\(\int \frac {x^{-1-\frac {n}{4}}}{a+b x^n+c x^{2 n}} \, dx\) [561]

Optimal result

Integrand size = 26, antiderivative size = 414 \[ \int \frac {x^{-1-\frac {n}{4}}}{a+b x^n+c x^{2 n}} \, dx=-\frac {4 x^{-n/4}}{a n}-\frac {2^{3/4} \left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{a^{5/4} \left (-b-\sqrt {b^2-4 a c}\right )^{3/4} n}-\frac {2^{3/4} \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{a^{5/4} \left (-b+\sqrt {b^2-4 a c}\right )^{3/4} n}-\frac {2^{3/4} \left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{a^{5/4} \left (-b-\sqrt {b^2-4 a c}\right )^{3/4} n}-\frac {2^{3/4} \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{a^{5/4} \left (-b+\sqrt {b^2-4 a c}\right )^{3/4} n} \]

[Out]

-4/a/n/(x^(1/4*n))-2^(3/4)*arctan(2^(1/4)*a^(1/4)/(x^(1/4*n))/(-b-(-4*a*c+b^2)^(1/2))^(1/4))*(b+(-2*a*c+b^2)/(
-4*a*c+b^2)^(1/2))/a^(5/4)/n/(-b-(-4*a*c+b^2)^(1/2))^(3/4)-2^(3/4)*arctanh(2^(1/4)*a^(1/4)/(x^(1/4*n))/(-b-(-4
*a*c+b^2)^(1/2))^(1/4))*(b+(-2*a*c+b^2)/(-4*a*c+b^2)^(1/2))/a^(5/4)/n/(-b-(-4*a*c+b^2)^(1/2))^(3/4)-2^(3/4)*ar
ctan(2^(1/4)*a^(1/4)/(x^(1/4*n))/(-b+(-4*a*c+b^2)^(1/2))^(1/4))*(b+(2*a*c-b^2)/(-4*a*c+b^2)^(1/2))/a^(5/4)/n/(
-b+(-4*a*c+b^2)^(1/2))^(3/4)-2^(3/4)*arctanh(2^(1/4)*a^(1/4)/(x^(1/4*n))/(-b+(-4*a*c+b^2)^(1/2))^(1/4))*(b+(2*
a*c-b^2)/(-4*a*c+b^2)^(1/2))/a^(5/4)/n/(-b+(-4*a*c+b^2)^(1/2))^(3/4)

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {1395, 1354, 1381, 1436, 218, 214, 211} \[ \int \frac {x^{-1-\frac {n}{4}}}{a+b x^n+c x^{2 n}} \, dx=-\frac {2^{3/4} \left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{a^{5/4} n \left (-\sqrt {b^2-4 a c}-b\right )^{3/4}}-\frac {2^{3/4} \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{a^{5/4} n \left (\sqrt {b^2-4 a c}-b\right )^{3/4}}-\frac {2^{3/4} \left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{a^{5/4} n \left (-\sqrt {b^2-4 a c}-b\right )^{3/4}}-\frac {2^{3/4} \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{a^{5/4} n \left (\sqrt {b^2-4 a c}-b\right )^{3/4}}-\frac {4 x^{-n/4}}{a n} \]

[In]

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

[Out]

