3.6.56 \(\int \frac {x^{-1+\frac {n}{4}}}{a+b x^n+c x^{2 n}} \, dx\) [556]
Optimal. Leaf size=353 \[ \frac {2\ 2^{3/4} c^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \left (-b-\sqrt {b^2-4 a c}\right )^{3/4} n}-\frac {2\ 2^{3/4} c^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \left (-b+\sqrt {b^2-4 a c}\right )^{3/4} n}+\frac {2\ 2^{3/4} c^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \left (-b-\sqrt {b^2-4 a c}\right )^{3/4} n}-\frac {2\ 2^{3/4} c^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \left (-b+\sqrt {b^2-4 a c}\right )^{3/4} n} \]
[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)
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Rubi [A]
time = 0.35, antiderivative size = 353, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1395, 1361,
218, 214, 211} \begin {gather*} \frac {2\ 2^{3/4} c^{3/4} \text {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 {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} \tanh ^{-1}\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} \tanh ^{-1}\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}} \end {gather*}
Antiderivative was successfully verified.
[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*} \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+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*}
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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 = 62, normalized size = 0.18 \begin {gather*} \frac {\text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {-n \log (x)+4 \log \left (x^{n/4}-\text {$\#$1}\right )}{b \text {$\#$1}^3+2 c \text {$\#$1}^7}\&\right ]}{4 n} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[x^(-1 + n/4)/(a + b*x^n + c*x^(2*n)),x]
[Out]
RootSum[a + b*#1^4 + c*#1^8 & , (-(n*Log[x]) + 4*Log[x^(n/4) - #1])/(b*#1^3 + 2*c*#1^7) & ]/(4*n)
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.49, size = 280, normalized size = 0.79
| | |
| method |
result |
size |
| | |
| risch |
\(\munderset {\textit {\_R} =\RootOf \left (\left (256 a^{7} c^{4} n^{8}-256 a^{6} b^{2} c^{3} n^{8}+96 a^{5} b^{4} c^{2} n^{8}-16 a^{4} b^{6} c \,n^{8}+a^{3} b^{8} n^{8}\right ) \textit {\_Z}^{8}+\left (-48 a^{3} b \,c^{3} n^{4}+40 a^{2} b^{3} c^{2} n^{4}-11 a \,b^{5} c \,n^{4}+b^{7} n^{4}\right ) \textit {\_Z}^{4}+c^{3}\right )}{\sum }\textit {\_R} \ln \left (x^{\frac {n}{4}}+\left (\frac {16 n^{5} b \,a^{5} c^{2}}{c^{2} a -b^{2} c}-\frac {8 n^{5} b^{3} a^{4} c}{c^{2} a -b^{2} c}+\frac {n^{5} b^{5} a^{3}}{c^{2} a -b^{2} c}\right ) \textit {\_R}^{5}+\left (\frac {2 n \,a^{2} c^{2}}{c^{2} a -b^{2} c}-\frac {4 n \,b^{2} a c}{c^{2} a -b^{2} c}+\frac {n \,b^{4}}{c^{2} a -b^{2} c}\right ) \textit {\_R} \right )\) |
\(280\) |
| | |
|
|
|
|
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]
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))
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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]
integrate(x^(1/4*n - 1)/(c*x^(2*n) + b*x^n + a), x)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4426 vs.
\(2 (273) = 546\).
time = 0.63, size = 4426, normalized size = 12.54 \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]
-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)))*arctan(1/16*sqrt(2)*(2*sqrt(2)*((a^3*b^10*c - 15*a^4*b^8*c^2 + 86*a^5*b^6*c^3 - 232*a^6*b^4*c^4 + 288
*a^7*b^2*c^5 - 128*a^8*c^6)*n^7*x*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^9*c - 10*a*b^7*c^2 + 33*a^2*b^5*c^3 - 40*a^3*b^3*c^4 + 16*a^4*b*c^5)*n^3*x)*x^(1/4*n -
1)*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)) + sqrt(2)*((
a^3*b^8 - 14*a^4*b^6*c + 72*a^5*b^4*c^2 - 160*a^6*b^2*c^3 + 128*a^7*c^4)*n^7*x*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^7 - 9*a*b^5*c + 24*a^2*b^3*c^2 - 16*a^3*b
*c^3)*n^3*x)*sqrt((4*(b^4*c^2 - 2*a*b^2*c^3 + a^2*c^4)*x^2*x^(1/2*n - 2) - sqrt(2)*((a^3*b^9 - 13*a^4*b^7*c +
60*a^5*b^5*c^2 - 112*a^6*b^3*c^3 + 64*a^7*b*c^4)*n^6*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^8 - 8*a*b^6*c + 21*a^2*b^4*c^2 - 22*a^3*b^2*c^3 + 8*a^4*c^4)*n^2)*s
qrt(-((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^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)))*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)))/(b^4*c^3 - 2*a*b^2*c^4 + a^2
*c^5)) + 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)))*arctan(1/8*(2*((a^3*b^10*c - 15*a^4*b^8*c^2 + 86*a^5*b^6*c^3 - 232*a^6*b^4*c^4 + 288*a^7*b^2*c
^5 - 128*a^8*c^6)*n^7*x*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^9*c - 10*a*b^7*c^2 + 33*a^2*b^5*c^3 - 40*a^3*b^3*c^4 + 16*a^4*b*c^5)*n^3*x)*x^(1/4*n - 1)*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)))*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)) + ((a^3*b^8 - 14*a^4*b^6*c
+ 72*a^5*b^4*c^2 - 160*a^6*b^2*c^3 + 128*a^7*c^4)*n^7*x*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^7 - 9*a*b^5*c + 24*a^2*b^3*c^2 - 16*a^3*b*c^3)*n^3*x)*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)))*sqrt((4*(b^4
*c^2 - 2*a*b^2*c^3 + a^2*c^4)*x^2*x^(1/2*n - 2) + sqrt(2)*((a^3*b^9 - 13*a^4*b^7*c + 60*a^5*b^5*c^2 - 112*a^6*
b^3*c^3 + 64*a^7*b*c^4)*n^6*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^8 - 8*a*b^6*c + 21*a^2*b^4*c^2 - 22*a^3*b^2*c^3 + 8*a^4*c^4)*n^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^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)))/(b^4*c^3 - 2*a*b^2*c^4 + a^2*c^5)) + 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)*sq
rt(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^...
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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 4377 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 {x^{\frac {n}{4}-1}}{a+b\,x^n+c\,x^{2\,n}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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