3.327 \(\int \frac{1}{\left (a x^n+b x^n\right )^3} \, dx\)

Optimal. Leaf size=20 \[ \frac{x^{1-3 n}}{(1-3 n) (a+b)^3} \]

[Out]

x^(1 - 3*n)/((a + b)^3*(1 - 3*n))

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Rubi [A]  time = 0.0268827, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{x^{1-3 n}}{(1-3 n) (a+b)^3} \]

Antiderivative was successfully verified.

[In]  Int[(a*x^n + b*x^n)^(-3),x]

[Out]

x^(1 - 3*n)/((a + b)^3*(1 - 3*n))

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Rubi in Sympy [A]  time = 3.76444, size = 15, normalized size = 0.75 \[ \frac{x^{- 3 n + 1}}{\left (a + b\right )^{3} \left (- 3 n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**(-3*n + 1)/((a + b)**3*(-3*n + 1))

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Mathematica [A]  time = 0.00545859, size = 20, normalized size = 1. \[ \frac{x^{1-3 n}}{(1-3 n) (a+b)^3} \]

Antiderivative was successfully verified.

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

[Out]

x^(1 - 3*n)/((a + b)^3*(1 - 3*n))

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Maple [A]  time = 0.002, size = 21, normalized size = 1.1 \[ -{\frac{x}{ \left ( -1+3\,n \right ) \left ({x}^{n} \right ) ^{3} \left ( a+b \right ) ^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a*x^n+b*x^n)^3,x)

[Out]

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

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Maxima [A]  time = 1.38221, size = 69, normalized size = 3.45 \[ -\frac{x x^{-3 \, n}}{a^{3}{\left (3 \, n - 1\right )} + 3 \, a^{2} b{\left (3 \, n - 1\right )} + 3 \, a b^{2}{\left (3 \, n - 1\right )} + b^{3}{\left (3 \, n - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.239557, size = 70, normalized size = 3.5 \[ \frac{x}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3} - 3 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} n\right )} x^{3 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.46824, size = 119, normalized size = 5.95 \[ \begin{cases} - \frac{x}{3 a^{3} n x^{3 n} - a^{3} x^{3 n} + 9 a^{2} b n x^{3 n} - 3 a^{2} b x^{3 n} + 9 a b^{2} n x^{3 n} - 3 a b^{2} x^{3 n} + 3 b^{3} n x^{3 n} - b^{3} x^{3 n}} & \text{for}\: n \neq \frac{1}{3} \\\frac{\log{\left (x \right )}}{a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-x/(3*a**3*n*x**(3*n) - a**3*x**(3*n) + 9*a**2*b*n*x**(3*n) - 3*a**2*
b*x**(3*n) + 9*a*b**2*n*x**(3*n) - 3*a*b**2*x**(3*n) + 3*b**3*n*x**(3*n) - b**3*
x**(3*n)), Ne(n, 1/3)), (log(x)/(a**3 + 3*a**2*b + 3*a*b**2 + b**3), True))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a x^{n} + b x^{n}\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((a*x^n + b*x^n)^(-3), x)