c+b^2)^(1/2))^(
2/3)/(-4*a*c+b^2)^(1/2)-2^(2/3)*c^(2/3)*ln(2^(1/3)*c^(1/3)*x^(1/3*n)+(b+(-4*a*c+b^2)^(1/2))^(1/3))/n/(-4*a*c+b
^2)^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(2/3)+1/2*c^(2/3)*ln(2^(2/3)*c^(2/3)*x^(2/3*n)-2^(1/3)*c^(1/3)*x^(1/3*n)*(b+(
-4*a*c+b^2)^(1/2))^(1/3)+(b+(-4*a*c+b^2)^(1/2))^(2/3))*2^(2/3)/n/(-4*a*c+b^2)^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(2/
3)+2^(2/3)*c^(2/3)*arctan(1/3*(1-2*2^(1/3)*c^(1/3)*x^(1/3*n)/(b+(-4*a*c+b^2)^(1/2))^(1/3))*3^(1/2))*3^(1/2)/n/
(-4*a*c+b^2)^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(2/3)

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 610, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {1395, 1361, 206, 31, 648, 631, 210, 642} \[ \int \frac {x^{-1+\frac {n}{3}}}{a+b x^n+c x^{2 n}} \, dx=-\frac {2^{2/3} \sqrt {3} c^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x^{n/3}}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{n \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \sqrt {3} c^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x^{n/3}}{\sqrt [3]{\sqrt {b^2-4 a c}+b}}}{\sqrt {3}}\right )}{n \sqrt {b^2-4 a c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} c^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{n \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}-\frac {2^{2/3} c^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{n \sqrt {b^2-4 a c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}-\frac {c^{2/3} \log \left (-\sqrt [3]{2} \sqrt [3]{c} x^{n/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}+2^{2/3} c^{2/3} x^{2 n/3}\right )}{\sqrt [3]{2} n \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {c^{2/3} \log \left (-\sqrt [3]{2} \sqrt [3]{c} x^{n/3} \sqrt [3]{\sqrt {b^2-4 a c}+b}+\left (\sqrt {b^2-4 a c}+b\right )^{2/3}+2^{2/3} c^{2/3} x^{2 n/3}\right )}{\sqrt [3]{2} n \sqrt {b^2-4 a c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}} \]

[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 31

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

Rule 206

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 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 631

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 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 1361

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 1395

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]

Rubi steps \begin{align*} \text {integral}& = \frac {3 \text {Subst}\left (\int \frac {1}{a+b x^3+c x^6} \, dx,x,x^{n/3}\right )}{n} \\ & = \frac {(3 c) \text {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) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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} \text {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) \text {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} \text {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) \text {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 ) \text {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 ) \text {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] (verified)

Time = 0.64 (sec) , antiderivative size = 526, normalized size of antiderivative = 0.86 \[ \int \frac {x^{-1+\frac {n}{3}}}{a+b x^n+c x^{2 n}} \, dx=\frac {c^{2/3} \left (-2 \sqrt {3} \left (b+\sqrt {b^2-4 a c}\right )^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x^{n/3}}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )+2 \sqrt {3} \left (b-\sqrt {b^2-4 a c}\right )^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x^{n/3}}{\sqrt [3]{b+\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )+2 \left (b+\sqrt {b^2-4 a c}\right )^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )-2 \left (b-\sqrt {b^2-4 a c}\right )^{2/3} \log \left (\sqrt [3]{b+\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )-\left (b+\sqrt {b^2-4 a c}\right )^{2/3} \log \left (\left (b-\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x^{n/3}+2^{2/3} c^{2/3} x^{2 n/3}\right )+\left (b-\sqrt {b^2-4 a c}\right )^{2/3} \log \left (\left (b+\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x^{n/3}+2^{2/3} c^{2/3} x^{2 n/3}\right )\right )}{\sqrt [3]{2} \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3} \left (b+\sqrt {b^2-4 a c}\right )^{2/3} n} \]

[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)

