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

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

[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])*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)

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Rubi [A]  time = 1.56316, antiderivative size = 414, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ -\frac{2^{3/4} \left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{a^{5/4} n \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{2^{3/4} \left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{a^{5/4} n \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{2^{3/4} \left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{a^{5/4} n \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{2^{3/4} \left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{a^{5/4} n \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{4 x^{-n/4}}{a n} \]

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])*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)

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Rubi in Sympy [A]  time = 179.699, size = 403, normalized size = 0.97 \[ - \frac{4 x^{- \frac{n}{4}}}{a n} + \frac{2^{\frac{3}{4}} \left (- 2 a c + b^{2} - b \sqrt{- 4 a c + b^{2}}\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{2} \sqrt [4]{a} x^{- \frac{n}{4}}}{\sqrt [4]{- b + \sqrt{- 4 a c + b^{2}}}} \right )}}{a^{\frac{5}{4}} n \left (- b + \sqrt{- 4 a c + b^{2}}\right )^{\frac{3}{4}} \sqrt{- 4 a c + b^{2}}} + \frac{2^{\frac{3}{4}} \left (- 2 a c + b^{2} - b \sqrt{- 4 a c + b^{2}}\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{2} \sqrt [4]{a} x^{- \frac{n}{4}}}{\sqrt [4]{- b + \sqrt{- 4 a c + b^{2}}}} \right )}}{a^{\frac{5}{4}} n \left (- b + \sqrt{- 4 a c + b^{2}}\right )^{\frac{3}{4}} \sqrt{- 4 a c + b^{2}}} - \frac{2^{\frac{3}{4}} \left (- 2 a c + b^{2} + b \sqrt{- 4 a c + b^{2}}\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{2} \sqrt [4]{a} x^{- \frac{n}{4}}}{\sqrt [4]{- b - \sqrt{- 4 a c + b^{2}}}} \right )}}{a^{\frac{5}{4}} n \left (- b - \sqrt{- 4 a c + b^{2}}\right )^{\frac{3}{4}} \sqrt{- 4 a c + b^{2}}} - \frac{2^{\frac{3}{4}} \left (- 2 a c + b^{2} + b \sqrt{- 4 a c + b^{2}}\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{2} \sqrt [4]{a} x^{- \frac{n}{4}}}{\sqrt [4]{- b - \sqrt{- 4 a c + b^{2}}}} \right )}}{a^{\frac{5}{4}} n \left (- b - \sqrt{- 4 a c + b^{2}}\right )^{\frac{3}{4}} \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*x**(-n/4)/(a*n) + 2**(3/4)*(-2*a*c + b**2 - b*sqrt(-4*a*c + b**2))*atan(2**(1
/4)*a**(1/4)*x**(-n/4)/(-b + sqrt(-4*a*c + b**2))**(1/4))/(a**(5/4)*n*(-b + sqrt
(-4*a*c + b**2))**(3/4)*sqrt(-4*a*c + b**2)) + 2**(3/4)*(-2*a*c + b**2 - b*sqrt(
-4*a*c + b**2))*atanh(2**(1/4)*a**(1/4)*x**(-n/4)/(-b + sqrt(-4*a*c + b**2))**(1
/4))/(a**(5/4)*n*(-b + sqrt(-4*a*c + b**2))**(3/4)*sqrt(-4*a*c + b**2)) - 2**(3/
4)*(-2*a*c + b**2 + b*sqrt(-4*a*c + b**2))*atan(2**(1/4)*a**(1/4)*x**(-n/4)/(-b
- sqrt(-4*a*c + b**2))**(1/4))/(a**(5/4)*n*(-b - sqrt(-4*a*c + b**2))**(3/4)*sqr
t(-4*a*c + b**2)) - 2**(3/4)*(-2*a*c + b**2 + b*sqrt(-4*a*c + b**2))*atanh(2**(1
/4)*a**(1/4)*x**(-n/4)/(-b - sqrt(-4*a*c + b**2))**(1/4))/(a**(5/4)*n*(-b - sqrt
(-4*a*c + b**2))**(3/4)*sqrt(-4*a*c + b**2))

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Mathematica [C]  time = 0.109415, size = 105, normalized size = 0.25 \[ \frac{\text{RootSum}\left [\text{$\#$1}^8 a+\text{$\#$1}^4 b+c\&,\frac{4 \text{$\#$1}^4 b \log \left (x^{-n/4}-\text{$\#$1}\right )+\text{$\#$1}^4 b n \log (x)+4 c \log \left (x^{-n/4}-\text{$\#$1}\right )+c n \log (x)}{2 \text{$\#$1}^7 a+\text{$\#$1}^3 b}\&\right ]-16 x^{-n/4}}{4 a n} \]

