3.6.60 \(\int \frac {x^{-1-\frac {n}{3}}}{a+b x^n+c x^{2 n}} \, dx\) [560]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.93, antiderivative size = 699, normalized size of antiderivative = 1.00, 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 = {1395, 1354, 1381, 1436, 206, 31, 648, 631, 210, 642} \begin {gather*} -\frac {\sqrt {3} \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \text {ArcTan}\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 ) \text {ArcTan}\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 {\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 {3 x^{-n/3}}{a n} \end {gather*}

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 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 1354

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 1381

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 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]

Rule 1436

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])

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+\frac {c}{x^6}+\frac {b}{x^3}} \, dx,x,x^{-n/3}\right )}{n}\\ &=-\frac {3 \text {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 \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.14, size = 107, normalized size = 0.15 \begin {gather*} -\frac {9 x^{-n/3}-\text {RootSum}\left [c+b \text {$\#$1}^3+a \text {$\#$1}^6\&,\frac {c n \log (x)+3 c \log \left (x^{-n/3}-\text {$\#$1}\right )+b n \log (x) \text {$\#$1}^3+3 b \log \left (x^{-n/3}-\text {$\#$1}\right ) \text {$\#$1}^3}{b \text {$\#$1}^2+2 a \text {$\#$1}^5}\&\right ]}{3 a n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(9/x^(n/3) - RootSum[c + b*#1^3 + a*#1^6 & , (c*n*Log[x] + 3*c*Log[x^(-1/3*n) - #1] + b*n*Log[x]*#1^3 + 3
*b*Log[x^(-1/3*n) - #1]*#1^3)/(b*#1^2 + 2*a*#1^5) & ])/(a*n)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.50, size = 534, normalized size = 0.76

method	result	size
risch	\(-\frac {3 x^{-\frac {n}{3}}}{a n}+\left (\munderset {\textit {\_R} =\RootOf \left (\left (64 a^{7} c^{3} n^{6}-48 a^{6} b^{2} c^{2} n^{6}+12 a^{5} b^{4} c \,n^{6}-a^{4} b^{6} n^{6}\right ) \textit {\_Z}^{6}+\left (-32 a^{3} b \,c^{3} n^{3}+32 a^{2} b^{3} c^{2} n^{3}-10 a \,b^{5} c \,n^{3}+b^{7} n^{3}\right ) \textit {\_Z}^{3}+c^{4}\right )}{\sum }\textit {\_R} \ln \left (x^{\frac {n}{3}}+\left (-\frac {64 n^{5} a^{8} c^{4}}{2 a^{2} c^{5}-4 a \,b^{2} c^{4}+b^{4} c^{3}}+\frac {112 n^{5} b^{2} a^{7} c^{3}}{2 a^{2} c^{5}-4 a \,b^{2} c^{4}+b^{4} c^{3}}-\frac {60 n^{5} b^{4} a^{6} c^{2}}{2 a^{2} c^{5}-4 a \,b^{2} c^{4}+b^{4} c^{3}}+\frac {13 n^{5} b^{6} a^{5} c}{2 a^{2} c^{5}-4 a \,b^{2} c^{4}+b^{4} c^{3}}-\frac {n^{5} b^{8} a^{4}}{2 a^{2} c^{5}-4 a \,b^{2} c^{4}+b^{4} c^{3}}\right ) \textit {\_R}^{5}+\left (\frac {28 n^{2} b \,a^{4} c^{4}}{2 a^{2} c^{5}-4 a \,b^{2} c^{4}+b^{4} c^{3}}-\frac {63 n^{2} b^{3} a^{3} c^{3}}{2 a^{2} c^{5}-4 a \,b^{2} c^{4}+b^{4} c^{3}}+\frac {42 n^{2} b^{5} a^{2} c^{2}}{2 a^{2} c^{5}-4 a \,b^{2} c^{4}+b^{4} c^{3}}-\frac {11 n^{2} b^{7} a c}{2 a^{2} c^{5}-4 a \,b^{2} c^{4}+b^{4} c^{3}}+\frac {n^{2} b^{9}}{2 a^{2} c^{5}-4 a \,b^{2} c^{4}+b^{4} c^{3}}\right ) \textit {\_R}^{2}\right )\right )\)	\(534\)

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]

-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 = 0.00, 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]

-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] Leaf count of result is larger than twice the leaf count of optimal. 6279 vs. \(2 (567) = 1134\).
time = 1.74, size = 6279, normalized size = 8.98 \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]

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^...

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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]

Exception raised: SystemError >> excessive stack use: stack is 3006 deep

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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 {1}{x^{\frac {n}{3}+1}\,\left (a+b\,x^n+c\,x^{2\,n}\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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