3.2579 \(\int x^{-1-11 n} \left (a+b x^n\right )^8 \, dx\)

Optimal. Leaf size=77 \[ -\frac{b^2 x^{-9 n} \left (a+b x^n\right )^9}{495 a^3 n}+\frac{b x^{-10 n} \left (a+b x^n\right )^9}{55 a^2 n}-\frac{x^{-11 n} \left (a+b x^n\right )^9}{11 a n} \]

[Out]

-(a + b*x^n)^9/(11*a*n*x^(11*n)) + (b*(a + b*x^n)^9)/(55*a^2*n*x^(10*n)) - (b^2*
(a + b*x^n)^9)/(495*a^3*n*x^(9*n))

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Rubi [A]  time = 0.0871573, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\frac{b^2 x^{-9 n} \left (a+b x^n\right )^9}{495 a^3 n}+\frac{b x^{-10 n} \left (a+b x^n\right )^9}{55 a^2 n}-\frac{x^{-11 n} \left (a+b x^n\right )^9}{11 a n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - 11*n)*(a + b*x^n)^8,x]

[Out]

-(a + b*x^n)^9/(11*a*n*x^(11*n)) + (b*(a + b*x^n)^9)/(55*a^2*n*x^(10*n)) - (b^2*
(a + b*x^n)^9)/(495*a^3*n*x^(9*n))

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Rubi in Sympy [A]  time = 10.042, size = 63, normalized size = 0.82 \[ - \frac{x^{- 11 n} \left (a + b x^{n}\right )^{9}}{11 a n} + \frac{b x^{- 10 n} \left (a + b x^{n}\right )^{9}}{55 a^{2} n} - \frac{b^{2} x^{- 9 n} \left (a + b x^{n}\right )^{9}}{495 a^{3} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1-11*n)*(a+b*x**n)**8,x)

[Out]

-x**(-11*n)*(a + b*x**n)**9/(11*a*n) + b*x**(-10*n)*(a + b*x**n)**9/(55*a**2*n)
- b**2*x**(-9*n)*(a + b*x**n)**9/(495*a**3*n)

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Mathematica [A]  time = 0.0532717, size = 113, normalized size = 1.47 \[ -\frac{x^{-11 n} \left (45 a^8+396 a^7 b x^n+1540 a^6 b^2 x^{2 n}+3465 a^5 b^3 x^{3 n}+4950 a^4 b^4 x^{4 n}+4620 a^3 b^5 x^{5 n}+2772 a^2 b^6 x^{6 n}+990 a b^7 x^{7 n}+165 b^8 x^{8 n}\right )}{495 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 - 11*n)*(a + b*x^n)^8,x]

[Out]

-(45*a^8 + 396*a^7*b*x^n + 1540*a^6*b^2*x^(2*n) + 3465*a^5*b^3*x^(3*n) + 4950*a^
4*b^4*x^(4*n) + 4620*a^3*b^5*x^(5*n) + 2772*a^2*b^6*x^(6*n) + 990*a*b^7*x^(7*n)
+ 165*b^8*x^(8*n))/(495*n*x^(11*n))

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Maple [A]  time = 0.041, size = 136, normalized size = 1.8 \[ -{\frac{{b}^{8}}{3\,n \left ({x}^{n} \right ) ^{3}}}-2\,{\frac{a{b}^{7}}{n \left ({x}^{n} \right ) ^{4}}}-{\frac{28\,{a}^{2}{b}^{6}}{5\,n \left ({x}^{n} \right ) ^{5}}}-{\frac{28\,{a}^{3}{b}^{5}}{3\,n \left ({x}^{n} \right ) ^{6}}}-10\,{\frac{{a}^{4}{b}^{4}}{n \left ({x}^{n} \right ) ^{7}}}-7\,{\frac{{a}^{5}{b}^{3}}{n \left ({x}^{n} \right ) ^{8}}}-{\frac{28\,{a}^{6}{b}^{2}}{9\,n \left ({x}^{n} \right ) ^{9}}}-{\frac{4\,b{a}^{7}}{5\,n \left ({x}^{n} \right ) ^{10}}}-{\frac{{a}^{8}}{11\,n \left ({x}^{n} \right ) ^{11}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-11*n)*(a+b*x^n)^8,x)

[Out]

-1/3*b^8/n/(x^n)^3-2*a*b^7/n/(x^n)^4-28/5*a^2*b^6/n/(x^n)^5-28/3*a^3*b^5/n/(x^n)
^6-10*a^4*b^4/n/(x^n)^7-7*a^5*b^3/n/(x^n)^8-28/9*a^6*b^2/n/(x^n)^9-4/5*a^7*b/n/(
x^n)^10-1/11*a^8/n/(x^n)^11

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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^(-11*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227325, size = 153, normalized size = 1.99 \[ -\frac{165 \, b^{8} x^{8 \, n} + 990 \, a b^{7} x^{7 \, n} + 2772 \, a^{2} b^{6} x^{6 \, n} + 4620 \, a^{3} b^{5} x^{5 \, n} + 4950 \, a^{4} b^{4} x^{4 \, n} + 3465 \, a^{5} b^{3} x^{3 \, n} + 1540 \, a^{6} b^{2} x^{2 \, n} + 396 \, a^{7} b x^{n} + 45 \, a^{8}}{495 \, n x^{11 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^8*x^(-11*n - 1),x, algorithm="fricas")

[Out]

-1/495*(165*b^8*x^(8*n) + 990*a*b^7*x^(7*n) + 2772*a^2*b^6*x^(6*n) + 4620*a^3*b^
5*x^(5*n) + 4950*a^4*b^4*x^(4*n) + 3465*a^5*b^3*x^(3*n) + 1540*a^6*b^2*x^(2*n) +
396*a^7*b*x^n + 45*a^8)/(n*x^(11*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**(-1-11*n)*(a+b*x**n)**8,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.231535, size = 163, normalized size = 2.12 \[ -\frac{{\left (165 \, b^{8} e^{\left (8 \, n{\rm ln}\left (x\right )\right )} + 990 \, a b^{7} e^{\left (7 \, n{\rm ln}\left (x\right )\right )} + 2772 \, a^{2} b^{6} e^{\left (6 \, n{\rm ln}\left (x\right )\right )} + 4620 \, a^{3} b^{5} e^{\left (5 \, n{\rm ln}\left (x\right )\right )} + 4950 \, a^{4} b^{4} e^{\left (4 \, n{\rm ln}\left (x\right )\right )} + 3465 \, a^{5} b^{3} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 1540 \, a^{6} b^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 396 \, a^{7} b e^{\left (n{\rm ln}\left (x\right )\right )} + 45 \, a^{8}\right )} e^{\left (-11 \, n{\rm ln}\left (x\right )\right )}}{495 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^8*x^(-11*n - 1),x, algorithm="giac")

[Out]

-1/495*(165*b^8*e^(8*n*ln(x)) + 990*a*b^7*e^(7*n*ln(x)) + 2772*a^2*b^6*e^(6*n*ln
(x)) + 4620*a^3*b^5*e^(5*n*ln(x)) + 4950*a^4*b^4*e^(4*n*ln(x)) + 3465*a^5*b^3*e^
(3*n*ln(x)) + 1540*a^6*b^2*e^(2*n*ln(x)) + 396*a^7*b*e^(n*ln(x)) + 45*a^8)*e^(-1
1*n*ln(x))/n