3.112 \(\int \frac{x^{-1+n} \left (b+2 c x^n\right )}{\left (a+b x^n+c x^{2 n}\right )^8} \, dx\)

Optimal. Leaf size=23 \[ -\frac{1}{7 n \left (a+b x^n+c x^{2 n}\right )^7} \]

[Out]

-1/(7*n*(a + b*x^n + c*x^(2*n))^7)

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Rubi [A]  time = 0.0721155, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ -\frac{1}{7 n \left (a+b x^n+c x^{2 n}\right )^7} \]

Antiderivative was successfully verified.

[In]  Int[(x^(-1 + n)*(b + 2*c*x^n))/(a + b*x^n + c*x^(2*n))^8,x]

[Out]

-1/(7*n*(a + b*x^n + c*x^(2*n))^7)

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Rubi in Sympy [A]  time = 12.2668, size = 20, normalized size = 0.87 \[ - \frac{1}{7 n \left (a + b x^{n} + c x^{2 n}\right )^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+n)*(b+2*c*x**n)/(a+b*x**n+c*x**(2*n))**8,x)

[Out]

-1/(7*n*(a + b*x**n + c*x**(2*n))**7)

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Mathematica [A]  time = 0.0818763, size = 23, normalized size = 1. \[ -\frac{1}{7 n \left (a+b x^n+c x^{2 n}\right )^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(-1 + n)*(b + 2*c*x^n))/(a + b*x^n + c*x^(2*n))^8,x]

[Out]

-1/(7*n*(a + b*x^n + c*x^(2*n))^7)

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Maple [A]  time = 0.095, size = 22, normalized size = 1. \[ -{\frac{1}{7\,n \left ( a+b{x}^{n}+c \left ({x}^{n} \right ) ^{2} \right ) ^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+n)*(b+2*c*x^n)/(a+b*x^n+c*x^(2*n))^8,x)

[Out]

-1/7/n/(a+b*x^n+c*(x^n)^2)^7

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Maxima [A]  time = 1.21423, size = 562, normalized size = 24.43 \[ -\frac{1}{7 \,{\left (c^{7} n x^{14 \, n} + 7 \, b c^{6} n x^{13 \, n} + 7 \, a^{6} b n x^{n} + a^{7} n + 7 \,{\left (3 \, b^{2} c^{5} n + a c^{6} n\right )} x^{12 \, n} + 7 \,{\left (5 \, b^{3} c^{4} n + 6 \, a b c^{5} n\right )} x^{11 \, n} + 7 \,{\left (5 \, b^{4} c^{3} n + 15 \, a b^{2} c^{4} n + 3 \, a^{2} c^{5} n\right )} x^{10 \, n} + 7 \,{\left (3 \, b^{5} c^{2} n + 20 \, a b^{3} c^{3} n + 15 \, a^{2} b c^{4} n\right )} x^{9 \, n} + 7 \,{\left (b^{6} c n + 15 \, a b^{4} c^{2} n + 30 \, a^{2} b^{2} c^{3} n + 5 \, a^{3} c^{4} n\right )} x^{8 \, n} +{\left (b^{7} n + 42 \, a b^{5} c n + 210 \, a^{2} b^{3} c^{2} n + 140 \, a^{3} b c^{3} n\right )} x^{7 \, n} + 7 \,{\left (a b^{6} n + 15 \, a^{2} b^{4} c n + 30 \, a^{3} b^{2} c^{2} n + 5 \, a^{4} c^{3} n\right )} x^{6 \, n} + 7 \,{\left (3 \, a^{2} b^{5} n + 20 \, a^{3} b^{3} c n + 15 \, a^{4} b c^{2} n\right )} x^{5 \, n} + 7 \,{\left (5 \, a^{3} b^{4} n + 15 \, a^{4} b^{2} c n + 3 \, a^{5} c^{2} n\right )} x^{4 \, n} + 7 \,{\left (5 \, a^{4} b^{3} n + 6 \, a^{5} b c n\right )} x^{3 \, n} + 7 \,{\left (3 \, a^{5} b^{2} n + a^{6} c n\right )} x^{2 \, n}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x^n + b)*x^(n - 1)/(c*x^(2*n) + b*x^n + a)^8,x, algorithm="maxima")

[Out]

