3.325 \(\int \frac{1}{a x^n+b x^n} \, dx\)

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

[Out]

x^(1 - n)/((a + b)*(1 - n))

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Rubi [A]  time = 0.019008, 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 \[ \frac{x^{1-n}}{(1-n) (a+b)} \]

Antiderivative was successfully verified.

[In]  Int[(a*x^n + b*x^n)^(-1),x]

[Out]

x^(1 - n)/((a + b)*(1 - n))

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Rubi in Sympy [A]  time = 3.40945, size = 10, normalized size = 0.5 \[ \frac{x^{- n + 1}}{\left (a + b\right ) \left (- n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**(-n + 1)/((a + b)*(-n + 1))

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Mathematica [A]  time = 0.00582209, size = 20, normalized size = 1. \[ \frac{x^{1-n}}{(1-n) (a+b)} \]

Antiderivative was successfully verified.

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

[Out]

x^(1 - n)/((a + b)*(1 - n))

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Maple [A]  time = 0.003, size = 19, normalized size = 1. \[ -{\frac{x}{ \left ( -1+n \right ){x}^{n} \left ( a+b \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a*x^n+b*x^n),x)

[Out]

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

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Maxima [A]  time = 1.53807, size = 28, normalized size = 1.4 \[ -\frac{x x^{-n}}{a{\left (n - 1\right )} + b{\left (n - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a*x^n + b*x^n),x, algorithm="maxima")

[Out]

-x*x^(-n)/(a*(n - 1) + b*(n - 1))

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Fricas [A]  time = 0.238138, size = 30, normalized size = 1.5 \[ -\frac{x}{{\left ({\left (a + b\right )} n - a - b\right )} x^{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x/(((a + b)*n - a - b)*x^n)

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Sympy [A]  time = 1.93952, size = 32, normalized size = 1.6 \[ \begin{cases} - \frac{x}{a n x^{n} - a x^{n} + b n x^{n} - b x^{n}} & \text{for}\: n \neq 1 \\\frac{\log{\left (x \right )}}{a + b} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a*x**n+b*x**n),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(a*x^n + b*x^n), x)