3.2580 \(\int x^{-1-12 n} \left (a+b x^n\right )^8 \, dx\)

Optimal. Leaf size=104 \[ \frac{b^3 x^{-9 n} \left (a+b x^n\right )^9}{1980 a^4 n}-\frac{b^2 x^{-10 n} \left (a+b x^n\right )^9}{220 a^3 n}+\frac{b x^{-11 n} \left (a+b x^n\right )^9}{44 a^2 n}-\frac{x^{-12 n} \left (a+b x^n\right )^9}{12 a n} \]

[Out]

-(a + b*x^n)^9/(12*a*n*x^(12*n)) + (b*(a + b*x^n)^9)/(44*a^2*n*x^(11*n)) - (b^2*
(a + b*x^n)^9)/(220*a^3*n*x^(10*n)) + (b^3*(a + b*x^n)^9)/(1980*a^4*n*x^(9*n))

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Rubi [A]  time = 0.117756, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{b^3 x^{-9 n} \left (a+b x^n\right )^9}{1980 a^4 n}-\frac{b^2 x^{-10 n} \left (a+b x^n\right )^9}{220 a^3 n}+\frac{b x^{-11 n} \left (a+b x^n\right )^9}{44 a^2 n}-\frac{x^{-12 n} \left (a+b x^n\right )^9}{12 a n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - 12*n)*(a + b*x^n)^8,x]

[Out]

-(a + b*x^n)^9/(12*a*n*x^(12*n)) + (b*(a + b*x^n)^9)/(44*a^2*n*x^(11*n)) - (b^2*
(a + b*x^n)^9)/(220*a^3*n*x^(10*n)) + (b^3*(a + b*x^n)^9)/(1980*a^4*n*x^(9*n))

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Rubi in Sympy [A]  time = 14.3627, size = 87, normalized size = 0.84 \[ - \frac{x^{- 12 n} \left (a + b x^{n}\right )^{9}}{12 a n} + \frac{b x^{- 11 n} \left (a + b x^{n}\right )^{9}}{44 a^{2} n} - \frac{b^{2} x^{- 10 n} \left (a + b x^{n}\right )^{9}}{220 a^{3} n} + \frac{b^{3} x^{- 9 n} \left (a + b x^{n}\right )^{9}}{1980 a^{4} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1-12*n)*(a+b*x**n)**8,x)

[Out]

-x**(-12*n)*(a + b*x**n)**9/(12*a*n) + b*x**(-11*n)*(a + b*x**n)**9/(44*a**2*n)
- b**2*x**(-10*n)*(a + b*x**n)**9/(220*a**3*n) + b**3*x**(-9*n)*(a + b*x**n)**9/
(1980*a**4*n)

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Mathematica [A]  time = 0.0516853, size = 113, normalized size = 1.09 \[ -\frac{x^{-12 n} \left (165 a^8+1440 a^7 b x^n+5544 a^6 b^2 x^{2 n}+12320 a^5 b^3 x^{3 n}+17325 a^4 b^4 x^{4 n}+15840 a^3 b^5 x^{5 n}+9240 a^2 b^6 x^{6 n}+3168 a b^7 x^{7 n}+495 b^8 x^{8 n}\right )}{1980 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 - 12*n)*(a + b*x^n)^8,x]

[Out]

-(165*a^8 + 1440*a^7*b*x^n + 5544*a^6*b^2*x^(2*n) + 12320*a^5*b^3*x^(3*n) + 1732
5*a^4*b^4*x^(4*n) + 15840*a^3*b^5*x^(5*n) + 9240*a^2*b^6*x^(6*n) + 3168*a*b^7*x^
(7*n) + 495*b^8*x^(8*n))/(1980*n*x^(12*n))