-4/(a*n*x^(n/4)) - (2^(3/4)*(b + (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(2^(1/4)*a^(1/4))/((-b - Sqrt[b^2 - 4
*a*c])^(1/4)*x^(n/4))])/(a^(5/4)*(-b - Sqrt[b^2 - 4*a*c])^(3/4)*n) - (2^(3/4)*(b - (b^2 - 2*a*c)/Sqrt[b^2 - 4*
a*c])*ArcTan[(2^(1/4)*a^(1/4))/((-b + Sqrt[b^2 - 4*a*c])^(1/4)*x^(n/4))])/(a^(5/4)*(-b + Sqrt[b^2 - 4*a*c])^(3
/4)*n) - (2^(3/4)*(b + (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*ArcTanh[(2^(1/4)*a^(1/4))/((-b - Sqrt[b^2 - 4*a*c])^(1
/4)*x^(n/4))])/(a^(5/4)*(-b - Sqrt[b^2 - 4*a*c])^(3/4)*n) - (2^(3/4)*(b - (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*Arc
Tanh[(2^(1/4)*a^(1/4))/((-b + Sqrt[b^2 - 4*a*c])^(1/4)*x^(n/4))])/(a^(5/4)*(-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 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*} \text {integral}& = -\frac {4 \text {Subst}\left (\int \frac {1}{a+\frac {c}{x^8}+\frac {b}{x^4}} \, dx,x,x^{-n/4}\right )}{n} \\ & = -\frac {4 \text {Subst}\left (\int \frac {x^8}{c+b x^4+a x^8} \, dx,x,x^{-n/4}\right )}{n} \\ & = -\frac {4 x^{-n/4}}{a n}+\frac {4 \text {Subst}\left (\int \frac {c+b x^4}{c+b x^4+a x^8} \, dx,x,x^{-n/4}\right )}{a n} \\ & = -\frac {4 x^{-n/4}}{a n}+\frac {\left (2 \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^4} \, dx,x,x^{-n/4}\right )}{a n}+\frac {\left (2 \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^4} \, dx,x,x^{-n/4}\right )}{a n} \\ & = -\frac {4 x^{-n/4}}{a n}-\frac {\left (2 \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b+\sqrt {b^2-4 a c}}-\sqrt {2} \sqrt {a} x^2} \, dx,x,x^{-n/4}\right )}{a \sqrt {-b+\sqrt {b^2-4 a c}} n}-\frac {\left (2 \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b+\sqrt {b^2-4 a c}}+\sqrt {2} \sqrt {a} x^2} \, dx,x,x^{-n/4}\right )}{a \sqrt {-b+\sqrt {b^2-4 a c}} n}-\frac {\left (2 \left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}}-\sqrt {2} \sqrt {a} x^2} \, dx,x,x^{-n/4}\right )}{a \sqrt {-b-\sqrt {b^2-4 a c}} n}-\frac {\left (2 \left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}}+\sqrt {2} \sqrt {a} x^2} \, dx,x,x^{-n/4}\right )}{a \sqrt {-b-\sqrt {b^2-4 a c}} n} \\ & = -\frac {4 x^{-n/4}}{a n}-\frac {2^{3/4} \left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{a^{5/4} \left (-b-\sqrt {b^2-4 a c}\right )^{3/4} n}-\frac {2^{3/4} \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{a^{5/4} \left (-b+\sqrt {b^2-4 a c}\right )^{3/4} n}-\frac {2^{3/4} \left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{a^{5/4} \left (-b-\sqrt {b^2-4 a c}\right )^{3/4} n}-\frac {2^{3/4} \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{a^{5/4} \left (-b+\sqrt {b^2-4 a c}\right )^{3/4} n} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.17 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.31 \[ \int \frac {x^{-1-\frac {n}{4}}}{a+b x^n+c x^{2 n}} \, dx=\frac {8 c x^{-n/4} \left (\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{4},1,\frac {3}{4},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )}{b^2-4 a c-b \sqrt {b^2-4 a c}}+\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{4},1,\frac {3}{4},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{b^2-4 a c+b \sqrt {b^2-4 a c}}\right )}{n} \]

[In]

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

[Out]

(8*c*(Hypergeometric2F1[-1/4, 1, 3/4, (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])]/(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])
+ Hypergeometric2F1[-1/4, 1, 3/4, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]/(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])))/(n
*x^(n/4))

Maple [C] (verified)

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

Time = 1.54 (sec) , antiderivative size = 630, normalized size of antiderivative = 1.52

