\(\int \frac {x^{-1+\frac {n}{4}}}{a+b x^n+c x^{2 n}} \, dx\) [556]

Optimal result

Integrand size = 26, antiderivative size = 353 \[ \int \frac {x^{-1+\frac {n}{4}}}{a+b x^n+c x^{2 n}} \, dx=\frac {2\ 2^{3/4} c^{3/4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \left (-b-\sqrt {b^2-4 a c}\right )^{3/4} n}-\frac {2\ 2^{3/4} c^{3/4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \left (-b+\sqrt {b^2-4 a c}\right )^{3/4} n}+\frac {2\ 2^{3/4} c^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \left (-b-\sqrt {b^2-4 a c}\right )^{3/4} n}-\frac {2\ 2^{3/4} c^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \left (-b+\sqrt {b^2-4 a c}\right )^{3/4} n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1395, 1361, 218, 214, 211} \[ \int \frac {x^{-1+\frac {n}{4}}}{a+b x^n+c x^{2 n}} \, dx=\frac {2\ 2^{3/4} c^{3/4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{n \sqrt {b^2-4 a c} \left (-\sqrt {b^2-4 a c}-b\right )^{3/4}}-\frac {2\ 2^{3/4} c^{3/4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{n \sqrt {b^2-4 a c} \left (\sqrt {b^2-4 a c}-b\right )^{3/4}}+\frac {2\ 2^{3/4} c^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{n \sqrt {b^2-4 a c} \left (-\sqrt {b^2-4 a c}-b\right )^{3/4}}-\frac {2\ 2^{3/4} c^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{n \sqrt {b^2-4 a c} \left (\sqrt {b^2-4 a c}-b\right )^{3/4}} \]

[In]

Int[x^(-1 + n/4)/(a + b*x^n + c*x^(2*n)),x]

[Out]

(2*2^(3/4)*c^(3/4)*ArcTan[(2^(1/4)*c^(1/4)*x^(n/4))/(-b - Sqrt[b^2 - 4*a*c])^(1/4)])/(Sqrt[b^2 - 4*a*c]*(-b -
Sqrt[b^2 - 4*a*c])^(3/4)*n) - (2*2^(3/4)*c^(3/4)*ArcTan[(2^(1/4)*c^(1/4)*x^(n/4))/(-b + Sqrt[b^2 - 4*a*c])^(1/
4)])/(Sqrt[b^2 - 4*a*c]*(-b + Sqrt[b^2 - 4*a*c])^(3/4)*n) + (2*2^(3/4)*c^(3/4)*ArcTanh[(2^(1/4)*c^(1/4)*x^(n/4
))/(-b - Sqrt[b^2 - 4*a*c])^(1/4)])/(Sqrt[b^2 - 4*a*c]*(-b - Sqrt[b^2 - 4*a*c])^(3/4)*n) - (2*2^(3/4)*c^(3/4)*
ArcTanh[(2^(1/4)*c^(1/4)*x^(n/4))/(-b + Sqrt[b^2 - 4*a*c])^(1/4)])/(Sqrt[b^2 - 4*a*c]*(-b + Sqrt[b^2 - 4*a*c])
^(3/4)*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 214

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

Rule 218

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !Gt
Q[a/b, 0]

