3.2701 \(\int \left (a+b x^n\right )^{-1-\frac{1}{n}} \, dx\)

Optimal. Leaf size=18 \[ \frac{x \left (a+b x^n\right )^{-1/n}}{a} \]

[Out]

x/(a*(a + b*x^n)^n^(-1))

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Rubi [A]  time = 0.0118179, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{x \left (a+b x^n\right )^{-1/n}}{a} \]

Antiderivative was successfully verified.

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

[Out]

x/(a*(a + b*x^n)^n^(-1))

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Rubi in Sympy [A]  time = 1.39549, size = 12, normalized size = 0.67 \[ \frac{x \left (a + b x^{n}\right )^{- \frac{1}{n}}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(a + b*x**n)**(-1/n)/a

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Mathematica [A]  time = 0.036151, size = 18, normalized size = 1. \[ \frac{x \left (a+b x^n\right )^{-1/n}}{a} \]

Antiderivative was successfully verified.

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

[Out]

x/(a*(a + b*x^n)^n^(-1))

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Maple [B]  time = 0.036, size = 53, normalized size = 2.9 \[ x{{\rm e}^{ \left ( -1-{n}^{-1} \right ) \ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }}+{\frac{bx{{\rm e}^{n\ln \left ( x \right ) }}}{a}{{\rm e}^{ \left ( -1-{n}^{-1} \right ) \ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^n)^(-1-1/n),x)

[Out]

x*exp((-1-1/n)*ln(a+b*exp(n*ln(x))))+b/a*x*exp(n*ln(x))*exp((-1-1/n)*ln(a+b*exp(
n*ln(x))))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^n + a)^(-1/n - 1), x)

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Fricas [A]  time = 0.239941, size = 42, normalized size = 2.33 \[ \frac{b x x^{n} + a x}{{\left (b x^{n} + a\right )}^{\frac{n + 1}{n}} a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: RecursionError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^n + a)^(-1/n - 1), x)