3.6.59 \(\int \frac {x^{-1-\frac {n}{2}}}{a+b x^n+c x^{2 n}} \, dx\) [559]

Optimal. Leaf size=205 \[ -\frac {2 x^{-n/2}}{a n}+\frac {\sqrt {2} \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x^{-n/2}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{a^{3/2} \sqrt {b-\sqrt {b^2-4 a c}} n}+\frac {\sqrt {2} \left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x^{-n/2}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{a^{3/2} \sqrt {b+\sqrt {b^2-4 a c}} n} \]

[Out]

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

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Rubi [A]
time = 0.27, antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1395, 1354, 1136, 1180, 211} \begin {gather*} \frac {\sqrt {2} \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {a} x^{-n/2}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{a^{3/2} n \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {a} x^{-n/2}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{a^{3/2} n \sqrt {\sqrt {b^2-4 a c}+b}}-\frac {2 x^{-n/2}}{a n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 211

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

Rule 1136

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[d^3*(d*x)^(m - 3)*((a + b*
x^2 + c*x^4)^(p + 1)/(c*(m + 4*p + 1))), x] - Dist[d^4/(c*(m + 4*p + 1)), Int[(d*x)^(m - 4)*Simp[a*(m - 3) + b
*(m + 2*p - 1)*x^2, x]*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b^2 - 4*a*c, 0] && Gt
Q[m, 3] && NeQ[m + 4*p + 1, 0] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])

Rule 1180

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[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 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}{2}}}{a+b x^n+c x^{2 n}} \, dx &=-\frac {2 \text {Subst}\left (\int \frac {1}{a+\frac {c}{x^4}+\frac {b}{x^2}} \, dx,x,x^{-n/2}\right )}{n}\\ &=-\frac {2 \text {Subst}\left (\int \frac {x^4}{c+b x^2+a x^4} \, dx,x,x^{-n/2}\right )}{n}\\ &=-\frac {2 x^{-n/2}}{a n}+\frac {2 \text {Subst}\left (\int \frac {c+b x^2}{c+b x^2+a x^4} \, dx,x,x^{-n/2}\right )}{a n}\\ &=-\frac {2 x^{-n/2}}{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 {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+a x^2} \, dx,x,x^{-n/2}\right )}{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 {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+a x^2} \, dx,x,x^{-n/2}\right )}{a n}\\ &=-\frac {2 x^{-n/2}}{a n}+\frac {\sqrt {2} \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x^{-n/2}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{a^{3/2} \sqrt {b-\sqrt {b^2-4 a c}} n}+\frac {\sqrt {2} \left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x^{-n/2}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{a^{3/2} \sqrt {b+\sqrt {b^2-4 a c}} 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 = 0.13, size = 105, normalized size = 0.51 \begin {gather*} -\frac {4 x^{-n/2}-\text {RootSum}\left [c+b \text {$\#$1}^2+a \text {$\#$1}^4\&,\frac {c n \log (x)+2 c \log \left (x^{-n/2}-\text {$\#$1}\right )+b n \log (x) \text {$\#$1}^2+2 b \log \left (x^{-n/2}-\text {$\#$1}\right ) \text {$\#$1}^2}{b \text {$\#$1}+2 a \text {$\#$1}^3}\&\right ]}{2 a n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(4/x^(n/2) - RootSum[c + b*#1^2 + a*#1^4 & , (c*n*Log[x] + 2*c*Log[x^(-1/2*n) - #1] + b*n*Log[x]*#1^2 + 2
*b*Log[x^(-1/2*n) - #1]*#1^2)/(b*#1 + 2*a*#1^3) & ])/(a*n)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.23, size = 268, normalized size = 1.31

