3.2542 \(\int x^{-1-6 n} \left (a+b x^n\right )^3 \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0666669, 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 \[ -\frac{a^3 x^{-6 n}}{6 n}-\frac{3 a^2 b x^{-5 n}}{5 n}-\frac{3 a b^2 x^{-4 n}}{4 n}-\frac{b^3 x^{-3 n}}{3 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - 6*n)*(a + b*x^n)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 10.8654, size = 54, normalized size = 0.86 \[ - \frac{a^{3} x^{- 6 n}}{6 n} - \frac{3 a^{2} b x^{- 5 n}}{5 n} - \frac{3 a b^{2} x^{- 4 n}}{4 n} - \frac{b^{3} x^{- 3 n}}{3 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1-6*n)*(a+b*x**n)**3,x)

[Out]

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

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Mathematica [A]  time = 0.0276948, size = 48, normalized size = 0.76 \[ -\frac{x^{-6 n} \left (10 a^3+36 a^2 b x^n+45 a b^2 x^{2 n}+20 b^3 x^{3 n}\right )}{60 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 - 6*n)*(a + b*x^n)^3,x]

[Out]

-(10*a^3 + 36*a^2*b*x^n + 45*a*b^2*x^(2*n) + 20*b^3*x^(3*n))/(60*n*x^(6*n))

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Maple [A]  time = 0.031, size = 56, normalized size = 0.9 \[ -{\frac{{b}^{3}}{3\,n \left ({x}^{n} \right ) ^{3}}}-{\frac{3\,a{b}^{2}}{4\,n \left ({x}^{n} \right ) ^{4}}}-{\frac{3\,{a}^{2}b}{5\,n \left ({x}^{n} \right ) ^{5}}}-{\frac{{a}^{3}}{6\,n \left ({x}^{n} \right ) ^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-6*n)*(a+b*x^n)^3,x)

[Out]

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

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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^(-6*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.223979, size = 65, normalized size = 1.03 \[ -\frac{20 \, b^{3} x^{3 \, n} + 45 \, a b^{2} x^{2 \, n} + 36 \, a^{2} b x^{n} + 10 \, a^{3}}{60 \, n x^{6 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^3*x^(-6*n - 1),x, algorithm="fricas")

[Out]

-1/60*(20*b^3*x^(3*n) + 45*a*b^2*x^(2*n) + 36*a^2*b*x^n + 10*a^3)/(n*x^(6*n))

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Sympy [A]  time = 117.865, size = 63, normalized size = 1. \[ \begin{cases} - \frac{a^{3} x^{- 6 n}}{6 n} - \frac{3 a^{2} b x^{- 5 n}}{5 n} - \frac{3 a b^{2} x^{- 4 n}}{4 n} - \frac{b^{3} x^{- 3 n}}{3 n} & \text{for}\: n \neq 0 \\\left (a + b\right )^{3} \log{\left (x \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(-1-6*n)*(a+b*x**n)**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.22154, size = 69, normalized size = 1.1 \[ -\frac{{\left (20 \, b^{3} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 45 \, a b^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 36 \, a^{2} b e^{\left (n{\rm ln}\left (x\right )\right )} + 10 \, a^{3}\right )} e^{\left (-6 \, n{\rm ln}\left (x\right )\right )}}{60 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^3*x^(-6*n - 1),x, algorithm="giac")

[Out]

-1/60*(20*b^3*e^(3*n*ln(x)) + 45*a*b^2*e^(2*n*ln(x)) + 36*a^2*b*e^(n*ln(x)) + 10
*a^3)*e^(-6*n*ln(x))/n