3.2556 \(\int x^{-1-8 n} \left (a+b x^n\right )^5 \, dx\)

Optimal. Leaf size=77 \[ -\frac{b^2 x^{-6 n} \left (a+b x^n\right )^6}{168 a^3 n}+\frac{b x^{-7 n} \left (a+b x^n\right )^6}{28 a^2 n}-\frac{x^{-8 n} \left (a+b x^n\right )^6}{8 a n} \]

[Out]

-(a + b*x^n)^6/(8*a*n*x^(8*n)) + (b*(a + b*x^n)^6)/(28*a^2*n*x^(7*n)) - (b^2*(a
+ b*x^n)^6)/(168*a^3*n*x^(6*n))

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Rubi [A]  time = 0.0835373, 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 \[ -\frac{b^2 x^{-6 n} \left (a+b x^n\right )^6}{168 a^3 n}+\frac{b x^{-7 n} \left (a+b x^n\right )^6}{28 a^2 n}-\frac{x^{-8 n} \left (a+b x^n\right )^6}{8 a n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - 8*n)*(a + b*x^n)^5,x]

[Out]

-(a + b*x^n)^6/(8*a*n*x^(8*n)) + (b*(a + b*x^n)^6)/(28*a^2*n*x^(7*n)) - (b^2*(a
+ b*x^n)^6)/(168*a^3*n*x^(6*n))

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Rubi in Sympy [A]  time = 17.499, size = 87, normalized size = 1.13 \[ - \frac{a^{5} x^{- 8 n}}{8 n} - \frac{5 a^{4} b x^{- 7 n}}{7 n} - \frac{5 a^{3} b^{2} x^{- 6 n}}{3 n} - \frac{2 a^{2} b^{3} x^{- 5 n}}{n} - \frac{5 a b^{4} x^{- 4 n}}{4 n} - \frac{b^{5} x^{- 3 n}}{3 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1-8*n)*(a+b*x**n)**5,x)

[Out]

-a**5*x**(-8*n)/(8*n) - 5*a**4*b*x**(-7*n)/(7*n) - 5*a**3*b**2*x**(-6*n)/(3*n) -
2*a**2*b**3*x**(-5*n)/n - 5*a*b**4*x**(-4*n)/(4*n) - b**5*x**(-3*n)/(3*n)

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Mathematica [A]  time = 0.0378441, size = 74, normalized size = 0.96 \[ -\frac{x^{-8 n} \left (21 a^5+120 a^4 b x^n+280 a^3 b^2 x^{2 n}+336 a^2 b^3 x^{3 n}+210 a b^4 x^{4 n}+56 b^5 x^{5 n}\right )}{168 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 - 8*n)*(a + b*x^n)^5,x]

[Out]

-(21*a^5 + 120*a^4*b*x^n + 280*a^3*b^2*x^(2*n) + 336*a^2*b^3*x^(3*n) + 210*a*b^4
*x^(4*n) + 56*b^5*x^(5*n))/(168*n*x^(8*n))

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Maple [A]  time = 0.036, size = 88, normalized size = 1.1 \[ -{\frac{{b}^{5}}{3\,n \left ({x}^{n} \right ) ^{3}}}-{\frac{5\,a{b}^{4}}{4\,n \left ({x}^{n} \right ) ^{4}}}-2\,{\frac{{a}^{2}{b}^{3}}{n \left ({x}^{n} \right ) ^{5}}}-{\frac{5\,{a}^{3}{b}^{2}}{3\,n \left ({x}^{n} \right ) ^{6}}}-{\frac{5\,{a}^{4}b}{7\,n \left ({x}^{n} \right ) ^{7}}}-{\frac{{a}^{5}}{8\,n \left ({x}^{n} \right ) ^{8}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-8*n)*(a+b*x^n)^5,x)

[Out]

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

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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)^5*x^(-8*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.225728, size = 100, normalized size = 1.3 \[ -\frac{56 \, b^{5} x^{5 \, n} + 210 \, a b^{4} x^{4 \, n} + 336 \, a^{2} b^{3} x^{3 \, n} + 280 \, a^{3} b^{2} x^{2 \, n} + 120 \, a^{4} b x^{n} + 21 \, a^{5}}{168 \, n x^{8 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^5*x^(-8*n - 1),x, algorithm="fricas")

[Out]

-1/168*(56*b^5*x^(5*n) + 210*a*b^4*x^(4*n) + 336*a^2*b^3*x^(3*n) + 280*a^3*b^2*x
^(2*n) + 120*a^4*b*x^n + 21*a^5)/(n*x^(8*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**(-1-8*n)*(a+b*x**n)**5,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.228996, size = 107, normalized size = 1.39 \[ -\frac{{\left (56 \, b^{5} e^{\left (5 \, n{\rm ln}\left (x\right )\right )} + 210 \, a b^{4} e^{\left (4 \, n{\rm ln}\left (x\right )\right )} + 336 \, a^{2} b^{3} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 280 \, a^{3} b^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 120 \, a^{4} b e^{\left (n{\rm ln}\left (x\right )\right )} + 21 \, a^{5}\right )} e^{\left (-8 \, n{\rm ln}\left (x\right )\right )}}{168 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^5*x^(-8*n - 1),x, algorithm="giac")

[Out]

-1/168*(56*b^5*e^(5*n*ln(x)) + 210*a*b^4*e^(4*n*ln(x)) + 336*a^2*b^3*e^(3*n*ln(x
)) + 280*a^3*b^2*e^(2*n*ln(x)) + 120*a^4*b*e^(n*ln(x)) + 21*a^5)*e^(-8*n*ln(x))/
n