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 {4 \text {Subst}\left (\int \frac {1}{a+b x^4+c x^8} \, dx,x,x^{n/4}\right )}{n} \\ & = \frac {(4 c) \text {Subst}\left (\int \frac {1}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^4} \, dx,x,x^{n/4}\right )}{\sqrt {b^2-4 a c} n}-\frac {(4 c) \text {Subst}\left (\int \frac {1}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^4} \, dx,x,x^{n/4}\right )}{\sqrt {b^2-4 a c} n} \\ & = \frac {(4 c) \text {Subst}\left (\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}}-\sqrt {2} \sqrt {c} x^2} \, dx,x,x^{n/4}\right )}{\sqrt {b^2-4 a c} \sqrt {-b-\sqrt {b^2-4 a c}} n}+\frac {(4 c) \text {Subst}\left (\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}}+\sqrt {2} \sqrt {c} x^2} \, dx,x,x^{n/4}\right )}{\sqrt {b^2-4 a c} \sqrt {-b-\sqrt {b^2-4 a c}} n}-\frac {(4 c) \text {Subst}\left (\int \frac {1}{\sqrt {-b+\sqrt {b^2-4 a c}}-\sqrt {2} \sqrt {c} x^2} \, dx,x,x^{n/4}\right )}{\sqrt {b^2-4 a c} \sqrt {-b+\sqrt {b^2-4 a c}} n}-\frac {(4 c) \text {Subst}\left (\int \frac {1}{\sqrt {-b+\sqrt {b^2-4 a c}}+\sqrt {2} \sqrt {c} x^2} \, dx,x,x^{n/4}\right )}{\sqrt {b^2-4 a c} \sqrt {-b+\sqrt {b^2-4 a c}} n} \\ & = \frac {2\ 2^{3/4} c^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \left (-b-\sqrt {b^2-4 a c}\right )^{3/4} n}-\frac {2\ 2^{3/4} c^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \left (-b+\sqrt {b^2-4 a c}\right )^{3/4} n}+\frac {2\ 2^{3/4} c^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \left (-b-\sqrt {b^2-4 a c}\right )^{3/4} n}-\frac {2\ 2^{3/4} c^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \left (-b+\sqrt {b^2-4 a c}\right )^{3/4} n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.70 (sec) , antiderivative size = 340, normalized size of antiderivative = 0.96 \[ \int \frac {x^{-1+\frac {n}{4}}}{a+b x^n+c x^{2 n}} \, dx=\frac {2\ 2^{3/4} c^{3/4} \left (-\frac {\sqrt [4]{-b-\sqrt {b^2-4 a c}} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{b^2-4 a c+b \sqrt {b^2-4 a c}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \left (-b+\sqrt {b^2-4 a c}\right )^{3/4}}-\frac {\sqrt [4]{-b-\sqrt {b^2-4 a c}} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{b^2-4 a c+b \sqrt {b^2-4 a c}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \left (-b+\sqrt {b^2-4 a c}\right )^{3/4}}\right )}{n} \]

[In]

Integrate[x^(-1 + n/4)/(a + b*x^n + c*x^(2*n)),x]

[Out]

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

Maple [C] (verified)

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

Time = 1.04 (sec) , antiderivative size = 280, normalized size of antiderivative = 0.79

method	result	size
risch	\(\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (256 a^{7} c^{4} n^{8}-256 a^{6} b^{2} c^{3} n^{8}+96 a^{5} b^{4} c^{2} n^{8}-16 a^{4} b^{6} c \,n^{8}+a^{3} b^{8} n^{8}\right ) \textit {\_Z}^{8}+\left (-48 a^{3} b \,c^{3} n^{4}+40 a^{2} b^{3} c^{2} n^{4}-11 a \,b^{5} c \,n^{4}+b^{7} n^{4}\right ) \textit {\_Z}^{4}+c^{3}\right )}{\sum }\textit {\_R} \ln \left (x^{\frac {n}{4}}+\left (\frac {16 n^{5} b \,a^{5} c^{2}}{a \,c^{2}-b^{2} c}-\frac {8 n^{5} b^{3} a^{4} c}{a \,c^{2}-b^{2} c}+\frac {n^{5} b^{5} a^{3}}{a \,c^{2}-b^{2} c}\right ) \textit {\_R}^{5}+\left (\frac {2 n \,a^{2} c^{2}}{a \,c^{2}-b^{2} c}-\frac {4 n \,b^{2} a c}{a \,c^{2}-b^{2} c}+\frac {n \,b^{4}}{a \,c^{2}-b^{2} c}\right ) \textit {\_R} \right )\)	\(280\)

[In]

int(x^(-1+1/4*n)/(a+b*x^n+c*x^(2*n)),x,method=_RETURNVERBOSE)

[Out]

sum(_R*ln(x^(1/4*n)+(16/(a*c^2-b^2*c)*n^5*b*a^5*c^2-8/(a*c^2-b^2*c)*n^5*b^3*a^4*c+1/(a*c^2-b^2*c)*n^5*b^5*a^3)
*_R^5+(2/(a*c^2-b^2*c)*n*a^2*c^2-4/(a*c^2-b^2*c)*n*b^2*a*c+1/(a*c^2-b^2*c)*n*b^4)*_R),_R=RootOf((256*a^7*c^4*n
^8-256*a^6*b^2*c^3*n^8+96*a^5*b^4*c^2*n^8-16*a^4*b^6*c*n^8+a^3*b^8*n^8)*_Z^8+(-48*a^3*b*c^3*n^4+40*a^2*b^3*c^2
*n^4-11*a*b^5*c*n^4+b^7*n^4)*_Z^4+c^3))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3481 vs. \(2 (273) = 546\).

