3.2583 \(\int x^{-1-15 n} \left (a+b x^n\right )^8 \, dx\)

Optimal. Leaf size=151 \[ -\frac{a^8 x^{-15 n}}{15 n}-\frac{4 a^7 b x^{-14 n}}{7 n}-\frac{28 a^6 b^2 x^{-13 n}}{13 n}-\frac{14 a^5 b^3 x^{-12 n}}{3 n}-\frac{70 a^4 b^4 x^{-11 n}}{11 n}-\frac{28 a^3 b^5 x^{-10 n}}{5 n}-\frac{28 a^2 b^6 x^{-9 n}}{9 n}-\frac{a b^7 x^{-8 n}}{n}-\frac{b^8 x^{-7 n}}{7 n} \]

[Out]

-a^8/(15*n*x^(15*n)) - (4*a^7*b)/(7*n*x^(14*n)) - (28*a^6*b^2)/(13*n*x^(13*n)) -
(14*a^5*b^3)/(3*n*x^(12*n)) - (70*a^4*b^4)/(11*n*x^(11*n)) - (28*a^3*b^5)/(5*n*
x^(10*n)) - (28*a^2*b^6)/(9*n*x^(9*n)) - (a*b^7)/(n*x^(8*n)) - b^8/(7*n*x^(7*n))

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Rubi [A]  time = 0.165057, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ -\frac{a^8 x^{-15 n}}{15 n}-\frac{4 a^7 b x^{-14 n}}{7 n}-\frac{28 a^6 b^2 x^{-13 n}}{13 n}-\frac{14 a^5 b^3 x^{-12 n}}{3 n}-\frac{70 a^4 b^4 x^{-11 n}}{11 n}-\frac{28 a^3 b^5 x^{-10 n}}{5 n}-\frac{28 a^2 b^6 x^{-9 n}}{9 n}-\frac{a b^7 x^{-8 n}}{n}-\frac{b^8 x^{-7 n}}{7 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - 15*n)*(a + b*x^n)^8,x]

[Out]

-a^8/(15*n*x^(15*n)) - (4*a^7*b)/(7*n*x^(14*n)) - (28*a^6*b^2)/(13*n*x^(13*n)) -
(14*a^5*b^3)/(3*n*x^(12*n)) - (70*a^4*b^4)/(11*n*x^(11*n)) - (28*a^3*b^5)/(5*n*
x^(10*n)) - (28*a^2*b^6)/(9*n*x^(9*n)) - (a*b^7)/(n*x^(8*n)) - b^8/(7*n*x^(7*n))

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Rubi in Sympy [A]  time = 28.2679, size = 136, normalized size = 0.9 \[ - \frac{a^{8} x^{- 15 n}}{15 n} - \frac{4 a^{7} b x^{- 14 n}}{7 n} - \frac{28 a^{6} b^{2} x^{- 13 n}}{13 n} - \frac{14 a^{5} b^{3} x^{- 12 n}}{3 n} - \frac{70 a^{4} b^{4} x^{- 11 n}}{11 n} - \frac{28 a^{3} b^{5} x^{- 10 n}}{5 n} - \frac{28 a^{2} b^{6} x^{- 9 n}}{9 n} - \frac{a b^{7} x^{- 8 n}}{n} - \frac{b^{8} x^{- 7 n}}{7 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1-15*n)*(a+b*x**n)**8,x)

[Out]

-a**8*x**(-15*n)/(15*n) - 4*a**7*b*x**(-14*n)/(7*n) - 28*a**6*b**2*x**(-13*n)/(1
3*n) - 14*a**5*b**3*x**(-12*n)/(3*n) - 70*a**4*b**4*x**(-11*n)/(11*n) - 28*a**3*
b**5*x**(-10*n)/(5*n) - 28*a**2*b**6*x**(-9*n)/(9*n) - a*b**7*x**(-8*n)/n - b**8
*x**(-7*n)/(7*n)

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Mathematica [A]  time = 0.0510606, size = 113, normalized size = 0.75 \[ -\frac{x^{-15 n} \left (3003 a^8+25740 a^7 b x^n+97020 a^6 b^2 x^{2 n}+210210 a^5 b^3 x^{3 n}+286650 a^4 b^4 x^{4 n}+252252 a^3 b^5 x^{5 n}+140140 a^2 b^6 x^{6 n}+45045 a b^7 x^{7 n}+6435 b^8 x^{8 n}\right )}{45045 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 - 15*n)*(a + b*x^n)^8,x]

[Out]