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.70 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.43

method	result	size
risch	\(\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (64 a^{5} c^{3} n^{6}-48 a^{4} b^{2} c^{2} n^{6}+12 a^{3} b^{4} c \,n^{6}-a^{2} b^{6} n^{6}\right ) \textit {\_Z}^{6}+\left (16 a^{2} b \,c^{2} n^{3}-8 a \,b^{3} c \,n^{3}+b^{5} n^{3}\right ) \textit {\_Z}^{3}+c^{2}\right )}{\sum }\textit {\_R} \ln \left (x^{\frac {n}{3}}+\left (-\frac {16 n^{4} b \,a^{4} c^{2}}{2 a \,c^{2}-b^{2} c}+\frac {8 n^{4} b^{3} a^{3} c}{2 a \,c^{2}-b^{2} c}-\frac {n^{4} b^{5} a^{2}}{2 a \,c^{2}-b^{2} c}\right ) \textit {\_R}^{4}+\left (\frac {4 n \,a^{2} c^{2}}{2 a \,c^{2}-b^{2} c}-\frac {5 n \,b^{2} a c}{2 a \,c^{2}-b^{2} c}+\frac {n \,b^{4}}{2 a \,c^{2}-b^{2} c}\right ) \textit {\_R} \right )\)	\(260\)

[In]

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

[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))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2461 vs. \(2 (465) = 930\).

Time = 0.35 (sec) , antiderivative size = 2461, normalized size of antiderivative = 4.03 \[ \int \frac {x^{-1+\frac {n}{3}}}{a+b x^n+c x^{2 n}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/2*(1/2)^(1/3)*(sqrt(-3) + 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)*log(-(4*(b^2*c - 2*a*c^2)*x
*x^(1/3*n - 1) + (1/2)^(1/3)*(sqrt(-3)*(b^4 - 6*a*b^2*c + 8*a^2*c^2)*n + (b^4 - 6*a*b^2*c + 8*a^2*c^2)*n - (sq
rt(-3)*(a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*n^4 + (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)))*(((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*(1/2)^(1/3)*(sqrt(-3) - 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)*log(-(4*(b^2*c - 2*a*c^2)*x*x^(1/3*n - 1) - (1/2)^(1/3)*(sqrt(-3)*(b^4 - 6*a*b^2*c + 8*a^2*c^2)*n - (b^4 -
6*a*b^2*c + 8*a^2*c^2)*n - (sqrt(-3)*(a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*n^4 - (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
)))*(((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)*(sqrt(-3) + 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)*log(-(4*(b^2*c - 2*a*c^2)*x*x^(1/3*n - 1) + (1/2)^(1/3)*(sqrt(-3)*(b^4 - 6*a
*b^2*c + 8*a^2*c^2)*n + (b^4 - 6*a*b^2*c + 8*a^2*c^2)*n + (sqrt(-3)*(a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*n^4
+ (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 + 4
8*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)))*(-((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 - 1
2*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*(1/2)^(1/3)*(
sqrt(-3) - 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)*log(-(4*(b^2*c - 2*a*c^2)*x*x^(1/3*n - 1) -
(1/2)^(1/3)*(sqrt(-3)*(b^4 - 6*a*b^2*c + 8*a^2*c^2)*n - (b^4 - 6*a*b^2*c + 8*a^2*c^2)*n + (sqrt(-3)*(a^2*b^5 -
8*a^3*b^3*c + 16*a^4*b*c^2)*n^4 - (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)))*(-((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 - 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)

Sympy [F]

\[ \int \frac {x^{-1+\frac {n}{3}}}{a+b x^n+c x^{2 n}} \, dx=\int \frac {x^{\frac {n}{3} - 1}}{a + b x^{n} + c x^{2 n}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {x^{-1+\frac {n}{3}}}{a+b x^n+c x^{2 n}} \, dx=\int { \frac {x^{\frac {1}{3} \, n - 1}}{c x^{2 \, n} + b x^{n} + a} \,d x } \]

[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)

Giac [F]

\[ \int \frac {x^{-1+\frac {n}{3}}}{a+b x^n+c x^{2 n}} \, dx=\int { \frac {x^{\frac {1}{3} \, n - 1}}{c x^{2 \, n} + b x^{n} + a} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{-1+\frac {n}{3}}}{a+b x^n+c x^{2 n}} \, dx=\int \frac {x^{\frac {n}{3}-1}}{a+b\,x^n+c\,x^{2\,n}} \,d x \]

[In]

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

[Out]

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