3.2703 \(\int \left (a+b x^n\right )^{1-\frac{1}{n}} \, dx\)

Optimal. Leaf size=53 \[ a x \left (a+b x^n\right )^{-1/n} \left (\frac{b x^n}{a}+1\right )^{\frac{1}{n}} \, _2F_1\left (\frac{1}{n}-1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0351041, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ a x \left (a+b x^n\right )^{-1/n} \left (\frac{b x^n}{a}+1\right )^{\frac{1}{n}} \, _2F_1\left (\frac{1}{n}-1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^n)^(1 - n^(-1)),x]

[Out]

(a*x*(1 + (b*x^n)/a)^n^(-1)*Hypergeometric2F1[-1 + 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 = 4.40203, size = 42, normalized size = 0.79 \[ a x \left (1 + \frac{b x^{n}}{a}\right )^{\frac{1}{n}} \left (a + b x^{n}\right )^{- \frac{1}{n}}{{}_{2}F_{1}\left (\begin{matrix} -1 + \frac{1}{n}, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0370572, size = 53, normalized size = 1. \[ a x \left (a+b x^n\right )^{-1/n} \left (\frac{b x^n}{a}+1\right )^{\frac{1}{n}} \, _2F_1\left (\frac{1}{n}-1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.105, size = 0, normalized size = 0. \[ \int \left ( a+b{x}^{n} \right ) ^{1-{n}^{-1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^n)^(1-1/n),x)

[Out]

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

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError

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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)