method	result	size
risch	\(-\frac {4 x^{-\frac {n}{4}}}{a n}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (256 a^{9} c^{4} n^{8}-256 a^{8} b^{2} c^{3} n^{8}+96 a^{7} b^{4} c^{2} n^{8}-16 a^{6} b^{6} c \,n^{8}+a^{5} b^{8} n^{8}\right ) \textit {\_Z}^{8}+\left (80 a^{4} b \,c^{4} n^{4}-120 a^{3} b^{3} c^{3} n^{4}+61 a^{2} b^{5} c^{2} n^{4}-13 a \,b^{7} c \,n^{4}+b^{9} n^{4}\right ) \textit {\_Z}^{4}+c^{5}\right )}{\sum }\textit {\_R} \ln \left (x^{\frac {n}{4}}+\left (-\frac {128 n^{7} a^{10} c^{5}}{a^{2} c^{6}-3 a \,b^{2} c^{5}+b^{4} c^{4}}+\frac {352 n^{7} b^{2} a^{9} c^{4}}{a^{2} c^{6}-3 a \,b^{2} c^{5}+b^{4} c^{4}}-\frac {280 n^{7} b^{4} a^{8} c^{3}}{a^{2} c^{6}-3 a \,b^{2} c^{5}+b^{4} c^{4}}+\frac {98 n^{7} b^{6} a^{7} c^{2}}{a^{2} c^{6}-3 a \,b^{2} c^{5}+b^{4} c^{4}}-\frac {16 n^{7} b^{8} a^{6} c}{a^{2} c^{6}-3 a \,b^{2} c^{5}+b^{4} c^{4}}+\frac {n^{7} b^{10} a^{5}}{a^{2} c^{6}-3 a \,b^{2} c^{5}+b^{4} c^{4}}\right ) \textit {\_R}^{7}+\left (-\frac {36 n^{3} b \,a^{5} c^{5}}{a^{2} c^{6}-3 a \,b^{2} c^{5}+b^{4} c^{4}}+\frac {129 n^{3} b^{3} a^{4} c^{4}}{a^{2} c^{6}-3 a \,b^{2} c^{5}+b^{4} c^{4}}-\frac {138 n^{3} b^{5} a^{3} c^{3}}{a^{2} c^{6}-3 a \,b^{2} c^{5}+b^{4} c^{4}}+\frac {63 n^{3} b^{7} a^{2} c^{2}}{a^{2} c^{6}-3 a \,b^{2} c^{5}+b^{4} c^{4}}-\frac {13 n^{3} b^{9} a c}{a^{2} c^{6}-3 a \,b^{2} c^{5}+b^{4} c^{4}}+\frac {n^{3} b^{11}}{a^{2} c^{6}-3 a \,b^{2} c^{5}+b^{4} c^{4}}\right ) \textit {\_R}^{3}\right )\right )\)	\(630\)

[In]

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

[Out]

-4/a/n/(x^(1/4*n))+sum(_R*ln(x^(1/4*n)+(-128/(a^2*c^6-3*a*b^2*c^5+b^4*c^4)*n^7*a^10*c^5+352/(a^2*c^6-3*a*b^2*c
^5+b^4*c^4)*n^7*b^2*a^9*c^4-280/(a^2*c^6-3*a*b^2*c^5+b^4*c^4)*n^7*b^4*a^8*c^3+98/(a^2*c^6-3*a*b^2*c^5+b^4*c^4)
*n^7*b^6*a^7*c^2-16/(a^2*c^6-3*a*b^2*c^5+b^4*c^4)*n^7*b^8*a^6*c+1/(a^2*c^6-3*a*b^2*c^5+b^4*c^4)*n^7*b^10*a^5)*
_R^7+(-36/(a^2*c^6-3*a*b^2*c^5+b^4*c^4)*n^3*b*a^5*c^5+129/(a^2*c^6-3*a*b^2*c^5+b^4*c^4)*n^3*b^3*a^4*c^4-138/(a
^2*c^6-3*a*b^2*c^5+b^4*c^4)*n^3*b^5*a^3*c^3+63/(a^2*c^6-3*a*b^2*c^5+b^4*c^4)*n^3*b^7*a^2*c^2-13/(a^2*c^6-3*a*b
^2*c^5+b^4*c^4)*n^3*b^9*a*c+1/(a^2*c^6-3*a*b^2*c^5+b^4*c^4)*n^3*b^11)*_R^3),_R=RootOf((256*a^9*c^4*n^8-256*a^8
*b^2*c^3*n^8+96*a^7*b^4*c^2*n^8-16*a^6*b^6*c*n^8+a^5*b^8*n^8)*_Z^8+(80*a^4*b*c^4*n^4-120*a^3*b^3*c^3*n^4+61*a^
2*b^5*c^2*n^4-13*a*b^7*c*n^4+b^9*n^4)*_Z^4+c^5))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4375 vs. \(2 (342) = 684\).