-1/7/(c^7*n*x^(14*n) + 7*b*c^6*n*x^(13*n) + 7*a^6*b*n*x^n + a^7*n + 7*(3*b^2*c^5
*n + a*c^6*n)*x^(12*n) + 7*(5*b^3*c^4*n + 6*a*b*c^5*n)*x^(11*n) + 7*(5*b^4*c^3*n
+ 15*a*b^2*c^4*n + 3*a^2*c^5*n)*x^(10*n) + 7*(3*b^5*c^2*n + 20*a*b^3*c^3*n + 15
*a^2*b*c^4*n)*x^(9*n) + 7*(b^6*c*n + 15*a*b^4*c^2*n + 30*a^2*b^2*c^3*n + 5*a^3*c
^4*n)*x^(8*n) + (b^7*n + 42*a*b^5*c*n + 210*a^2*b^3*c^2*n + 140*a^3*b*c^3*n)*x^(
7*n) + 7*(a*b^6*n + 15*a^2*b^4*c*n + 30*a^3*b^2*c^2*n + 5*a^4*c^3*n)*x^(6*n) + 7
*(3*a^2*b^5*n + 20*a^3*b^3*c*n + 15*a^4*b*c^2*n)*x^(5*n) + 7*(5*a^3*b^4*n + 15*a
^4*b^2*c*n + 3*a^5*c^2*n)*x^(4*n) + 7*(5*a^4*b^3*n + 6*a^5*b*c*n)*x^(3*n) + 7*(3
*a^5*b^2*n + a^6*c*n)*x^(2*n))

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Fricas [A]  time = 0.362501, size = 532, normalized size = 23.13 \[ -\frac{1}{7 \,{\left (c^{7} n x^{14 \, n} + 7 \, b c^{6} n x^{13 \, n} + 7 \, a^{6} b n x^{n} + a^{7} n + 7 \,{\left (3 \, b^{2} c^{5} + a c^{6}\right )} n x^{12 \, n} + 7 \,{\left (5 \, b^{3} c^{4} + 6 \, a b c^{5}\right )} n x^{11 \, n} + 7 \,{\left (5 \, b^{4} c^{3} + 15 \, a b^{2} c^{4} + 3 \, a^{2} c^{5}\right )} n x^{10 \, n} + 7 \,{\left (3 \, b^{5} c^{2} + 20 \, a b^{3} c^{3} + 15 \, a^{2} b c^{4}\right )} n x^{9 \, n} + 7 \,{\left (b^{6} c + 15 \, a b^{4} c^{2} + 30 \, a^{2} b^{2} c^{3} + 5 \, a^{3} c^{4}\right )} n x^{8 \, n} +{\left (b^{7} + 42 \, a b^{5} c + 210 \, a^{2} b^{3} c^{2} + 140 \, a^{3} b c^{3}\right )} n x^{7 \, n} + 7 \,{\left (a b^{6} + 15 \, a^{2} b^{4} c + 30 \, a^{3} b^{2} c^{2} + 5 \, a^{4} c^{3}\right )} n x^{6 \, n} + 7 \,{\left (3 \, a^{2} b^{5} + 20 \, a^{3} b^{3} c + 15 \, a^{4} b c^{2}\right )} n x^{5 \, n} + 7 \,{\left (5 \, a^{3} b^{4} + 15 \, a^{4} b^{2} c + 3 \, a^{5} c^{2}\right )} n x^{4 \, n} + 7 \,{\left (5 \, a^{4} b^{3} + 6 \, a^{5} b c\right )} n x^{3 \, n} + 7 \,{\left (3 \, a^{5} b^{2} + a^{6} c\right )} n x^{2 \, n}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x^n + b)*x^(n - 1)/(c*x^(2*n) + b*x^n + a)^8,x, algorithm="fricas")

[Out]

-1/7/(c^7*n*x^(14*n) + 7*b*c^6*n*x^(13*n) + 7*a^6*b*n*x^n + a^7*n + 7*(3*b^2*c^5
+ a*c^6)*n*x^(12*n) + 7*(5*b^3*c^4 + 6*a*b*c^5)*n*x^(11*n) + 7*(5*b^4*c^3 + 15*
a*b^2*c^4 + 3*a^2*c^5)*n*x^(10*n) + 7*(3*b^5*c^2 + 20*a*b^3*c^3 + 15*a^2*b*c^4)*
n*x^(9*n) + 7*(b^6*c + 15*a*b^4*c^2 + 30*a^2*b^2*c^3 + 5*a^3*c^4)*n*x^(8*n) + (b
^7 + 42*a*b^5*c + 210*a^2*b^3*c^2 + 140*a^3*b*c^3)*n*x^(7*n) + 7*(a*b^6 + 15*a^2
*b^4*c + 30*a^3*b^2*c^2 + 5*a^4*c^3)*n*x^(6*n) + 7*(3*a^2*b^5 + 20*a^3*b^3*c + 1
5*a^4*b*c^2)*n*x^(5*n) + 7*(5*a^3*b^4 + 15*a^4*b^2*c + 3*a^5*c^2)*n*x^(4*n) + 7*
(5*a^4*b^3 + 6*a^5*b*c)*n*x^(3*n) + 7*(3*a^5*b^2 + a^6*c)*n*x^(2*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+n)*(b+2*c*x**n)/(a+b*x**n+c*x**(2*n))**8,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.281919, size = 28, normalized size = 1.22 \[ -\frac{1}{7 \,{\left (c x^{2 \, n} + b x^{n} + a\right )}^{7} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x^n + b)*x^(n - 1)/(c*x^(2*n) + b*x^n + a)^8,x, algorithm="giac")

[Out]

-1/7/((c*x^(2*n) + b*x^n + a)^7*n)