3.2727 \(\int \left (a+b x^n\right )^{-1/n} \, dx\)

Optimal. Leaf size=50 \[ x \left (a+b x^n\right )^{-1/n} \left (\frac{b x^n}{a}+1\right )^{\frac{1}{n}} \, _2F_1\left (\frac{1}{n},\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right ) \]

[Out]

(x*(1 + (b*x^n)/a)^n^(-1)*Hypergeometric2F1[n^(-1), n^(-1), 1 + n^(-1), -((b*x^n
)/a)])/(a + b*x^n)^n^(-1)

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Rubi [A]  time = 0.0336286, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ x \left (a+b x^n\right )^{-1/n} \left (\frac{b x^n}{a}+1\right )^{\frac{1}{n}} \, _2F_1\left (\frac{1}{n},\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^n)^(-n^(-1)),x]

[Out]

(x*(1 + (b*x^n)/a)^n^(-1)*Hypergeometric2F1[n^(-1), n^(-1), 1 + n^(-1), -((b*x^n
)/a)])/(a + b*x^n)^n^(-1)

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Rubi in Sympy [A]  time = 4.02037, size = 39, normalized size = 0.78 \[ x \left (1 + \frac{b x^{n}}{a}\right )^{\frac{1}{n}} \left (a + b x^{n}\right )^{- \frac{1}{n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{n}, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(1 + b*x**n/a)**(1/n)*(a + b*x**n)**(-1/n)*hyper((1/n, 1/n), (1 + 1/n,), -b*x*
*n/a)

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Mathematica [A]  time = 0.0218699, size = 50, normalized size = 1. \[ x \left (a+b x^n\right )^{-1/n} \left (\frac{b x^n}{a}+1\right )^{\frac{1}{n}} \, _2F_1\left (\frac{1}{n},\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x*(1 + (b*x^n)/a)^n^(-1)*Hypergeometric2F1[n^(-1), n^(-1), 1 + n^(-1), -((b*x^n
)/a)])/(a + b*x^n)^n^(-1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/((a+b*x^n)^(1/n)),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^n + a)^(-1/n), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{n} + a\right )}^{\left (\frac{1}{n}\right )}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/((b*x^n + a)^(1/n)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((b*x^n + a)^(1/n)), x)