Time = 0.44 (sec) , antiderivative size = 4375, normalized size of antiderivative = 10.57 \[ \int \frac {x^{-1-\frac {n}{4}}}{a+b x^n+c x^{2 n}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/2*(sqrt(2)*a*n*sqrt(sqrt(2)*sqrt(-((a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*n^4*sqrt((b^8 - 6*a*b^6*c + 11*a^2*b
^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)/((a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)*n^8)) + b^5 - 5*
a*b^3*c + 5*a^2*b*c^2)/((a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*n^4)))*log((4*(b^4*c - 3*a*b^2*c^2 + a^2*c^3)*x*x
^(-1/4*n - 1) + sqrt(2)*((a^5*b^5 - 8*a^6*b^3*c + 16*a^7*b*c^2)*n^5*sqrt((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6
*a^3*b^2*c^3 + a^4*c^4)/((a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)*n^8)) - (b^6 - 7*a*b^4*c +
13*a^2*b^2*c^2 - 4*a^3*c^3)*n)*sqrt(sqrt(2)*sqrt(-((a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*n^4*sqrt((b^8 - 6*a*b
^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)/((a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)*n
^8)) + b^5 - 5*a*b^3*c + 5*a^2*b*c^2)/((a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*n^4))))/x) - sqrt(2)*a*n*sqrt(sqrt
(2)*sqrt(-((a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*n^4*sqrt((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a
^4*c^4)/((a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)*n^8)) + b^5 - 5*a*b^3*c + 5*a^2*b*c^2)/((a
^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*n^4)))*log((4*(b^4*c - 3*a*b^2*c^2 + a^2*c^3)*x*x^(-1/4*n - 1) - sqrt(2)*((
a^5*b^5 - 8*a^6*b^3*c + 16*a^7*b*c^2)*n^5*sqrt((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)/((
a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)*n^8)) - (b^6 - 7*a*b^4*c + 13*a^2*b^2*c^2 - 4*a^3*c^
3)*n)*sqrt(sqrt(2)*sqrt(-((a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*n^4*sqrt((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*
a^3*b^2*c^3 + a^4*c^4)/((a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)*n^8)) + b^5 - 5*a*b^3*c + 5
*a^2*b*c^2)/((a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*n^4))))/x) + sqrt(2)*a*n*sqrt(-sqrt(2)*sqrt(-((a^5*b^4 - 8*a
^6*b^2*c + 16*a^7*c^2)*n^4*sqrt((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)/((a^10*b^6 - 12*a
^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)*n^8)) + b^5 - 5*a*b^3*c + 5*a^2*b*c^2)/((a^5*b^4 - 8*a^6*b^2*c + 16
*a^7*c^2)*n^4)))*log((4*(b^4*c - 3*a*b^2*c^2 + a^2*c^3)*x*x^(-1/4*n - 1) + sqrt(2)*((a^5*b^5 - 8*a^6*b^3*c + 1
6*a^7*b*c^2)*n^5*sqrt((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)/((a^10*b^6 - 12*a^11*b^4*c
+ 48*a^12*b^2*c^2 - 64*a^13*c^3)*n^8)) - (b^6 - 7*a*b^4*c + 13*a^2*b^2*c^2 - 4*a^3*c^3)*n)*sqrt(-sqrt(2)*sqrt(
-((a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*n^4*sqrt((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)/(
(a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)*n^8)) + b^5 - 5*a*b^3*c + 5*a^2*b*c^2)/((a^5*b^4 -
8*a^6*b^2*c + 16*a^7*c^2)*n^4))))/x) - sqrt(2)*a*n*sqrt(-sqrt(2)*sqrt(-((a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*n
^4*sqrt((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)/((a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*
c^2 - 64*a^13*c^3)*n^8)) + b^5 - 5*a*b^3*c + 5*a^2*b*c^2)/((a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*n^4)))*log((4*
(b^4*c - 3*a*b^2*c^2 + a^2*c^3)*x*x^(-1/4*n - 1) - sqrt(2)*((a^5*b^5 - 8*a^6*b^3*c + 16*a^7*b*c^2)*n^5*sqrt((b
^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)/((a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a
^13*c^3)*n^8)) - (b^6 - 7*a*b^4*c + 13*a^2*b^2*c^2 - 4*a^3*c^3)*n)*sqrt(-sqrt(2)*sqrt(-((a^5*b^4 - 8*a^6*b^2*c
+ 16*a^7*c^2)*n^4*sqrt((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)/((a^10*b^6 - 12*a^11*b^4*
c + 48*a^12*b^2*c^2 - 64*a^13*c^3)*n^8)) + b^5 - 5*a*b^3*c + 5*a^2*b*c^2)/((a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2
)*n^4))))/x) - sqrt(2)*a*n*sqrt(sqrt(2)*sqrt(((a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*n^4*sqrt((b^8 - 6*a*b^6*c +