Time = 0.40 (sec) , antiderivative size = 3481, normalized size of antiderivative = 9.86 \[ \int \frac {x^{-1+\frac {n}{4}}}{a+b x^n+c x^{2 n}} \, dx=\text {Too large to display} \]

[In]

integrate(x^(-1+1/4*n)/(a+b*x^n+c*x^(2*n)),x, algorithm="fricas")

[Out]

1/2*sqrt(2)*sqrt(sqrt(2)*sqrt(-((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^
6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*n^8)) + b^3 - 3*a*b*c)/((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^
2)*n^4)))*log(-(4*(b^2*c - a*c^2)*x*x^(1/4*n - 1) + sqrt(2)*((a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*n^5*sqrt((
b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*n^8)) - (b^4 - 5*a*b^2*c +
4*a^2*c^2)*n)*sqrt(sqrt(2)*sqrt(-((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((
a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*n^8)) + b^3 - 3*a*b*c)/((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*
c^2)*n^4))))/x) - 1/2*sqrt(2)*sqrt(sqrt(2)*sqrt(-((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4*sqrt((b^4 - 2*a*b^2
*c + a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*n^8)) + b^3 - 3*a*b*c)/((a^3*b^4 - 8*a^4
*b^2*c + 16*a^5*c^2)*n^4)))*log(-(4*(b^2*c - a*c^2)*x*x^(1/4*n - 1) - sqrt(2)*((a^3*b^5 - 8*a^4*b^3*c + 16*a^5
*b*c^2)*n^5*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*n^8)) - (
b^4 - 5*a*b^2*c + 4*a^2*c^2)*n)*sqrt(sqrt(2)*sqrt(-((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4*sqrt((b^4 - 2*a*b
^2*c + a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*n^8)) + b^3 - 3*a*b*c)/((a^3*b^4 - 8*a
^4*b^2*c + 16*a^5*c^2)*n^4))))/x) + 1/2*sqrt(2)*sqrt(-sqrt(2)*sqrt(-((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4*
sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*n^8)) + b^3 - 3*a*b*c
)/((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4)))*log(-(4*(b^2*c - a*c^2)*x*x^(1/4*n - 1) + sqrt(2)*((a^3*b^5 - 8
*a^4*b^3*c + 16*a^5*b*c^2)*n^5*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64
*a^9*c^3)*n^8)) - (b^4 - 5*a*b^2*c + 4*a^2*c^2)*n)*sqrt(-sqrt(2)*sqrt(-((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n
^4*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*n^8)) + b^3 - 3*a*
b*c)/((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4))))/x) - 1/2*sqrt(2)*sqrt(-sqrt(2)*sqrt(-((a^3*b^4 - 8*a^4*b^2*
c + 16*a^5*c^2)*n^4*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*n
^8)) + b^3 - 3*a*b*c)/((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4)))*log(-(4*(b^2*c - a*c^2)*x*x^(1/4*n - 1) - s
qrt(2)*((a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*n^5*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c +
48*a^8*b^2*c^2 - 64*a^9*c^3)*n^8)) - (b^4 - 5*a*b^2*c + 4*a^2*c^2)*n)*sqrt(-sqrt(2)*sqrt(-((a^3*b^4 - 8*a^4*b
^2*c + 16*a^5*c^2)*n^4*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3
)*n^8)) + b^3 - 3*a*b*c)/((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4))))/x) - 1/2*sqrt(2)*sqrt(sqrt(2)*sqrt(((a^
3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c
^2 - 64*a^9*c^3)*n^8)) - b^3 + 3*a*b*c)/((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4)))*log(-(4*(b^2*c - a*c^2)*x
*x^(1/4*n - 1) + sqrt(2)*((a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*n^5*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^
6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*n^8)) + (b^4 - 5*a*b^2*c + 4*a^2*c^2)*n)*sqrt(sqrt(2)*sqrt(((a
^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*
c^2 - 64*a^9*c^3)*n^8)) - b^3 + 3*a*b*c)/((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4))))/x) + 1/2*sqrt(2)*sqrt(s
qrt(2)*sqrt(((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*
c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*n^8)) - b^3 + 3*a*b*c)/((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4)))*log(-(4*(
b^2*c - a*c^2)*x*x^(1/4*n - 1) - sqrt(2)*((a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*n^5*sqrt((b^4 - 2*a*b^2*c + a
^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*n^8)) + (b^4 - 5*a*b^2*c + 4*a^2*c^2)*n)*sqrt(
sqrt(2)*sqrt(((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^6 - 12*a^7*b^4
*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*n^8)) - b^3 + 3*a*b*c)/((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4))))/x) - 1/
2*sqrt(2)*sqrt(-sqrt(2)*sqrt(((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*
b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*n^8)) - b^3 + 3*a*b*c)/((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)
*n^4)))*log(-(4*(b^2*c - a*c^2)*x*x^(1/4*n - 1) + sqrt(2)*((a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*n^5*sqrt((b^
4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*n^8)) + (b^4 - 5*a*b^2*c + 4*
a^2*c^2)*n)*sqrt(-sqrt(2)*sqrt(((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^
6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*n^8)) - b^3 + 3*a*b*c)/((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^
2)*n^4))))/x) + 1/2*sqrt(2)*sqrt(-sqrt(2)*sqrt(((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4*sqrt((b^4 - 2*a*b^2*c
+ a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*n^8)) - b^3 + 3*a*b*c)/((a^3*b^4 - 8*a^4*b
^2*c + 16*a^5*c^2)*n^4)))*log(-(4*(b^2*c - a*c^2)*x*x^(1/4*n - 1) - sqrt(2)*((a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b
*c^2)*n^5*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*n^8)) + (b^
4 - 5*a*b^2*c + 4*a^2*c^2)*n)*sqrt(-sqrt(2)*sqrt(((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4*sqrt((b^4 - 2*a*b^2
*c + a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*n^8)) - b^3 + 3*a*b*c)/((a^3*b^4 - 8*a^4
*b^2*c + 16*a^5*c^2)*n^4))))/x)