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Maple [A]  time = 0.043, size = 136, normalized size = 1.3 \[ -{\frac{{b}^{8}}{4\,n \left ({x}^{n} \right ) ^{4}}}-{\frac{8\,a{b}^{7}}{5\,n \left ({x}^{n} \right ) ^{5}}}-{\frac{14\,{a}^{2}{b}^{6}}{3\,n \left ({x}^{n} \right ) ^{6}}}-8\,{\frac{{a}^{3}{b}^{5}}{n \left ({x}^{n} \right ) ^{7}}}-{\frac{35\,{a}^{4}{b}^{4}}{4\,n \left ({x}^{n} \right ) ^{8}}}-{\frac{56\,{a}^{5}{b}^{3}}{9\,n \left ({x}^{n} \right ) ^{9}}}-{\frac{14\,{a}^{6}{b}^{2}}{5\,n \left ({x}^{n} \right ) ^{10}}}-{\frac{8\,b{a}^{7}}{11\,n \left ({x}^{n} \right ) ^{11}}}-{\frac{{a}^{8}}{12\,n \left ({x}^{n} \right ) ^{12}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-12*n)*(a+b*x^n)^8,x)

[Out]

-1/4*b^8/n/(x^n)^4-8/5*a*b^7/n/(x^n)^5-14/3*a^2*b^6/n/(x^n)^6-8*a^3*b^5/n/(x^n)^
7-35/4*a^4*b^4/n/(x^n)^8-56/9*a^5*b^3/n/(x^n)^9-14/5*a^6*b^2/n/(x^n)^10-8/11*a^7
*b/n/(x^n)^11-1/12*a^8/n/(x^n)^12

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^8*x^(-12*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.229135, size = 153, normalized size = 1.47 \[ -\frac{495 \, b^{8} x^{8 \, n} + 3168 \, a b^{7} x^{7 \, n} + 9240 \, a^{2} b^{6} x^{6 \, n} + 15840 \, a^{3} b^{5} x^{5 \, n} + 17325 \, a^{4} b^{4} x^{4 \, n} + 12320 \, a^{5} b^{3} x^{3 \, n} + 5544 \, a^{6} b^{2} x^{2 \, n} + 1440 \, a^{7} b x^{n} + 165 \, a^{8}}{1980 \, n x^{12 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^8*x^(-12*n - 1),x, algorithm="fricas")

[Out]

-1/1980*(495*b^8*x^(8*n) + 3168*a*b^7*x^(7*n) + 9240*a^2*b^6*x^(6*n) + 15840*a^3
*b^5*x^(5*n) + 17325*a^4*b^4*x^(4*n) + 12320*a^5*b^3*x^(3*n) + 5544*a^6*b^2*x^(2
*n) + 1440*a^7*b*x^n + 165*a^8)/(n*x^(12*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**(-1-12*n)*(a+b*x**n)**8,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.233383, size = 163, normalized size = 1.57 \[ -\frac{{\left (495 \, b^{8} e^{\left (8 \, n{\rm ln}\left (x\right )\right )} + 3168 \, a b^{7} e^{\left (7 \, n{\rm ln}\left (x\right )\right )} + 9240 \, a^{2} b^{6} e^{\left (6 \, n{\rm ln}\left (x\right )\right )} + 15840 \, a^{3} b^{5} e^{\left (5 \, n{\rm ln}\left (x\right )\right )} + 17325 \, a^{4} b^{4} e^{\left (4 \, n{\rm ln}\left (x\right )\right )} + 12320 \, a^{5} b^{3} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 5544 \, a^{6} b^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 1440 \, a^{7} b e^{\left (n{\rm ln}\left (x\right )\right )} + 165 \, a^{8}\right )} e^{\left (-12 \, n{\rm ln}\left (x\right )\right )}}{1980 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^8*x^(-12*n - 1),x, algorithm="giac")

[Out]

-1/1980*(495*b^8*e^(8*n*ln(x)) + 3168*a*b^7*e^(7*n*ln(x)) + 9240*a^2*b^6*e^(6*n*
ln(x)) + 15840*a^3*b^5*e^(5*n*ln(x)) + 17325*a^4*b^4*e^(4*n*ln(x)) + 12320*a^5*b
^3*e^(3*n*ln(x)) + 5544*a^6*b^2*e^(2*n*ln(x)) + 1440*a^7*b*e^(n*ln(x)) + 165*a^8
)*e^(-12*n*ln(x))/n