3.2540 \(\int x^{-1-4 n} \left (a+b x^n\right )^3 \, dx\)

Optimal. Leaf size=24 \[ -\frac{x^{-4 n} \left (a+b x^n\right )^4}{4 a n} \]

[Out]

-(a + b*x^n)^4/(4*a*n*x^(4*n))

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Rubi [A]  time = 0.0216792, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ -\frac{x^{-4 n} \left (a+b x^n\right )^4}{4 a n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - 4*n)*(a + b*x^n)^3,x]

[Out]

-(a + b*x^n)^4/(4*a*n*x^(4*n))

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Rubi in Sympy [A]  time = 3.13279, size = 19, normalized size = 0.79 \[ - \frac{x^{- 4 n} \left (a + b x^{n}\right )^{4}}{4 a n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1-4*n)*(a+b*x**n)**3,x)

[Out]

-x**(-4*n)*(a + b*x**n)**4/(4*a*n)

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Mathematica [A]  time = 0.0273179, size = 46, normalized size = 1.92 \[ -\frac{x^{-4 n} \left (a^3+4 a^2 b x^n+6 a b^2 x^{2 n}+4 b^3 x^{3 n}\right )}{4 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 - 4*n)*(a + b*x^n)^3,x]

[Out]

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

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Maple [B]  time = 0.026, size = 63, normalized size = 2.6 \[{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}} \left ( -{\frac{{a}^{3}}{4\,n}}-{\frac{{b}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{n}}-{\frac{3\,a{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{2\,n}}-{\frac{{a}^{2}b{{\rm e}^{n\ln \left ( x \right ) }}}{n}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-4*n)*(a+b*x^n)^3,x)

[Out]

(-1/4*a^3/n-b^3/n*exp(n*ln(x))^3-3/2*a*b^2/n*exp(n*ln(x))^2-a^2*b/n*exp(n*ln(x))
)/exp(n*ln(x))^4

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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^(-4*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224071, size = 62, normalized size = 2.58 \[ -\frac{4 \, b^{3} x^{3 \, n} + 6 \, a b^{2} x^{2 \, n} + 4 \, a^{2} b x^{n} + a^{3}}{4 \, n x^{4 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^3*x^(-4*n - 1),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 117.792, size = 56, normalized size = 2.33 \[ \begin{cases} - \frac{a^{3} x^{- 4 n}}{4 n} - \frac{a^{2} b x^{- 3 n}}{n} - \frac{3 a b^{2} x^{- 2 n}}{2 n} - \frac{b^{3} x^{- n}}{n} & \text{for}\: n \neq 0 \\\left (a + b\right )^{3} \log{\left (x \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(-1-4*n)*(a+b*x**n)**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.220548, size = 66, normalized size = 2.75 \[ -\frac{{\left (4 \, b^{3} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 6 \, a b^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 4 \, a^{2} b e^{\left (n{\rm ln}\left (x\right )\right )} + a^{3}\right )} e^{\left (-4 \, n{\rm ln}\left (x\right )\right )}}{4 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^3*x^(-4*n - 1),x, algorithm="giac")

[Out]

-1/4*(4*b^3*e^(3*n*ln(x)) + 6*a*b^2*e^(2*n*ln(x)) + 4*a^2*b*e^(n*ln(x)) + a^3)*e
^(-4*n*ln(x))/n