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

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

[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)

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Rubi [A]
time = 0.71, antiderivative size = 610, normalized size of antiderivative = 1.00, 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 = {1395, 1361, 206, 31, 648, 631, 210, 642} \begin {gather*} -\frac {2^{2/3} \sqrt {3} c^{2/3} \text {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} \text {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}} \end {gather*}

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

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.12, size = 62, normalized size = 0.10 \begin {gather*} \frac {\text {RootSum}\left [a+b \text {$\#$1}^3+c \text {$\#$1}^6\&,\frac {-n \log (x)+3 \log \left (x^{n/3}-\text {$\#$1}\right )}{b \text {$\#$1}^2+2 c \text {$\#$1}^5}\&\right ]}{3 n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

RootSum[a + b*#1^3 + c*#1^6 & , (-(n*Log[x]) + 3*Log[x^(n/3) - #1])/(b*#1^2 + 2*c*#1^5) & ]/(3*n)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.31, size = 260, normalized size = 0.43

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

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,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))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 4699 vs. \(2 (465) = 930\).
time = 0.61, size = 4699, normalized size = 7.70 \begin {gather*} \text {Too large to display} \end {gather*}

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...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^{\frac {n}{3}-1}}{a+b\,x^n+c\,x^{2\,n}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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