3.2543 \(\int x^{-1-7 n} \left (a+b x^n\right )^3 \, dx\)

Optimal. Leaf size=63 \[ -\frac{a^3 x^{-7 n}}{7 n}-\frac{a^2 b x^{-6 n}}{2 n}-\frac{3 a b^2 x^{-5 n}}{5 n}-\frac{b^3 x^{-4 n}}{4 n} \]

[Out]

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

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Rubi [A]  time = 0.0686943, antiderivative size = 63, 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 \[ -\frac{a^3 x^{-7 n}}{7 n}-\frac{a^2 b x^{-6 n}}{2 n}-\frac{3 a b^2 x^{-5 n}}{5 n}-\frac{b^3 x^{-4 n}}{4 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - 7*n)*(a + b*x^n)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 10.6924, size = 53, normalized size = 0.84 \[ - \frac{a^{3} x^{- 7 n}}{7 n} - \frac{a^{2} b x^{- 6 n}}{2 n} - \frac{3 a b^{2} x^{- 5 n}}{5 n} - \frac{b^{3} x^{- 4 n}}{4 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1-7*n)*(a+b*x**n)**3,x)

[Out]

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

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Mathematica [A]  time = 0.0280766, size = 48, normalized size = 0.76 \[ -\frac{x^{-7 n} \left (20 a^3+70 a^2 b x^n+84 a b^2 x^{2 n}+35 b^3 x^{3 n}\right )}{140 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 - 7*n)*(a + b*x^n)^3,x]

[Out]

-(20*a^3 + 70*a^2*b*x^n + 84*a*b^2*x^(2*n) + 35*b^3*x^(3*n))/(140*n*x^(7*n))

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Maple [A]  time = 0.032, size = 56, normalized size = 0.9 \[ -{\frac{{b}^{3}}{4\,n \left ({x}^{n} \right ) ^{4}}}-{\frac{3\,a{b}^{2}}{5\,n \left ({x}^{n} \right ) ^{5}}}-{\frac{{a}^{2}b}{2\,n \left ({x}^{n} \right ) ^{6}}}-{\frac{{a}^{3}}{7\,n \left ({x}^{n} \right ) ^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-7*n)*(a+b*x^n)^3,x)

[Out]

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

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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)^3*x^(-7*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.223881, size = 65, normalized size = 1.03 \[ -\frac{35 \, b^{3} x^{3 \, n} + 84 \, a b^{2} x^{2 \, n} + 70 \, a^{2} b x^{n} + 20 \, a^{3}}{140 \, n x^{7 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^3*x^(-7*n - 1),x, algorithm="fricas")

[Out]

-1/140*(35*b^3*x^(3*n) + 84*a*b^2*x^(2*n) + 70*a^2*b*x^n + 20*a^3)/(n*x^(7*n))

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Sympy [A]  time = 118.801, size = 61, normalized size = 0.97 \[ \begin{cases} - \frac{a^{3} x^{- 7 n}}{7 n} - \frac{a^{2} b x^{- 6 n}}{2 n} - \frac{3 a b^{2} x^{- 5 n}}{5 n} - \frac{b^{3} x^{- 4 n}}{4 n} & \text{for}\: n \neq 0 \\\left (a + b\right )^{3} \log{\left (x \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(-1-7*n)*(a+b*x**n)**3,x)

[Out]

Piecewise((-a**3*x**(-7*n)/(7*n) - a**2*b*x**(-6*n)/(2*n) - 3*a*b**2*x**(-5*n)/(
5*n) - b**3*x**(-4*n)/(4*n), Ne(n, 0)), ((a + b)**3*log(x), True))

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GIAC/XCAS [A]  time = 0.221645, size = 69, normalized size = 1.1 \[ -\frac{{\left (35 \, b^{3} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 84 \, a b^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 70 \, a^{2} b e^{\left (n{\rm ln}\left (x\right )\right )} + 20 \, a^{3}\right )} e^{\left (-7 \, n{\rm ln}\left (x\right )\right )}}{140 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^3*x^(-7*n - 1),x, algorithm="giac")

[Out]

-1/140*(35*b^3*e^(3*n*ln(x)) + 84*a*b^2*e^(2*n*ln(x)) + 70*a^2*b*e^(n*ln(x)) + 2
0*a^3)*e^(-7*n*ln(x))/n