Sympy [F]

\[ \int \frac {x^{-1+\frac {n}{4}}}{a+b x^n+c x^{2 n}} \, dx=\int \frac {x^{\frac {n}{4} - 1}}{a + b x^{n} + c x^{2 n}}\, dx \]

[In]

integrate(x**(-1+1/4*n)/(a+b*x**n+c*x**(2*n)),x)

[Out]

Integral(x**(n/4 - 1)/(a + b*x**n + c*x**(2*n)), x)

Maxima [F]

\[ \int \frac {x^{-1+\frac {n}{4}}}{a+b x^n+c x^{2 n}} \, dx=\int { \frac {x^{\frac {1}{4} \, n - 1}}{c x^{2 \, n} + b x^{n} + a} \,d x } \]

[In]

integrate(x^(-1+1/4*n)/(a+b*x^n+c*x^(2*n)),x, algorithm="maxima")

[Out]

integrate(x^(1/4*n - 1)/(c*x^(2*n) + b*x^n + a), x)

Giac [F]

\[ \int \frac {x^{-1+\frac {n}{4}}}{a+b x^n+c x^{2 n}} \, dx=\int { \frac {x^{\frac {1}{4} \, n - 1}}{c x^{2 \, n} + b x^{n} + a} \,d x } \]

[In]

integrate(x^(-1+1/4*n)/(a+b*x^n+c*x^(2*n)),x, algorithm="giac")

[Out]

integrate(x^(1/4*n - 1)/(c*x^(2*n) + b*x^n + a), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{-1+\frac {n}{4}}}{a+b x^n+c x^{2 n}} \, dx=\int \frac {x^{\frac {n}{4}-1}}{a+b\,x^n+c\,x^{2\,n}} \,d x \]

[In]

int(x^(n/4 - 1)/(a + b*x^n + c*x^(2*n)),x)

[Out]

int(x^(n/4 - 1)/(a + b*x^n + c*x^(2*n)), x)