11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)/((a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)*n^8)) -
b^5 + 5*a*b^3*c - 5*a^2*b*c^2)/((a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*n^4)))*log((4*(b^4*c - 3*a*b^2*c^2 + a^2
*c^3)*x*x^(-1/4*n - 1) + sqrt(2)*((a^5*b^5 - 8*a^6*b^3*c + 16*a^7*b*c^2)*n^5*sqrt((b^8 - 6*a*b^6*c + 11*a^2*b^
4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)/((a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)*n^8)) + (b^6 - 7*
a*b^4*c + 13*a^2*b^2*c^2 - 4*a^3*c^3)*n)*sqrt(sqrt(2)*sqrt(((a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*n^4*sqrt((b^8
- 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)/((a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^1
3*c^3)*n^8)) - b^5 + 5*a*b^3*c - 5*a^2*b*c^2)/((a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*n^4))))/x) + sqrt(2)*a*n*s
qrt(sqrt(2)*sqrt(((a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*n^4*sqrt((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*
c^3 + a^4*c^4)/((a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)*n^8)) - b^5 + 5*a*b^3*c - 5*a^2*b*c
^2)/((a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*n^4)))*log((4*(b^4*c - 3*a*b^2*c^2 + a^2*c^3)*x*x^(-1/4*n - 1) - sqr
t(2)*((a^5*b^5 - 8*a^6*b^3*c + 16*a^7*b*c^2)*n^5*sqrt((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*
c^4)/((a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)*n^8)) + (b^6 - 7*a*b^4*c + 13*a^2*b^2*c^2 - 4
*a^3*c^3)*n)*sqrt(sqrt(2)*sqrt(((a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*n^4*sqrt((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^
2 - 6*a^3*b^2*c^3 + a^4*c^4)/((a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)*n^8)) - b^5 + 5*a*b^3
*c - 5*a^2*b*c^2)/((a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*n^4))))/x) - sqrt(2)*a*n*sqrt(-sqrt(2)*sqrt(((a^5*b^4
- 8*a^6*b^2*c + 16*a^7*c^2)*n^4*sqrt((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)/((a^10*b^6 -
12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)*n^8)) - b^5 + 5*a*b^3*c - 5*a^2*b*c^2)/((a^5*b^4 - 8*a^6*b^2*c
+ 16*a^7*c^2)*n^4)))*log((4*(b^4*c - 3*a*b^2*c^2 + a^2*c^3)*x*x^(-1/4*n - 1) + sqrt(2)*((a^5*b^5 - 8*a^6*b^3*
c + 16*a^7*b*c^2)*n^5*sqrt((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)/((a^10*b^6 - 12*a^11*b
^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)*n^8)) + (b^6 - 7*a*b^4*c + 13*a^2*b^2*c^2 - 4*a^3*c^3)*n)*sqrt(-sqrt(2)*
sqrt(((a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*n^4*sqrt((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^
4)/((a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)*n^8)) - b^5 + 5*a*b^3*c - 5*a^2*b*c^2)/((a^5*b^
4 - 8*a^6*b^2*c + 16*a^7*c^2)*n^4))))/x) + sqrt(2)*a*n*sqrt(-sqrt(2)*sqrt(((a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2
)*n^4*sqrt((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)/((a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b
^2*c^2 - 64*a^13*c^3)*n^8)) - b^5 + 5*a*b^3*c - 5*a^2*b*c^2)/((a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*n^4)))*log(
(4*(b^4*c - 3*a*b^2*c^2 + a^2*c^3)*x*x^(-1/4*n - 1) - sqrt(2)*((a^5*b^5 - 8*a^6*b^3*c + 16*a^7*b*c^2)*n^5*sqrt
((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)/((a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 6
4*a^13*c^3)*n^8)) + (b^6 - 7*a*b^4*c + 13*a^2*b^2*c^2 - 4*a^3*c^3)*n)*sqrt(-sqrt(2)*sqrt(((a^5*b^4 - 8*a^6*b^2
*c + 16*a^7*c^2)*n^4*sqrt((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)/((a^10*b^6 - 12*a^11*b^
4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)*n^8)) - b^5 + 5*a*b^3*c - 5*a^2*b*c^2)/((a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c
^2)*n^4))))/x) - 8*x*x^(-1/4*n - 1))/(a*n)

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]

-4/(a*n*x^(1/4*n)) - integrate((c*x^(7/4*n) + b*x^(3/4*n))/(a*c*x*x^(2*n) + a*b*x*x^n + a^2*x), 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. \[ \int \frac {x^{-1-\frac {n}{4}}}{a+b x^n+c x^{2 n}} \, dx=\int \frac {1}{x^{\frac {n}{4}+1}\,\left (a+b\,x^n+c\,x^{2\,n}\right )} \,d x \]

[In]

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

[Out]

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