method	result	size
risch	\(-\frac {2 x^{-\frac {n}{2}}}{a n}+\left (\munderset {\textit {\_R} =\RootOf \left (\left (16 a^{5} c^{2} n^{4}-8 a^{4} b^{2} c \,n^{4}+a^{3} b^{4} n^{4}\right ) \textit {\_Z}^{4}+\left (12 a^{2} b \,c^{2} n^{2}-7 a \,b^{3} c \,n^{2}+b^{5} n^{2}\right ) \textit {\_Z}^{2}+c^{3}\right )}{\sum }\textit {\_R} \ln \left (x^{\frac {n}{2}}+\left (-\frac {8 n^{3} a^{5} c^{2}}{a \,c^{3}-b^{2} c^{2}}+\frac {6 n^{3} b^{2} a^{4} c}{a \,c^{3}-b^{2} c^{2}}-\frac {n^{3} b^{4} a^{3}}{a \,c^{3}-b^{2} c^{2}}\right ) \textit {\_R}^{3}+\left (-\frac {5 n b \,a^{2} c^{2}}{a \,c^{3}-b^{2} c^{2}}+\frac {5 n \,b^{3} a c}{a \,c^{3}-b^{2} c^{2}}-\frac {n \,b^{5}}{a \,c^{3}-b^{2} c^{2}}\right ) \textit {\_R} \right )\right )\)	\(268\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/a/n/(x^(1/2*n))+sum(_R*ln(x^(1/2*n)+(-8/(a*c^3-b^2*c^2)*n^3*a^5*c^2+6/(a*c^3-b^2*c^2)*n^3*b^2*a^4*c-1/(a*c^
3-b^2*c^2)*n^3*b^4*a^3)*_R^3+(-5/(a*c^3-b^2*c^2)*n*b*a^2*c^2+5/(a*c^3-b^2*c^2)*n*b^3*a*c-1/(a*c^3-b^2*c^2)*n*b
^5)*_R),_R=RootOf((16*a^5*c^2*n^4-8*a^4*b^2*c*n^4+a^3*b^4*n^4)*_Z^4+(12*a^2*b*c^2*n^2-7*a*b^3*c*n^2+b^5*n^2)*_
Z^2+c^3))

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

[Out]

-2/(a*n*x^(1/2*n)) - integrate((c*x^(3/2*n) + b*x^(1/2*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. 1229 vs. \(2 (169) = 338\).
time = 0.41, size = 1229, normalized size = 6.00 \begin {gather*} \frac {\sqrt {2} a n \sqrt {-\frac {{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} n^{2} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{{\left (a^{6} b^{2} - 4 \, a^{7} c\right )} n^{4}}} + b^{3} - 3 \, a b c}{{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} n^{2}}} \log \left (-\frac {4 \, {\left (b^{2} c - a c^{2}\right )} x x^{-\frac {1}{2} \, n - 1} + \sqrt {2} {\left ({\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} n^{3} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{{\left (a^{6} b^{2} - 4 \, a^{7} c\right )} n^{4}}} - {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} n\right )} \sqrt {-\frac {{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} n^{2} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{{\left (a^{6} b^{2} - 4 \, a^{7} c\right )} n^{4}}} + b^{3} - 3 \, a b c}{{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} n^{2}}}}{x}\right ) - \sqrt {2} a n \sqrt {-\frac {{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} n^{2} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{{\left (a^{6} b^{2} - 4 \, a^{7} c\right )} n^{4}}} + b^{3} - 3 \, a b c}{{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} n^{2}}} \log \left (-\frac {4 \, {\left (b^{2} c - a c^{2}\right )} x x^{-\frac {1}{2} \, n - 1} - \sqrt {2} {\left ({\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} n^{3} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{{\left (a^{6} b^{2} - 4 \, a^{7} c\right )} n^{4}}} - {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} n\right )} \sqrt {-\frac {{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} n^{2} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{{\left (a^{6} b^{2} - 4 \, a^{7} c\right )} n^{4}}} + b^{3} - 3 \, a b c}{{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} n^{2}}}}{x}\right ) - \sqrt {2} a n \sqrt {\frac {{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} n^{2} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{{\left (a^{6} b^{2} - 4 \, a^{7} c\right )} n^{4}}} - b^{3} + 3 \, a b c}{{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} n^{2}}} \log \left (-\frac {4 \, {\left (b^{2} c - a c^{2}\right )} x x^{-\frac {1}{2} \, n - 1} + \sqrt {2} {\left ({\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} n^{3} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{{\left (a^{6} b^{2} - 4 \, a^{7} c\right )} n^{4}}} + {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} n\right )} \sqrt {\frac {{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} n^{2} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{{\left (a^{6} b^{2} - 4 \, a^{7} c\right )} n^{4}}} - b^{3} + 3 \, a b c}{{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} n^{2}}}}{x}\right ) + \sqrt {2} a n \sqrt {\frac {{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} n^{2} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{{\left (a^{6} b^{2} - 4 \, a^{7} c\right )} n^{4}}} - b^{3} + 3 \, a b c}{{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} n^{2}}} \log \left (-\frac {4 \, {\left (b^{2} c - a c^{2}\right )} x x^{-\frac {1}{2} \, n - 1} - \sqrt {2} {\left ({\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} n^{3} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{{\left (a^{6} b^{2} - 4 \, a^{7} c\right )} n^{4}}} + {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} n\right )} \sqrt {\frac {{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} n^{2} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{{\left (a^{6} b^{2} - 4 \, a^{7} c\right )} n^{4}}} - b^{3} + 3 \, a b c}{{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} n^{2}}}}{x}\right ) - 4 \, x x^{-\frac {1}{2} \, n - 1}}{2 \, a n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

[Out]

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

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