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

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Maple [C]  time = 0.841, size = 630, normalized size = 1.5 \[ -4\,{\frac{1}{an{x}^{n/4}}}+\sum _{{\it \_R}={\it RootOf} \left ( \left ( 256\,{a}^{9}{c}^{4}{n}^{8}-256\,{a}^{8}{b}^{2}{c}^{3}{n}^{8}+96\,{a}^{7}{b}^{4}{c}^{2}{n}^{8}-16\,{a}^{6}{b}^{6}c{n}^{8}+{a}^{5}{b}^{8}{n}^{8} \right ){{\it \_Z}}^{8}+ \left ( 80\,{a}^{4}b{c}^{4}{n}^{4}-120\,{a}^{3}{b}^{3}{c}^{3}{n}^{4}+61\,{a}^{2}{b}^{5}{c}^{2}{n}^{4}-13\,a{b}^{7}c{n}^{4}+{b}^{9}{n}^{4} \right ){{\it \_Z}}^{4}+{c}^{5} \right ) }{\it \_R}\,\ln \left ({x}^{{\frac{n}{4}}}+ \left ( -128\,{\frac{{a}^{10}{n}^{7}{c}^{5}}{{a}^{2}{c}^{6}-3\,a{b}^{2}{c}^{5}+{b}^{4}{c}^{4}}}+352\,{\frac{{n}^{7}{b}^{2}{a}^{9}{c}^{4}}{{a}^{2}{c}^{6}-3\,a{b}^{2}{c}^{5}+{b}^{4}{c}^{4}}}-280\,{\frac{{n}^{7}{b}^{4}{a}^{8}{c}^{3}}{{a}^{2}{c}^{6}-3\,a{b}^{2}{c}^{5}+{b}^{4}{c}^{4}}}+98\,{\frac{{n}^{7}{b}^{6}{a}^{7}{c}^{2}}{{a}^{2}{c}^{6}-3\,a{b}^{2}{c}^{5}+{b}^{4}{c}^{4}}}-16\,{\frac{{n}^{7}{b}^{8}{a}^{6}c}{{a}^{2}{c}^{6}-3\,a{b}^{2}{c}^{5}+{b}^{4}{c}^{4}}}+{\frac{{n}^{7}{b}^{10}{a}^{5}}{{a}^{2}{c}^{6}-3\,a{b}^{2}{c}^{5}+{b}^{4}{c}^{4}}} \right ){{\it \_R}}^{7}+ \left ( -36\,{\frac{{n}^{3}b{a}^{5}{c}^{5}}{{a}^{2}{c}^{6}-3\,a{b}^{2}{c}^{5}+{b}^{4}{c}^{4}}}+129\,{\frac{{n}^{3}{b}^{3}{a}^{4}{c}^{4}}{{a}^{2}{c}^{6}-3\,a{b}^{2}{c}^{5}+{b}^{4}{c}^{4}}}-138\,{\frac{{n}^{3}{b}^{5}{a}^{3}{c}^{3}}{{a}^{2}{c}^{6}-3\,a{b}^{2}{c}^{5}+{b}^{4}{c}^{4}}}+63\,{\frac{{n}^{3}{b}^{7}{a}^{2}{c}^{2}}{{a}^{2}{c}^{6}-3\,a{b}^{2}{c}^{5}+{b}^{4}{c}^{4}}}-13\,{\frac{{n}^{3}{b}^{9}ac}{{a}^{2}{c}^{6}-3\,a{b}^{2}{c}^{5}+{b}^{4}{c}^{4}}}+{\frac{{n}^{3}{b}^{11}}{{a}^{2}{c}^{6}-3\,a{b}^{2}{c}^{5}+{b}^{4}{c}^{4}}} \right ){{\it \_R}}^{3} \right ) \]

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)

[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., size = 0, normalized size = 0. \[ -\frac{4 \, x^{-\frac{1}{4} \, n}}{a n} - \int \frac{c x^{\frac{7}{4} \, n} + b x^{\frac{3}{4} \, n}}{a c x x^{2 \, n} + a b x x^{n} + a^{2} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*x^(-1/4*n)/(a*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 [A]  time = 0.626246, size = 5652, normalized size = 13.65 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(-1/4*n - 1)/(c*x^(2*n) + b*x^n + a),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(1/2*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)))/(2*(b^4*c - 3*a*b^2*c^2 + a^2*c^3)*x*x^(-1/4*n - 1) + 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*s
qrt((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))) - 4*sqrt(2)*a*n*sqrt(sqr
t(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(1/2*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^1
0*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)))/(2*(b^4*c - 3*a*b^2*c
^2 + a^2*c^3)*x*x^(-1/4*n - 1) + 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*s
qrt((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 + 1
6*a^7*c^2)*n^4)))/x^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*s
qrt((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^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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{-\frac{1}{4} \, n - 1}}{c x^{2 \, n} + b x^{n} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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