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

Optimal. Leaf size=414 \[ -\frac {4 x^{-n/4}}{a n}-\frac {2^{3/4} \left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{a^{5/4} \left (-b-\sqrt {b^2-4 a c}\right )^{3/4} n}-\frac {2^{3/4} \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{a^{5/4} \left (-b+\sqrt {b^2-4 a c}\right )^{3/4} n}-\frac {2^{3/4} \left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{a^{5/4} \left (-b-\sqrt {b^2-4 a c}\right )^{3/4} n}-\frac {2^{3/4} \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{a^{5/4} \left (-b+\sqrt {b^2-4 a c}\right )^{3/4} n} \]

[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)

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Rubi [A]
time = 0.52, antiderivative size = 414, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {1395, 1354, 1381, 1436, 218, 214, 211} \begin {gather*} -\frac {2^{3/4} \left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) \text {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 ) \text {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 ) \tanh ^{-1}\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 ) \tanh ^{-1}\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} \end {gather*}

Antiderivative was successfully verified.

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

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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 = 105, normalized size = 0.25 \begin {gather*} \frac {-16 x^{-n/4}+\text {RootSum}\left [c+b \text {$\#$1}^4+a \text {$\#$1}^8\&,\frac {c n \log (x)+4 c \log \left (x^{-n/4}-\text {$\#$1}\right )+b n \log (x) \text {$\#$1}^4+4 b \log \left (x^{-n/4}-\text {$\#$1}\right ) \text {$\#$1}^4}{b \text {$\#$1}^3+2 a \text {$\#$1}^7}\&\right ]}{4 a n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-16/x^(n/4) + RootSum[c + b*#1^4 + a*#1^8 & , (c*n*Log[x] + 4*c*Log[x^(-1/4*n) - #1] + b*n*Log[x]*#1^4 + 4*b*
Log[x^(-1/4*n) - #1]*#1^4)/(b*#1^3 + 2*a*#1^7) & ])/(4*a*n)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.77, size = 630, normalized size = 1.52

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

Verification of antiderivative is not currently implemented for this CAS.

[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))

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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^(-1-1/4*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)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 5712 vs. \(2 (342) = 684\).
time = 0.98, size = 5712, normalized size = 13.80 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(4*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)))*arctan(1/16*sqrt(2)*(2*sqrt(2)*((a^5*b^14
*c - 19*a^6*b^12*c^2 + 147*a^7*b^10*c^3 - 590*a^8*b^8*c^4 + 1290*a^9*b^6*c^5 - 1464*a^10*b^4*c^6 + 736*a^11*b^
2*c^7 - 128*a^12*c^8)*n^7*x*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^15*c - 16*a*b^13*c^2 + 103*a^2*b^11*c^3 - 340*a^3*b^9*c
^4 + 605*a^4*b^7*c^5 - 554*a^5*b^5*c^6 + 224*a^6*b^3*c^7 - 32*a^7*b*c^8)*n^3*x)*x^(-1/4*n - 1)*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)) - sqrt(2)*((a^5*b^10 - 16*a^6*b^8*c + 98*a^7*b^6*c^2 - 280*a^8*b^4*c^3 + 352*a^9*b^2*c^4
- 128*a^10*c^5)*n^7*x*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^11 - 13*a*b^9*c + 63*a^2*b^7*c^2 - 138*a^3*b^5*c^3 + 128*a^4*
b^3*c^4 - 32*a^5*b*c^5)*n^3*x)*sqrt((4*(b^8*c^2 - 6*a*b^6*c^3 + 11*a^2*b^4*c^4 - 6*a^3*b^2*c^5 + a^4*c^6)*x^2*
x^(-1/2*n - 2) + sqrt(2)*((a^5*b^11 - 15*a^6*b^9*c + 85*a^7*b^7*c^2 - 220*a^8*b^5*c^3 + 240*a^9*b^3*c^4 - 64*a
^10*b*c^5)*n^6*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^12 - 12*a*b^10*c + 55*a^2*b^8*c^2 - 120*a^3*b^6*c^3 + 125*a^4*b^4*c^
4 - 54*a^5*b^2*c^5 + 8*a^6*c^6)*n^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^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)))*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)))/(b^8*c^5 - 6*a*b^6*c^6 + 11*a^2*b^4*c^7 - 6*a^3*b^2*c^
8 + a^4*c^9)) - 4*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)))*arctan(1/8*(2*((a^5*b^14*c -
19*a^6*b^12*c^2 + 147*a^7*b^10*c^3 - 590*a^8*b^8*c^4 + 1290*a^9*b^6*c^5 - 1464*a^10*b^4*c^6 + 736*a^11*b^2*c^7
- 128*a^12*c^8)*n^7*x*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^15*c - 16*a*b^13*c^2 + 103*a^2*b^11*c^3 - 340*a^3*b^9*c^4 +
605*a^4*b^7*c^5 - 554*a^5*b^5*c^6 + 224*a^6*b^3*c^7 - 32*a^7*b*c^8)*n^3*x)*x^(-1/4*n - 1)*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)))*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)) - ((a^5*b^10 - 16*a^6*b^8*c + 98*a^7*b^6*c
^2 - 280*a^8*b^4*c^3 + 352*a^9*b^2*c^4 - 128*a^10*c^5)*n^7*x*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^11 - 13*a*b^9*c + 63*a
^2*b^7*c^2 - 138*a^3*b^5*c^3 + 128*a^4*b^3*c^4 - 32*a^5*b*c^5)*n^3*x)*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)))*sqrt((4*(b^8*c^2 - 6*a*b^6*c^3 + 11*a^2*b^4*c^4 - 6*a^3*b^2*c^5 + a^4*c^6)*x^2*x^(-1/2*n - 2) - sqr
t(2)*((a^5*b^11 - 15*a^6*b^9*c + 85*a^7*b^7*c^2 - 220*a^8*b^5*c^3 + 240*a^9*b^3*c^4 - 64*a^10*b*c^5)*n^6*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^12 - 12*a*b^10*c + 55*a^2*b^8*c^2 - 120*a^3*b^6*c^3 + 125*a^4*b^4*c^4 - 54*a^5*b^2*c^5 +
8*a^6*c^6)*n^2)*sqrt(-((a^5*b^4 - 8*a^6*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**(-1-1/4*n)/(a+b*x**n+c*x**(2*n)),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 6191 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^(-1-1/4*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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{x^{\frac {n}{4}+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/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)

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