-(3003*a^8 + 25740*a^7*b*x^n + 97020*a^6*b^2*x^(2*n) + 210210*a^5*b^3*x^(3*n) +
286650*a^4*b^4*x^(4*n) + 252252*a^3*b^5*x^(5*n) + 140140*a^2*b^6*x^(6*n) + 45045
*a*b^7*x^(7*n) + 6435*b^8*x^(8*n))/(45045*n*x^(15*n))

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Maple [A]  time = 0.042, size = 136, normalized size = 0.9 \[ -{\frac{{b}^{8}}{7\,n \left ({x}^{n} \right ) ^{7}}}-{\frac{a{b}^{7}}{n \left ({x}^{n} \right ) ^{8}}}-{\frac{28\,{a}^{2}{b}^{6}}{9\,n \left ({x}^{n} \right ) ^{9}}}-{\frac{28\,{a}^{3}{b}^{5}}{5\,n \left ({x}^{n} \right ) ^{10}}}-{\frac{70\,{a}^{4}{b}^{4}}{11\,n \left ({x}^{n} \right ) ^{11}}}-{\frac{14\,{a}^{5}{b}^{3}}{3\,n \left ({x}^{n} \right ) ^{12}}}-{\frac{28\,{a}^{6}{b}^{2}}{13\,n \left ({x}^{n} \right ) ^{13}}}-{\frac{4\,b{a}^{7}}{7\,n \left ({x}^{n} \right ) ^{14}}}-{\frac{{a}^{8}}{15\,n \left ({x}^{n} \right ) ^{15}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-15*n)*(a+b*x^n)^8,x)

[Out]

-1/7*b^8/n/(x^n)^7-a*b^7/n/(x^n)^8-28/9*a^2*b^6/n/(x^n)^9-28/5*a^3*b^5/n/(x^n)^1
0-70/11*a^4*b^4/n/(x^n)^11-14/3*a^5*b^3/n/(x^n)^12-28/13*a^6*b^2/n/(x^n)^13-4/7*
a^7*b/n/(x^n)^14-1/15*a^8/n/(x^n)^15

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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^(-15*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.226939, size = 153, normalized size = 1.01 \[ -\frac{6435 \, b^{8} x^{8 \, n} + 45045 \, a b^{7} x^{7 \, n} + 140140 \, a^{2} b^{6} x^{6 \, n} + 252252 \, a^{3} b^{5} x^{5 \, n} + 286650 \, a^{4} b^{4} x^{4 \, n} + 210210 \, a^{5} b^{3} x^{3 \, n} + 97020 \, a^{6} b^{2} x^{2 \, n} + 25740 \, a^{7} b x^{n} + 3003 \, a^{8}}{45045 \, n x^{15 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^8*x^(-15*n - 1),x, algorithm="fricas")

[Out]

-1/45045*(6435*b^8*x^(8*n) + 45045*a*b^7*x^(7*n) + 140140*a^2*b^6*x^(6*n) + 2522
52*a^3*b^5*x^(5*n) + 286650*a^4*b^4*x^(4*n) + 210210*a^5*b^3*x^(3*n) + 97020*a^6
*b^2*x^(2*n) + 25740*a^7*b*x^n + 3003*a^8)/(n*x^(15*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**(-1-15*n)*(a+b*x**n)**8,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.233031, size = 163, normalized size = 1.08 \[ -\frac{{\left (6435 \, b^{8} e^{\left (8 \, n{\rm ln}\left (x\right )\right )} + 45045 \, a b^{7} e^{\left (7 \, n{\rm ln}\left (x\right )\right )} + 140140 \, a^{2} b^{6} e^{\left (6 \, n{\rm ln}\left (x\right )\right )} + 252252 \, a^{3} b^{5} e^{\left (5 \, n{\rm ln}\left (x\right )\right )} + 286650 \, a^{4} b^{4} e^{\left (4 \, n{\rm ln}\left (x\right )\right )} + 210210 \, a^{5} b^{3} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 97020 \, a^{6} b^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 25740 \, a^{7} b e^{\left (n{\rm ln}\left (x\right )\right )} + 3003 \, a^{8}\right )} e^{\left (-15 \, n{\rm ln}\left (x\right )\right )}}{45045 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^8*x^(-15*n - 1),x, algorithm="giac")

[Out]

-1/45045*(6435*b^8*e^(8*n*ln(x)) + 45045*a*b^7*e^(7*n*ln(x)) + 140140*a^2*b^6*e^
(6*n*ln(x)) + 252252*a^3*b^5*e^(5*n*ln(x)) + 286650*a^4*b^4*e^(4*n*ln(x)) + 2102
10*a^5*b^3*e^(3*n*ln(x)) + 97020*a^6*b^2*e^(2*n*ln(x)) + 25740*a^7*b*e^(n*ln(x))
+ 3003*a^8)*e^(-15*n*ln(x))/n